partial transposition in quantum information theory · 2017. 8. 14. · partial transposition in...

Partial Transposition

in Quantum Information Theory

Von der Gemeinsamen Naturwissenschaftlichen Fakultat

der Technischen Universitat Carolo-Wilhelmina

zu Braunschweig

zur Erlangung des Grades eines

Doktors der Naturwissenschaften

(Dr.rer.nat.)

genehmigte

D i s s e r t a t i o n

Kumulative Arbeit

von: Michael Marc Wolf

aus: Schongau

1. Referent: Prof. Dr. Reinhard F. Werner

2. Referent: Prof. Dr. J. Ignacio Cirac

eingereicht am: 18.11.2002

mundliche Prufung (Disputation) am: 29.01.2003

Druckjahr: 2003

2

Vorveroffentlichungen der Dissertation

Teilergebnisse aus dieser Arbeit wurden mit Genehmigung der Gemeinsamen Natur-wissenschaftlichen Fakultat, vertreten durch den Mentor der Arbeit, in folgendenBeitragen vorab veroffentlicht:

Publikationen

1. K.G.H. Vollbrecht, M.M. Wolf, Efficient distillation beyond qubits, Phys. Rev.A 67, 012303 (2003).

2. A. Acin, V. Scarani, M.M. Wolf, Bell inequalities and distillability in N-quantum-bit systems, Phys. Rev. A 66, 042323 (2002).

3. K.G.H. Vollbrecht, M.M. Wolf, Conditional entropies and their relation toentanglement criteria, J. Math. Phys. 43, 4299 (2002).

4. M.M. Wolf, J. Eisert, M.B. Plenio, The entangling power of passive opti-cal elements, LANL e-print quant-ph/0206171 (2002); zur Veroffentlichungangenommen bei Phys. Rev. Lett.

5. F. Verstraete, M.M. Wolf, Entanglement versus Bell violations and their be-haviour under local filtering operations, Phys. Rev. Lett. 89, 170401 (2002).

6. K.G.H. Vollbrecht, M.M. Wolf, Activating distillation with an infinitesimalamount of bound entanglement, Phys. Rev. Lett. 88, 247901 (2002).

7. A. Acin, V. Scarani, M.M. Wolf, Violation of Bell’s inequalities implies dis-tillability for N qubits, J. Phys. A 36 L21, (2003).

8. R.F. Werner, M.M. Wolf, Bell inequalities and Entanglement, Quant. Inf.Comp. 1(3), 1 (2001).

9. T. Eggeling, K.G.H. Vollbrecht, R.F. Werner, M.M. Wolf, Distillability viaprotocols respecting the positivity of partial transpose, Phys. Rev. Lett. 87,257902 (2001).

10. R.F. Werner, M.M. Wolf, All multipartite Bell correlation inequalities for twodichotomic observables per site, Phys. Rev. A 64, 032112 (2001).

11. K.G.H. Vollbrecht, M.M. Wolf, Can spectral and local information decide sep-arability?, LANL e-print quant-ph/0107014 (2001).

12. R.F. Werner, M.M. Wolf, Bound entangled Gaussian states, Phys. Rev. Lett.86, 3658 (2000).

13. R.F. Werner, M.M. Wolf, Bell’s inequalities for states with positive partialtranspose, Phys. Rev. A 61, 062102 (1999).

Tagungsbeitrage

14. M.M. Wolf, Distillability and the irreversibility of entanglement distillation,IQING, Imperial College, London, 19.-22. September 2002.

15. F. Verstraete, R.F. Werner, M.M. Wolf, Bell inequalities and Entanglement(Poster), ESF Euresco Conference on Quantum Information, San Feliu deGuixols (Spanien), 23.-28. Marz 2002.

16. K.G.H. Vollbrecht, M.M. Wolf, Activating distillation with an infinitesimalamount of bound entanglement (Poster), ESF Euresco Conference on Quan-tum Information, San Feliu de Guixols (Spanien), 23.-28. Marz 2002.

17. M.M. Wolf, F. Verstraete, Entanglement versus Bell violations and their be-haviour under local filtering operations, Fruhjahrstagung der Deutschen Physikalis-chen Gesellschaft 2002, Osnabruck, 04.-08. Marz 2002.

18. O. Kruger, R.F. Werner, M.M. Wolf, Entanglement measures for Gaussianstates with 1× 1 modes (Poster), Fruhjahrstagung der Deutschen Physikalis-chen Gesellschaft 2002, Osnabruck, 04.-08. Marz 2002.

19. M.M. Wolf, Activating distillation with bound entangled states, QRandom II,MPI PKS Dresden, 27.Januar - 1.Februar 2002.

20. M.M. Wolf, On Bell’s inequalities, Coherent Evolution in Noisy Environments,MPI PKS Dresden, 2.April - 30.May 2001.

21. T. Eggeling, K.G.H. Vollbrecht, R.F. Werner, M.M. Wolf, Relations betweenvarious degrees of classicalness (Poster), ESF Conference Quantum Informa-tion, Gdansk (Polen), 10.-18. Juli 2001.

22. M.M. Wolf, R.F. Werner, Neues von Bell’s Ungleichungen, Fruhjahrstagungder Deutschen Physikalischen Gesellschaft 2000, Dresden, 20.-24. Marz 2000.

23. M.M. Wolf, Neues von Bell’s Ungleichungen, QIV-Kolloquium, Bad Honnef,10.-12. Januar 2000.

Contents

The references in square brackets are attached as reprints and represent the main part of

the thesis. The references in parenthesis are used implicitly.

Summary 7

1 Introduction 9

2 Basic concepts 13

2.1 Separable and entangled states . . . . . . . . . . . . . . . . . . . . . 13

2.2 Positive maps as separability criteria . . . . . . . . . . . . . . . . . . 16

2.3 Entangled PPT states and Hilbert’s 17th problem . . . . . . . . . . 20

2.4 Characterization of instruments . . . . . . . . . . . . . . . . . . . . . 23

2.5 Matrix reorderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Transposition as time reversal . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Symmetric states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.1 Werner symmetry . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7.2 Isotropic symmetry . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7.3 GHZ and Bell type symmetries . . . . . . . . . . . . . . . . . 30

3 Distillability 33

3.1 Distillation protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Distillable and undistillable states . . . . . . . . . . . . . . . . . . . 35

3.3 Distillability via PPT preserving protocols . . . . . . . . . . . . . . . 38

[Phys. Rev. Lett. 87, 257902 (2001)] . . . . . . . . . . . . . . . . . . 39

3.4 Activating distillation with symmetric states . . . . . . . . . . . . . 43

[Phys. Rev. Lett. 88, 247901 (2002)] . . . . . . . . . . . . . . . . . . 45

4 Conditional entropies 49

[J. Math. Phys. 43, 4299 (2002)] . . . . . . . . . . . . . . . . . . . . 51

5 Bell inequalities 59

(Quant. Inf. Comp. 1, 1 (2001))

5.1 Hidden variables and joint distributions . . . . . . . . . . . . . . . . 59

5.2 Bell inequalities and Bell polytopes . . . . . . . . . . . . . . . . . . . 63

5.3 Quantum states admitting a local classical description . . . . . . . . 65

5.4 Bell inequalities for states with positive partial transpose . . . . . . 67

[Phys. Rev. A 61, 062102 (1999)] . . . . . . . . . . . . . . . . . . . . 69

5.5 A complete set of Bell inequalities for multipartite systems . . . . . 73

[Phys. Rev. A 64, 032112 (2001)] . . . . . . . . . . . . . . . . . . . . 75

5.6 Bell inequalities and distillability in n-quantum-bit systems . . . . . 85

(Phys. Rev. A 66, 042323 (2002))

5

CONTENTS

6 Gaussian states 876.1 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2 Phase space functions and Gaussian states . . . . . . . . . . . . . . . 896.3 Symplectic Transformations . . . . . . . . . . . . . . . . . . . . . . . 926.4 Bound entangled Gaussian states . . . . . . . . . . . . . . . . . . . . 95

[Phys. Rev. Lett. 86, 3658 (2000)] . . . . . . . . . . . . . . . . . . . 976.5 The entangling power of passive optical elements . . . . . . . . . . . 101

[LANL e-print quant-ph/0206171 (2002)] . . . . . . . . . . . . . . . 103

References 107

6

Summary

The present thesis is concerned with the theory of entanglement. It investigatesvarious aspects of one of the most important mathematical tools in entanglementtheory — the partial transposition. The questions posed are of rather abstractnature, at least from an experimentalists point of view. The purpose behind themis to understand the structure of the state space of composite quantum systems withrespect to various entanglement properties. The physical carriers of the consideredquantum systems will usually not be specified, very much like classical informationtheory does not specify the carriers of classical information.

A large part of this thesis deals with with investigating the properties of stateshaving a “positive partial transpose” (PPT states). With respect to various aspectsthese states behave like classical systems, although they may be entangled. Thefollowing gives a summary of the main parts of this thesis chapter by chapter:

• 3. Distillability: “Entanglement distillation” is crucial for quantum com-munication, since it enables all the fascinating protocols in the presence of anoisy and interacting environment. The classical properties of entangled PPTstates may suggest that they are useless for quantum information processingpurposes. In fact, the entanglement contained in them is “bound” in the sensethat it cannot be extracted via entanglement distillation. However, it is shownin Sec.3.3 and 3.4 that they can serve as “activators” for the distillation ofevery non-PPT state. At the same time, this gives a partial answer to one ofthe big open questions in the theory of entanglement, namely whether everynon-PPT state is distillable. We prove that this is indeed the case when thedistillation process is supplemented by an arbitrarily small amount of PPTbound entanglement.

• 4. Conditional entropies: This chapter shows that all PPT states havea common property with classical probability distributions: their subsystemscan never have a larger Renyi entropy than the state on the whole, i.e., the“conditional” Renyi entropy is positive. Conversely, it is proven that anynegative conditional Renyi entropy implies distillability.

• 5. Bell inequalities: In Sec.5.5 we provide the first complete set of “Bellinequalities” for multipartite systems and give a detailed discussion of theirproperties including their violations within quantum mechanics. It is shownthat PPT states satisfy all these inequalities, such that the considered correla-tions admit a classical description within a local hidden variable model. For asubset of the inequalities a finer distinction of the PPT property is made andinvestigated in Sec.5.4. An application of the obtained results is mentionedin Sec.5.6, which states that the amount of the violation of a Bell inequalitygives us information about the usefulness of the state for quantum informationapplications.

• 6. Gaussian states: This chapter discusses Gaussian continuous variablestates. These states play an important role in quantum optics, and have

7

SUMMARY

attracted more and more attention in quantum information in recent years.In Sec.6.4 it is proven that every Gaussian state of 1×nmodes is unentangled ifand only if it is PPT. Moreover, we show that there exist entangled PPT statesfor bipartite systems with at least two modes per site. In Sec.6.5 we investigatethe possibility of generating non-PPT states from squeezed states by meansof passive optical elements (beam splitters, mirrors and phase shifters). Forthe case of two modes the optimal entangling procedure is provided.

How to read this thesis: This thesis is “cumulative”. Its main part are theattached papers. In principal, each paper is self-contained. However, some papersare limited in length and all of them refrain from giving a detailed discussion ofthe background of the tackled problem. So each paper is embedded into a chapter,which provides some additional information and shows many of the things explicitlythat are just cited in the papers. Most of the time this additional information canbe regarded as a summary of what is more or less “well known” in the field. TheBasic concepts chapter contains prerequisites which are common to more than oneof the following chapters. Moreover, it introduces basic notions as well as funda-mental tools and illuminates the “partial transposition” from several perspectives.Although the chapters 3-6 are linked via the partial transposition, they representrather different topics from entanglement theory. Since they do not really build upon each other it is not necessary to read them in the predetermined order.

The following Introduction is intended to motivate the work on a very basic leveland describes quantum information theory from a broader view.

8

Chapter 1

Introduction

In recent years quantum information theory has received a lot of public attentionand it has become a vastly growing field attracting researchers from physics, com-puter science, pure mathematics and electrical engineering. Much of the fascinationof this field is coming from the idea of putting the apparent strangeness and myster-ies of quantum theory into physical applications. Quite often these applications havesome science fiction overtones, like “quantum teleportation”, which simply forcesone to think of captain Kirk with his crew of 200 men. Similarly, the advertisedquantum computer, which will perform certain tasks faster than any present andfuture “classical” computer, and quantum cryptography, which supposedly workswith “the security of nature’s laws” may at first cause some wonderment. Despitethe fact that quantum information theory was born only about ten years ago, it has,however, become a well-established field, which fills yet its own section in PhysicalReview A, on an equal level with “Atomic and molecular structure and dynamics”.

One of the main innovations of quantum information theory is to consider “en-tanglement” as a physical resource, which is used up during quantum informationprocesses very much like fuel is consumed by an engine. This new perspective en-tails many challenging problems for both experimentalist and theoreticians. Onthe experimental side it has launched intensive effort to create entangled states,distribute their parts over large distances and manipulate them in a very precisecoherent way, always faced with unwanted decoherence effects and interactions witha noisy environment.

On the theoretical side our knowledge about entanglement has increased dra-matically within the last few years, and the “theory of entanglement” has become avery richly structured research field on its own. However, many of the problems inentanglement theory are burdened with a very rapid (often exponential) increase ofdimensions. In fact, this is roughly speaking what will give the quantum computerits power one day, but it also implies that brute force approaches to theoreticalproblems are doomed to failure right from the beginning. This is the reason whynumerics plays almost no role in quantum information theory, in contrast to otherfields like high energy physics or solid state physics. In particular in entanglementtheory therefore we rely on powerful mathematical tools which help us to overcomethese obstacles at least in parts. An outstanding tool for the theory of entangle-ment, which is as simple as important, is the “partial transposition”. In order tounderstand its meaning, we have to consider a typical situation in entanglementtheory:

We will always deal with correlations between two or more (spatially separated)“parties” or sites. It is helpful to imagine that the experiments at the sites areconducted by physicists, traditionally named Alice and Bob in the bipartite (two

9

CHAPTER 1. INTRODUCTION

party) case. Assume that each of the two has a particle1 in her/his lab, and thatthese particles are initially uncorrelated. We let the two physicists operate locallyon their particles and we additionally allow them to communicate with each othervia a “classical channel”, say the internet. In this way they can establish corre-lations between certain properties of their particles. The correlations which canbe generated by means of these “local operations with classical communication”(LOCC) are “classical” and in this case the two particles are called “unentangled”.

However, if the two particles were initially obtained from a common source, saytwo photons from a non-linear crystal for instance, their correlations may in somecases be qualitatively different from the ones describe above — the particles maybe entangled.

Assume now that we have a complete theoretical description of the preparationof the two particles, i.e., we know the joint density matrix. Unfortunately, decidingwhether the respective state is entangled is in general a highly non-trivial problem.However, we may carry out the following mathematical operation: We apply amatrix transposition to that part of the density matrix which corresponds to Bob’sparticle and leave Alice’s part unchanged. Physically this “partial transposition”would correspond to a time reversal in Bob’s lab. If the state was unentangled thenits partial transpose is again an admissible density matrix. However, the partialtransposition may lead to negative and thence unphysical eigenvalues of the densitymatrix if the state is entangled. Looking at whether or not the density matrix hasa “positive partial transpose”(PPT) can in this way help us to answer whether agiven state is entangled.

Depending on the size of the two subsystems there may be states which have this“PPT property” although they are entangled. Hence, their entanglement is hiddenfor us with respect to the above mathematical procedure. Quite surprisingly, how-ever, their entanglement is also “hidden” with respect to various physical aspects.In fact, these states behave very much like classical states: they supposedly admita local classical description, their entanglement cannot be “distilled” and any partof them does never have a larger entropy than the state on the whole. All theseproperties will be investigated in detail within this thesis.

One may think that entangled PPT states are useless for quantum informationprocessing purposes. In fact, this is the case if we consider these states on theirown. However, we will see that they can serve as useful “catalysts” or “activators”,in particular in the context of “entanglement distillation”.

Before going into the details of this thesis, we will briefly discuss several aspectsof entanglement, and some of its “applications”. The list below is not intended tobe complete. It should illuminate the role of entanglement and its relation to otherparts of quantum information theory and to the foundations of quantum mechanics:

• Teleportation: “Entanglement assisted teleportation” is the process oftransmitting the state of a quantum system from Alice to Bob without sendinga physical carrier. The requirements are a classical channel and a “maximallyentangled” state. Teleportation is arguably the first major discovery in quan-tum information theory, and its implementation in the lab was beyond doubtone of the early experimental breakthroughs in the field.

• Super dense coding is closely related to teleportation. It denotes the pos-sibility of transmitting two bits of classical information from Alice to Bob bysending only one two-level quantum system — a “qubit”. It is required thatthis system is initially maximally entangled with a system on Bob’s side.

1The “particle” may be any physical system.

10


• Quantum key distribution: Entanglement can also be used for distribut-ing a secure key for cryptographic schemes.2 The main ingredients are theperfect correlations of maximally entangled states and the “no-cloning theo-rem”, which states that an unknown quantum state cannot be copied. Thisenables one to distribute a cryptographic key in such a way that every eaves-dropper attack be discovered. Quantum key distribution is probably the firstquantum information application which is ready for the market.

• Quantum computation: Computers working with “quantum information”rather than with classical information can be emulated on classical computersof course. Hence, every problem, which is not solvable on classical computersin principle, is an impossible task for quantum computers as well. The mainpromise of quantum computation lies in the reduction of running time. Themost prominent example is Shor’s factorization algorithm, which gives rise toan exponential speed-up compared to known classical algorithms. This is ofparticular importance, since inverting the encryption stage of RSA public keycryptosystems is a problem closely related to factoring.

For quantum algorithms operating on pure states entanglement is crucial forsuch a speed-up.3

• Quantum operations: Entanglement is closely related to the mathematicaldescription of operations within the framework of quantum mechanics. Inparticular, “complete positivity” and (together with a locality assumption)linearity of quantum operations are connected with entanglement.

• Violation of Bell inequalities: The fact that entanglement is qualitativelydifferent from classical correlations can be verified experimentally. In 1964 Bellshowed that assuming an underlying local classical model leads to falsifiablepredictions for certain correlation experiments. These “Bell inequalities” canindeed be violated in quantum mechanics and their violation is confirmed bymany experiments.

For a more detailed discussion of these topics see [ABH+01].

2However, there exist other schemes which do not rely on entanglement.3For mixed states this might be different (see [JL02]).

11


12

Chapter 2

Basic concepts

This section will introduce the basic notions and provide some prerequisites forthe following chapters. Most of the statements below are “well known” in theliterature, and they are presented here with the purpose of giving a more self-contained introduction for the subsequent papers.

We will mainly restrict ourselves to finite dimensional bipartite quantum sys-tems, described by (and often identified with) a density operator ρ ∈ B(H(1)⊗H(2)),with dimH(1) = dimH(2) = d.1 Cases different from that will always be mentionedexplicitly or treated in greater detail in subsequent chapters.

2.1 Separable and entangled states

The independence of degrees of freedom is within the framework of quantum me-chanics reflected by a tensor product of Hilbert spaces. Together with the superpo-sition principle this tensor product structure is the root of “entanglement”, which inturn represents one of the deepest departures from classical physics, in particular,if the degrees of freedom correspond to spatially separated systems.

We will begin our discussion of entanglement with putting it into mathematicalterms. For mixed states it is most easily defined in terms of its negation:

Definition 1 (Separability [Wer89a]) Let ρ be a density matrix describing acomposite quantum system with respective Hilbert space H = H(1)⊗H(2). The statecharacterized by ρ is called “separable”, “classically correlated” or “unentangled” ifthe density operator can be written (or approximated, e.g., in trace norm) as

ρ =∑

j

pj ρ(1)j ⊗ ρ(2)j , (2.1)

where the positive weights pj sum up to one and each ρ(i)j describes a state on H(i).

Remark: For density operators of finite rank it follows from Caratheodory’s theorem[Alf71] that a separable density matrix can always be decomposed into at mostrank(ρ)2 pure product states.

The terminology “classically correlated” is justified due to the fact that thepreparation leading to the correlations can be assumed to be classical in the fol-lowing sense: Suppose we have two independent preparing devices, one for eachsubsystem, which prepare a certain state ρj depending on some classical input j.Then, in order to obtain a state of the form (2.1), we just have to add a randomgenerator, which produces numbers j with probability pj . Combining these three

1B(H) denotes the set of linear bounded operators B : H → H.

13

Separable and entangled states

devices thus leads to such a “separable” state and the joint expectation value oftwo observables A(1), A(2) is then given by

tr[ρA(1) ⊗A(2)

]=∑

j

pjtr[ρ(1)j A(1)]tr[ρ

(2)j A(2)]. (2.2)

Thus the correlations depend on the random generator only, which can however bechosen to be a purely classical device. We note that classically correlated does notmean that the state has actually been prepared in the manner described, but onlythat its statistical properties can be reproduced by a classical mechanism.

States for which there is no decomposition of the form (2.1) are called “entan-gled”, and in the above sense the correlations of their subsystems are different fromclassical correlations. Note that correlations are independent of the choice of localbases. In particular, two states ρ and ρ′ = (U1 ⊗ U2)ρ(U1 ⊗ U2)

∗ have the sameentanglement properties, qualitative as well as quantitative, if U1, U2 are unitaries.In the following we will always assume that the subsystems corresponding to thedifferent tensor factors are situated at distant locations. Such a “distant locationparadigm” is not really necessary. However, it prevents us from misconceptionsappearing when the apparently entangled systems are not spatially separated (see[vE02]).

Obviously, pure states are separable if and only if they are product states. Thatis, any kind of correlation between subsystems of a pure state are due to entangle-ment and thus non-classical. Since entanglement properties are independent of thechoice of local bases, we may bring a pure state into a simple normal form:

Proposition 2 (Schmidt decomposition [Sch07]) Let Ψ ∈ H(1) ⊗ H(2) corre-spond to a bipartite pure state, and d = mindimH(1),dimH(2). Then there existorthonormal bases ej, fj in H(1) resp. H(2) such that

|Ψ〉 =d∑

j=1

√λj |ej〉 ⊗ |fj〉, with λj ≥ 0 . (2.3)

The coefficients √λj are called “Schmidt coefficients” and the number of non-

zero λj is the “Schmidt rank” of Ψ. Prop.2 is easily proven by noting that a changeof local bases in a general decomposition |Ψ〉 = ∑

ij cij |ij〉 corresponds to left andright multiplication of the coefficient matrix c with two different unitaries. Hence,we can choose unitaries diagonalizing c such that

√λj are the singular values

of c.2 We will briefly summarize some of the most important implications of theSchmidt decomposition:

• Reductions and purifications: In general the “reduced state” ρ1 of a den-sity matrix ρ with respect to the first tensor factor is obtained by taking thepartial trace over the second subsystem, ρ1 = tr2[ρ]. For every operator Aacting on H(1) we have then tr [ρ1A] = tr [ρ(A⊗ 1)]. The reduced states of apure state given in Schmidt decomposition as in Eq.(2.3) are

ρ1 =∑

j

λj |ej〉〈ej | , ρ2 =∑

j

λj |fj〉〈fj |, (2.4)

such that both reduced states have the same spectrum λj. Conversely, if weare given an arbitrary mixed state ρ, we can by Eqs.(2.3,2.4) easily constructa “purification” Ψ, which is a pure state such that ρ = tr1[|Ψ〉〈Ψ|].

2This makes clear that the bases ej and fj depend on the vector Ψ.

14

Separable and entangled states

• Entanglement properties of pure states only depend on their Schmidt co-efficients. A pure state is separable iff3 it has Schmidt rank one, i.e., if allbut one λj are zero. The other extreme case of all λj being equal to 1

d corre-sponds to “maximally entangled” states, which have thus completely chaoticreductions.

Under reasonable assumptions the von Neumann entropy4 of the reducedstate, i.e. the Shannon entropy of the probability distribution λj, is theunique measure of entanglement for pure states (cf.[DHR02]).

• Cyclicity: It can be read off from Eq.(2.3), that every pure state Ψ can bewritten as

|Ψ〉 =(1⊗X

)|Ω〉 , with (2.5)

|Ω〉 =1√d

d∑

j=1

|ej〉 ⊗ |fj〉, (2.6)

where Ω is a maximally entangled state and X =√d ρ

12

2 , with ρ2 being thereduced density matrix of Ψ. More general, there always exists an operatorX such that Eq.(2.5) holds if we replace Ω by any vector of full Schmidtrank. This property of vectors with strictly positive Schmidt coefficients issometimes called “cyclicity”.5 It implies that two parties, Alice and Bob,which are given such an entangled state, can prepare every other state with acertain probability by means of local operations and classical communication.

If Ω in Eq.(2.5) is any maximally entangled state (not necessarily constructedfrom the Schmidt basis of Ψ), then tr [X∗X] = d. In particular, any twomaximally entangled states Ω and Ω′ are always related by a unitary U via

|Ω′〉 = (1⊗ U)|Ω〉. (2.7)

Hence, for Hilbert spaces of equal dimension we can construct bases of max-imally entangled states from unitary operator bases6, which are orthonormalwith respect to the Hilbert Schmidt scalar product (U, V ) 7→ tr [U ∗V ] /d. Foran example see Sec.2.7.3.

• Mixed states: Density matrices for mixed states can be considered as ele-ments of the Hilbert space of Hilbert-Schmidt class operators.7 Analogous toEq.(2.3) there exists a Schmidt-decomposition of the form

ρ =∑

j

√λj Ej ⊗ Fj , with λj ≥ 0 , (2.8)

where Ej and Fj are orthonormal operator bases satisfying tr [E∗i Ej ] =δij and tr [F ∗i Fj ] = δij respectively. It was shown in [Rud02], that if ρ isseparable, then

∑j

√λj ≤ 1 (see Sec.2.5).

3“iff” is a common abbreviation for “if and only if”.4The von Neumann entropy of ρ is given by S(ρ) = −tr [ρ log2 ρ] = −

∑jλj log2 λj , where

λj is the spectrum of ρ.5Here we have assumed that the two Hilbert spaces are of the same dimension. If the dimensions

are different, then there exist only vectors, which are cyclic with respect to the larger Hilbertspace, i.e., X has to act on the larger tensor factor in order to generate every vector Ψ of thetensor product.

6Orthonormal bases of unitary operators are not completely characterized. However, thereexists a fairly general construction scheme involving complex Hadamard matrices and Latin squares[Wer01]. Unfortunately, these are again not completely characterized combinatorial designs.

7The set of operators X ∈ B(H) with finite tr [X∗X] is called Hilbert-Schmidt class. Equippedwith the scalar product (X,Y ) 7→ tr [X∗Y ] this forms a Hilbert space.

15

Positive maps as separability criteria

The separability problem: The question whether a given state is entangled ormerely classically correlated is easy to answer for pure states: we just have to lookat the rank of the reduced density matrix. For general mixed states, however, this isstill an open problem, and it is even hard to solve numerically for an apparently smalldimension like d = 4. Counting parameters in a straight forward numerical attemptof finding a decomposition into product states makes this clearer: for d = 4 we mayneed 28 product vectors in our decomposition, where each of them is characterizedby 15 real parameters. Hence, in a naive numerical approach we would have to varyover about 4000 parameters. Moreover, if we fail to find a convex decompositioninto product states by numerical means, it may be hard to find out whether this isdue to numerical problems or due the fact that the state is entangled.

Fortunately, however, the entanglement of many mixed states can easily bedetected with the help of positive, but not completely positive maps, which we willintroduce in the next section.8

2.2 Positive maps as separability criteria

The existence of entanglement imposes some additional constraints on quantumoperations. Evidently, every linear9 map ρ 7→ Λ(ρ), which corresponds to an ad-missible physical transformation of a density matrix, has to be trace preserving andpositive in the sense that it maps positive semi-definite operators again onto pos-itive semi-definite operators.10 However, positivity is not sufficient since (Λ ⊗ id)has to be positive semi-definite as well. In other words, if we apply the operation Λonly to one part of a composite system, and leave the other parts unchanged, thenthe overall state after the operation has to be described by an admissible densitymatrix, i.e., a positive semi-definite operator. The additional unchanged part of thesystem is sometimes called the “innocent bystander”. A map for which (Λ ⊗ id)is positive for any dimension of the additional second tensor factor is called “com-pletely positive” (cp), and it is implied by Eqs.(2.3,2.5) that Λ : B(H1)→ B(H2) iscp iff (Λ ⊗ id)(|Ω〉〈Ω|) is positive, where Ω ∈ H1 ⊗H1 is any maximally entangledstate. Moreover, a map is cp iff it can be written as

Λ(ρ) =∑

i

KiρK∗i , (2.9)

for suitable “Kraus operators” Ki [Kra83]. We will discuss cp maps in greater detailin Sec.2.4.

The paradigm of a map, which is positive but not cp, is the usual matrixtransposition A 7→ θ(A) = AT which is defined with respect to a given basis by〈i|AT |j〉 = 〈j|A|i〉. Evidently, the transposition is a positive and trace preservingmap. However, we have that

(θ ⊗ id

) (|Ω〉〈Ω|

)=

1

dF , F =

d∑

i,j=1

|ei ⊗ fj〉〈ej ⊗ fi| , (2.10)

where F is the “flip operator”, which has eigenvalues +1 and −1 corresponding tosymmetric and antisymmetric eigenvectors respectively. Hence, the transposition isnot cp.

8These maps lead also to more refined numerical methods for deciding the separability ques-tion. Currently, the most powerful numerical method for deciding separability for low-dimensionalbipartite systems (d ∼ 3, 4, 5) can be found in [DPS02].

9Linearity can also be connected to entanglement via an additional locality assumption. It wasproven in [SBG01] that under basic assumptions about the description of states and measurements,non-linear operations are not consistent with locality and the existence of entanglement.

10This is an immediate consequence of the minimal statistical interpretation of quantum me-chanics and the fact that probabilities have to be non-negative and sum up to one.

16


A map, which is positive but not cp, like the transposition, does not correspondto a physically implementable operation. Nevertheless, positive non-cp maps, andin particular the transposition, have become important mathematical tools in thetheory of entanglement. In fact, such maps applied as (Λ ⊗ id) to a bipartitedensity matrix are powerful tools for “detecting” entanglement. Suppose that ρ is aseparable state of the form in Eq.(2.1), then positivity of Λ implies that (Λ⊗ id)(ρ)remains positive semi-definite (psd), since tensor products and convex combinationsof psd operators are again psd. Hence, for any positive non-cp map Λ, a negativeeigenvalue of (Λ⊗ id)(ρ) proves that ρ is entangled:

∃Ψ : 〈Ψ|(Λ⊗ id)(ρ)|Ψ〉 < 0 ⇒ ρ is entangled. (2.11)

Conversely, for any entangled state ρ, there is a positive map, which “detects” theentanglement:

Proposition 3 [HHH96] A density matrix ρ ∈ B(H1 ⊗ H2) corresponds to aseparable bipartite state if and only if (Λ⊗ id)(ρ) ≥ 0 for every positive linear mapΛ : B(H1)→ B(H2).

Proof: As discussed above, we have for every positive map Λ and every separa-ble state ρ, that (Λ ⊗ id)(ρ) ≥ 0. It therefore remains to show, that we can finda positive map Λ for every entangled state ρ, such that (Λ ⊗ id)(ρ) has at leastone negative eigenvalue. To this end it is useful to introduce the notion of “en-tanglement witnesses”. First note that the set of separable density matrices, sayS, is a convex compact set. It is then a consequence of the Hahn-Banach theorem(cf.[HHH96]), that for every entangled ρ 6∈ S there exists a hyperplane, described bya Hermitian operator W , which separates ρ from S, such that tr [ρW ] < 0, whereas∀σ ∈ S : tr [σW ] ≥ 0. Such an operator W is called “entanglement witness”, and itcorresponds to a positive map via the equation

tr [(A⊗B)W ] ≡ tr[BTΛ(A)

], (2.12)

where A,B are arbitrary bounded operators on the respective Hilbert spaces. Ther.h.s. of Eq.(2.12) can be written as11 tr

[F(Λ(A)⊗BT )

]= d 〈Ω|(Λ⊗id)(A⊗B)|Ω〉.

We have therefore

tr [ρW ] = d 〈Ω|(Λ⊗ id)(ρ)|Ω〉, (2.13)

such that the existence of an entanglement witness, for which tr [ρW ] < 0, impliesthe existence of the sought positive map.

Note that Eq.(2.13) shows that a positive map detects more entangled statesthan the respective entanglement witness, since non-negativity of the expectationvalue on the r.h.s. does not imply positive semi-definiteness of the operator (Λ ⊗id)(ρ).

The separability criteria based on entanglement witnesses and positive maps areabstract theoretical concepts, which help us to “detect” entanglement with pen andpaper if we are given the density matrix. However, one can derive physical methodsfrom them, which may lead to “entanglement detectors” in the literal sense. Wewill just sketch two such ideas:

• Entanglement witnesses are Hermitian operators and we may thus considerthem as observables (cf.[GHB+02]). In order to assure that we can measurethe expectation value tr [ρW ] with local apparatus situated at distant loca-tions, we may decompose W =

∑ij cijAi ⊗ Bj with respect to a product of

11Here we use Eq.(2.10), tr [FX ⊗ Y ] = tr [XY ], and tr[XT ⊗ Y

]= tr [X ⊗ Y ].

17


operator bases Ai, Bj of Hermitian operators. Then every basis elementAi resp. Bj may correspond to one measurement apparatus, such that tr [ρW ]is nothing but a linear combination of measured correlations. A particular in-stance are correlation functions corresponding to Bell inequalities, which wewill discuss in detail in chapter 5.

• As said above, a positive non-cp map Λ does not correspond to a physicallyimplementable operation. However, if we add a sufficiently large amount ofnoise parameterized by p ∈ [0, 1], the map

A 7→ tr [A]p

d21d ⊗ 1d + (1− p)(Λ⊗ id)(A) (2.14)

becomes a cp map and can thus in principle be implemented in the lab. Ifwe apply this operation to a separable state, the minimal eigenvalue of theresulting state has to be larger than a certain threshold [HE02]. If it is belowthat threshold,12 we know that the initial state was entangled. In this wayany method for estimating the smallest eigenvalue of a density matrix canlead to a physically implementable detector of entanglement.13

Let us now have a closer look at the transposition θ. The map ρ 7→ ρT1 =(θ ⊗ id)(ρ) is called “partial transposition” and it is defined with respect to a cer-tain product basis as 〈ij|ρT1 |kl〉 = 〈kj|ρ|il〉 (and analogous for ρT2). If a state isseparable, then its density operator has a positive partial transpose, we say it is“PPT”. Hence, the “PPT criterion” is a necessary criterion for separability [Per96].In fact, for some special cases it turns out to be sufficient as well, and it is to dateby far the most important criterion for separability. By Eq.(2.13) the entangle-ment witness corresponding to the transposition is given by the flip operator, andtherefore every separable state has to satisfy

tr [ρF] ≥ 0 . (2.15)

Note that the partial transpose of a density operator depends on the definingproduct basis. If we take the partial transposition with respect to a different (local)

basis, labeled by the superscript T1 , we get

ρT1 = (U ⊗ 1)[(U∗ ⊗ 1)ρ(U ⊗ 1)

]T1(U∗ ⊗ 1) (2.16)

=[(UUT )⊗ 1

]ρT1[(UUT )∗ ⊗ 1

](2.17)

6= ρT1 . (2.18)

However, Eqs.(2.16-2.18) show that the eigenvalues of the partial transpose arebasis independent, and so is the positivity of ρT1 . Moreover, ρT1 ≥ 0 is equivalentto ρT2 ≥ 0 since the two operators only differ by a global transposition.

We can now divide the set Λ of all positive maps into two classes, dependingon whether or not Λ can be obtained in a simple manner from the transpositionmap θ. A positive map is denoted “decomposable” if it can be written as

Λ = Λ1 + Λ2 θ , (2.19)

where Λ1 and Λ2 are completely positive maps.14 If there is no decomposition ofthis form, then Λ is called “non-decomposable”. Inserting Eq.(2.19) into Eq.(2.13)

12Let −λ be the smallest eigenvalue of (idd ⊗ idd)⊗ (idd ⊗Λ) acting on a maximally entangledstate Ω. Then p = (d4λ)/(d4λ + 1) is the smallest possible factor such that Eq.(2.14) is a cpmap. With this value of p the threshold for the minimal eigenvalue of any separable state becomes(d2λ)/(d4λ+ 1).

13A particularly simple upper bound for the smallest eigenvalue of a density matrix is given bythe probability of any outcome of an arbitrary von Neumann measurement.

14Since θ Λ2 θ is a cp map if Λ2 is cp, the decomposition in Eq.(2.19) is equivalent toΛ = Λ1 + θ Λ2.

18


we obtain that the entanglement witness corresponding to a decomposable map isof the form

W = Q1 +QT1

2 , (2.20)

where Qi = d(Λ∗i ⊗id)(|Ω〉〈Ω|) are psd operators.15 Conversely, every decomposablewitness of the form in Eq.(2.20) with arbitrary Qi ≥ 0 corresponds to a decompos-able positive map. When considered as separability criteria the latter are weakerthan the transposition, since ρT1 ≥ 0 implies (Λ⊗ id)(ρ) ≥ 0 for every decomposablemap.

An important example of a decomposable map is given by

Λred(A) = tr [A]1−A, (2.21)

which is clearly positive since the trace of a psd operator is never smaller than itslargest eigenvalue. The separability criterion (Λred ⊗ id)(ρ) ≥ 0 reads then

ρ1 ⊗ 1 ≥ ρ, resp. 1⊗ ρ2 ≥ ρ, (2.22)

which is known as the “reduction criterion” [HH99a, CAG99]. The respective wit-ness is Wred = (1−F)T1 = 2PT1

− , where P− is the projector onto the antisymmetric(Bose) subspace. Wred is of the form in Eq.(2.20) and Wred resp. Λred are there-fore indeed decomposable. Moreover, the witness property of Wred together withEq.(2.10) implies that every separable state has to satisfy

〈Ω|ρ|Ω〉 ≤ 1

d, (2.23)

for any maximally entangled state Ω ∈ Cd ⊗ C

d.For the case H = C

2⊗C2 the projector onto the antisymmetric subspace is one

dimensional and corresponds to a maximally entangled state. Hence, the entangle-ment witnesses corresponding to the reduction and the PPT criterion are equal upto a different choice of the local bases. This implies that in dimension 2 ⊗ 2 thereduction criterion is satisfied if and only if ρT1 ≥ 0.

An even more important peculiarity of the low dimensional cases 2 ⊗ 2 and2⊗ 3 is that every positive map is decomposable, which was proven in [Str63] and[Wor76]. Due to Prop.3 this implies:

Proposition 4 [HHH96] Let ρ be a density matrix on C2⊗C

2 or C2⊗C

3. Thenρ corresponds to a separable state if and only if ρT1 ≥ 0.

In the following section we will discuss, that the equivalence of separability on theone hand and the positivity of the partial transpose on the other fails to be truein general for higher dimensions. A non-constructive way to show this is to find annon-decomposable positive map:

Proposition 5 Entangled states with positive partial transpose exist iff there arenon-decomposable positive maps.

Proof: By Prop.3 there exists a positive map Λ for every entangled PPT state ρsuch that (Λ ⊗ id)(ρ) is not psd. However, this map must be non-decomposablesince decomposable positive maps cannot detect PPT states.

Conversely, every non-decomposable positive map leads to a non-decomposableentanglement witness Wnd. By applying the Hahn-Banach theorem again, we seethat there is a hyperplane characterized by an operator R, which separates Wnd

from the closed convex set of decomposable entanglement witnesses, such thattr [RWnd] < 0, whereas tr [RW ] ≥ 0 for every decomposable witness. R has thus tobe psd and PPT, and since tr [RWnd] < 0, it corresponds (up to normalization) toan entangled PPT state.

15Here Λ∗ is the dual map with respect to Λ, which corresponds to the Heisenberg picture. Thatis, we have by definition tr [XΛ(Y )] = tr [Λ∗(X)Y ] for every pair of admissible operators X,Y .

19

Entangled PPT states and Hilbert’s 17th problem

2.3 Entangled PPT states and Hilbert’s 17th prob-

lem

The question, whether there exist entangled states having a positive partial trans-pose (PPT), arose immediately when Peres observed that PPT is an easy com-putable necessary criterion for separability [Per96]. Later on it was proven thatPPT entangled states surprisingly have many classical properties (see Ch.3,4,5),although classical mechanisms are not sufficient for their preparation.

The first examples for PPT entangled states were constructed in 1997 by P.Horodecki [Hor97]. After that elegant ways of generating larger families of thesestates for 3 ⊗ 3 were provided in [BDM+99, BP00]. We will briefly discuss one ofthese methods based on “unextendable product bases” at the end of this section.However, rather than following the chronology of PPT entangled states within quan-tum information theory, we will begin one century earlier and discuss the relationbetween our question and “sum of squares” problems in pure mathematics, startingwith Hilbert’s work in 1888, which led to “Hilbert’s 17th problem”.

The link between quantum information theory on the one hand and Hilbert’sproblem on the other is a work of Choi [Cho75], in which he investigated real psdbiquadratic forms, which are the real analogues of entanglement witnesses. Therelation between these two fields was already mentioned in [Ter01] and appearedagain in a natural way in [DPS02], where a semidefinite programming approach tothe separability problem was provided. For an overview of Hilbert’s 17th problemsee [Rez00].

Hilbert’s problem: A sum of squares (sos) of real polynomials is obviouslypositive semi-definite. The converse question, whether a real homogeneous psdpolynomial of even degree has a sos decomposition, was posed and answered byDavid Hilbert in 1888. Let Hh(R

n) be the set of homogeneous polynomials of evendegree h in n real variables, and

Ph(Rn) = p ∈ Hh(R

n)∣∣∀x ∈ R

n : p(x) ≥ 0, (2.24)

Σh(Rn) =

p ∈ Ph(Rn)

∣∣∃hk ∈ Hh/2(Rn) : p =

∑

k

h2k

, (2.25)

the set of psd polynomials in H and the subset of polynomials having an sos de-composition, respectively.16 It is not difficult to see that P = Σ if n = 2 or h = 2.In fact, the latter is nothing but the spectral decomposition of psd matrices. In1888 Hilbert showed at first, that every p ∈ P4(R

3) can be written as the sum ofsquares of three quadratic forms, i.e., P4(R

3) = Σ4(R3). Moreover, he proved, al-

beit in a non-constructive way, that (h, n) = (2, n), (h, 2), (4, 3) are in fact the onlycases where P = Σ. In 1900 Hilbert posed a generalization of these questions ashis “17th problem” at the International Congress of Mathematics in Paris17: Doesevery homogeneous psd form admit a decomposition into a sum of squares of ra-tional functions? This was answered in the affirmative by Artin in 1927 using theArtin-Schreier theory of real closed fields [Art27] .18

Choi’s example: Concerning Hilbert’s original 1888 work it took nearly 80 yearsfor explicit polynomials p ∈ P\Σ to appear in the literature. One of these exampleswas provided by Choi, while investigating one of the numerous “sum of squares”

16Note that P and Σ are both convex cones. Moreover, they are closed, i.e., if pn → p coeffi-cientwise, and each pn is in P resp. Σ, then so is p.

17At this congress Hilbert outlined 23 major mathematical problems. Whereas some are broad,such as the axiomatization of physics (6th problem), others were very specific and could be solvedquickly afterwards.

18Artin’s proof holds for forms with coefficients from a field with unique order, i.e., in particularfor R.

20


variants of Hilbert’s problem. Choi showed in 1975 that there are psd biquadraticforms that cannot be expressed as the sum of squares of bilinear forms. Moreprecisely, let

F (x, y) =

n∑

i,j,k,l=1

Fijklxixjykyl , ∀x, y : F (x, y) ≥ 0 (2.26)

be a real psd biquadratic form in x, y ∈ Rn. For n = 3 Choi found a counterexample

to the (wrongly proven19) conjecture that there is always a decomposition into asum of squares of bilinear forms, i.e.

F (x, y)?=∑

α

(f (α)(x, y)

)2=∑

α

( n∑

ij=1

f(α)ij xiyj

)2. (2.27)

Choi’s counterexample, for which he gave very elementary proofs, is

F (x, y) = 2

3∑

i=1

x2i (y2i + y2i+1)−

( 3∑

j=1

xjyj

)2, (2.28)

with y3+1 ≡ y1.Note that F ∈ P4(R

6) and that every quadratic form appearing in an sos de-composition of F has to be bilinear. Hence, Eq.(2.28) is an element of P\Σ for(h, n) = (4, 6).20

Relation to positive maps: The main significance of biquadratic forms lies intheir relation to linear maps on symmetric matrices. This relation is in fact the realanalogue of the correspondence between entanglement witnesses and positive mapsexpressed in Eq.(2.12). Let Sn be the set of all real symmetric n× n matrices, andlet Φ : Sn → Sn be a positive linear map on Sn, i.e. ∀x ∈ R

n : Φ(|x〉〈x|

)≥ 0. Then

Φ corresponds to a psd biquadratic form F and vice versa via21

F (x, y) = 〈y|Φ(|x〉〈x|

)|y〉. (2.29)

Particular instances of such maps are congruence maps of the form Φ(A) = KTAK,which in turn correspond to psd biquadratic forms F (x, y) = f(x, y)2, where f(x, y) =∑

ij Kijxiyj is a bilinear form. The existence of a psd biquadratic form, whichdoes not admit an sos decomposition, thus implies that the set of positive mapsΦ : Sn → Sn properly contains the convex hull of all congruence maps (for n ≥ 3).The latter is, however, the real analogue of the set of (complex) decomposablepositive maps. Let us now see, whether the map

Φ(S) = 2[tr [S]1− diag(S33, S11, S22)

]− S, (2.30)

which corresponds to the psd biquadratic form in Eq.(2.28), represents an admissiblecomplex non-decomposable positive map as well. Let Λ be a complex extension of Φdefined as in Eq.(2.30) but on arbitrary complex 3×3 matrices. Obviously, Λ mapsHermitian matrices onto Hermitian matrices. Moreover, if Λ is a decomposablepositive map, it acts on symmetric matrices as Λ(S) =

∑j K

∗j SKj for some complex

19For the case x ∈ Rn, y ∈ R

m with n = 2 or m = 2 Calderon [Cal73] proved that there isalways a sos decomposition of the form in Eq.(2.27). In [Kog68] Koga claimed that this is alsotrue for arbitrary n,m. Finding the flaw in Koga’s proof, was apparently Choi’s motivation for[Cho75].

20By choosing dependent x and y, Choi also specified F (x, y), such that it yielded elements ofP\Σ for (h, n) = (4, 4) and (6, 3) respectively.

21F ≥ 0 implies Φ ≥ 0 since any symmetric matrix S ∈ Sn is psd if 〈y|S|y〉 ≥ 0 for every realvector y ∈ R

n.

21


matrices Kj. If we decompose the latter with respect to their real and imaginaryparts Kj = Rj + iIj we get

Λ(S) =∑

j

RTj SRj + ITj SIj

!= Φ(S), (2.31)

contradicting the fact that Φ in Eq.(2.30) has no such decomposition into congruencemaps. Hence, Λ is either not decomposable or not a positive map at all. The lattercan, however, be excluded, since we have for every vector z ∈ C

3 with zj = xjϕj ,xj = |zj |:

Λ(|z〉〈z|

)= D∗Φ

(|x〉〈x|

)D ≥ 0 , D = diag(ϕ1, ϕ2, ϕ3). (2.32)

By Prop.(5) Choi’s map thus proves, albeit in a non-constructive way, the existenceof PPT entangled states in 3⊗ 3.

The question whether an entanglement witness W is decomposable or not, isequivalent to asking whether the biquadratic Hermitian psd form

W (Φ,Ψ) = 〈Φ⊗Ψ|W |Φ⊗Ψ〉 =∑

ijkl

WijklΦiΨjΦkΨl (2.33)

has an sos decomposition. It is therefore just the complex analog of Eq.(2.27). Apartfrom examples there is, however, not much known about forms which do not admitsuch a decomposition. In Ref.[DPS02] a necessary and sufficient22 separability cri-terion was provided, which is based on constructing symmetric PPT extensions ofa state via a semidefinite program. The dual problem yields (non-decomposable)witnesses, which are such that W (Φ,Ψ)〈Φ|Φ〉k〈Ψ|Ψ〉l has a sos decomposition forsome k, l. It is conjectured that this may be a general property of all biquadraticHermitian psd forms, but there is no proof yet.

States from unextendable product bases: An elegant way of constructingPPT entangled states explicitly is to utilize an “unextendable product basis” (UPB)as introduced in [BDM+99]. A set S = |αj〉 ⊗ |βj〉 of normalized and mutuallyorthogonal product vectors in H is called a UPB if there is no product vector,which is orthogonal to every element of S, and |S| < dimH.23 An example of sucha UPB in 3 ⊗ 3 can be constructed from five real vectors forming the apex of aregular pentagonal pyramid, where the height h is chosen such that nonadjacentapex vectors are orthogonal. These vectors are

vj = N

(cos

2πj

5, sin

2πj

5, h

), j = 0, . . . , 4 , (2.34)

with N = 2/√

5 +√5 and h = 1

2

√1 +

√5. The UPB is then given by |αj〉⊗ |βj〉 =

|vj〉⊗|v2j mod5〉. Since any subset of three vectors on either side spans the full space,there cannot be a product vector orthogonal to all these states. Based on this orany other UPB, one can easily construct entangled PPT states:

Proposition 6 [BDM+99] Let S = |αj〉 ⊗ |βj〉 be a UPB of product vectors inH with dimH = d. Then the density matrix

ρS =(1d −

|S|∑

j=1

|αj〉〈αj | ⊗ |βj〉〈βj |)/(d− |S|) (2.35)

corresponds to an entangled PPT state.

22Sufficiency follows from [Wer89b], where is was shown that only separable states have sym-metric extension of arbitrary size.

23Note that this is only possible if |αj〉 (resp. |βj〉) is not a set of mutually orthogonalvectors.

22

Characterization of instruments

Proof : By definition every separable state has a convex decomposition into pureproduct states. Every vector occurring in such a decomposition lies in the range ofthe density matrix. However, ρS has no product vector in its range and thereforehas to be entangled. Moreover, ρT1 ≥ 0, since |αj〉⊗|βj〉 is again a set of mutuallyorthogonal and normalized vectors.

It was shown in [Ter00] that every real unextendable product basis (as above)can be used to construct a real-valued non-decomposable entanglement witness.These witnesses thus correspond to biquadratic psd forms, which do not admit ansos decomposition and are therefore elements of P\Σ.

2.4 Characterization of instruments

As already discussed in Sec.2.2, entanglement imposes constraints on the mathe-matical description of physically implementable transformations of quantum states.Such a transformation, called “channel”, has to be a completely positive linear map.In the following we will use the Schrodinger picture, in which a channel is describedas a map on density operators rather than on observables, and we will assume thatmaps are always linear maps. If a cp map Λ gives a complete description of a phys-ical process, it has to be trace preserving. However, if Λ involves measurements, ithas a natural decomposition Λ =

∑α Λα into trace non-increasing operations Λα,

each of them corresponding to a possible sequence of measurement outcomes α. Theprobability of obtaining α for a state ρ is then given by the trace tr [Λα(ρ)]. Dueto Davies [Dav76], such an operation with quantum and classical output is calledan “instrument”. Instruments have two “marginals”: trace preserving overall statechanges, if we disregard the classical output, and measurements if we disregard thequantum output. Measurements will be discussed at the end of this section.

Every linear cp map Λ : B(Hin) → B(Hout) has a Kraus decomposition of theform

Λ(X) =∑

i

K∗i XKi , (2.36)

where the Kraus operators Ki have to satisfy∑

iKiK∗i ≤ 1 with equality iff Λ is

trace preserving [Kra83]. Conversely, every map of the form in Eq.(2.36) is cp.

Note that the Kraus decomposition is not unique. In fact, two sets of Krausoperators Ki and K ′j correspond to the same map Λ iff there is a unitary usuch that K ′j =

∑i ujiKi, where we have appended zero operators to the shorter

list of Kraus operators. A simple corollary of this fact is, that there always exists aKraus decomposition with at most dim(Hin) · dim(Hout) terms.24

Quantum information theory mostly deals with systems, which are composedout of two or more (spatially separated) subsystems. The natural operations in thiscase are therefore “local operations” possibly depending on classical information,for instance measurement outcomes of foregoing local operations, which have beendistributed among the parties via classical communication. This set of “local quan-tum operations with classical communication” (LOCC) is, however, mathematicallyrather intractable and we do not have a simple characterization of this importantset at the moment. For this reason sets of operations have become important whichare tractable on the one hand and include the LOCC set but not too many non-localoperations on the other hand:

24We just have to expand a given set of Kraus operators in terms of an operator basis ofdim(Hin) ·dim(Hout) elements, and then apply the singular value decomposition to the coefficientmatrix.

23


• Separable superoperators (SSO) form an important class of such opera-tions. A cp map Λ is called a SSO if it is of the form

Λ(X) =∑

i

(Ai ⊗Bi)∗X(Ai ⊗Bi), (2.37)

with∑

iAiA∗i ⊗ BiB

∗i ≤ 1. Assume that we consecutively apply two instru-

ments of separable superoperators Λα and Σ(α)β to our system, wherethe second instrument depends on the outcome α of the first one. Then the

composition Σ(α)β Λα again is an element of an instrument of separable su-

peroperators. This implies that every LOCC operation can be written as aSSO, since it is a concatenation of local operations of the form

Λ(X) =∑

i

(Ai ⊗ 1)∗X(Ai ⊗ 1) (2.38)

with analogous operations acting on the second tensor factor, where againevery operation is allowed to depend on the preceding measurement results.Conversely, every SSO can probabilistically be implemented by a LOCC oper-ation, since every summand in Eq.(2.37) can as well appear in a compositionof two local operations. However, the inclusion LOCC ⊂ SSO is strict, since itwas shown in [DMS+99] that there exist SSO which can not be implementedby LOCC operations with unit probability.

If one considers just a single term in Eq.(2.38), one sometimes talks about“local filtering operations”.

• PPT preserving maps: Another important class of operations is the setof cp maps preserving the positivity of the partial transpose. We say thatΛ : B(Hin) → B(Hout), with Hin and Hout again corresponding to bipartitesystems, is PPT preserving if

∀Ψ ∈ Hin : Λ(|Ψ〉〈Ψ|T1

)T1 ≥ 0 . (2.39)

Utilizing the state-channel duality discussed below it easy to see, that if a cpmap Λ is PPT preserving, then so is Λ⊗ id for an arbitrary additional system.Moreover, every SSO is PPT preserving since

Λ(PT1

)T1=

∑

i

[(Ai ⊗Bi)∗PT1(Ai ⊗Bi)

]T1(2.40)

=∑

i

(Ai ⊗Bi)∗P (Ai ⊗Bi) (2.41)

is positive semi-definite for every P ≥ 0. The converse relation is, however,not even true qualitatively, since in contrast to separable superoperators PPTpreserving maps can generate entangled (PPT) states.

A very useful tool, when dealing with channels acting on distributed quantumsystems, is the duality between channels and states introduced by Jamiolkowski[Jam72]. There are various ways of formalizing this duality. We will state the onepresented in [CDKL01]:

Proposition 7 (operator-map dualism) Consider a bipartite system with re-

spective Hilbert space H =⊗2

i=0(HAi⊗HBi), composed out of three subsystems withHi = HAi ⊗HBi , each of dimension d⊗d. Then every cp map Λ : B(H0)→ B(H2)corresponds to a psd operator σ ∈ B(H12) and vice versa, via the relations

σ = (Λ⊗ id1)(ωA01⊗ ωB01

), (2.42)

Λ(ρ) = d4 tr01 [(ρ⊗ σ)(ωA01⊗ ωB01

)] , (2.43)

24


where ω ∈ B(Cd ⊗ Cd) are maximally entangled states and the indices label the

subsystems on which the respective operators act.25

Proof : Recall that dtr [(A⊗B)ω] = tr[(AT ⊗B)F

]= tr

[ATB

]for every pair

of admissible operators A,B. Utilizing these relations step by step we can write

tr [Λ(ρ)(A⊗B)] = tr [ρ0Λ∗0(A⊗B)] (2.44)

= tr [ρ1 ⊗ Λ∗0(A⊗B)F01] (2.45)

= d2tr[ρT1 ⊗ (A⊗B)σ12

](2.46)

= d2tr[ρT0 ⊗

((1⊗A⊗B)σ12

)(F01 ⊗ 1)

](2.47)

= d4tr [(ρ⊗ σ)(ωA01⊗ ωB01

⊗A⊗B)] , (2.48)

where we have again labeled the subsystems by indices where necessary. Hence,every cp map Λ can be written in the form of Eq.(2.43). Conversely, due to Eq.(2.5)every positive operator acting on a bipartite Hilbert space can probabilistically beobtained by a local operation applied to one side of a maximally entangled stateand is thus of the form in Eq.(2.42).

The Eqs.(2.42,2.43) have obvious interpretations: σ is up to normalization thedensity matrix resulting from applying the operation Λ (as an element of an in-strument) to one side of a maximally entangled state. On the other hand, if weare given the state σ, Eq.(2.43) tells us that we can probabilistically implement theoperation Λ by performing local measurements in a maximally entangled basis onthe input state ρ and one part of σ. If both measurement outcomes then correspondto ω, the final state on the part of the system, which has not been measured, isgiven by Λ(ρ).

Let us now apply the above one-to-one correspondence between cp maps andpsd operators to the above two important cases [CDKL01]. We will consider entan-glement properties of σ resp. Λ with respect to the split A|B:

• Separable superoperators correspond to separable operators σ. That isΛ is a separable superoperator iff σ is a convex combination of products.Moreover, Λ is capable of creating entanglement iff σ is entangled.

• PPT preserving maps correspond to operators σ with positive partial trans-pose. In Eq.(2.42) every PPT preserving map Λ leads to a PPT state. Con-versely, since Eq.(2.43) corresponds to a LOCC operation acting on σ and ρ,every PPT operator σ gives rise to a PPT preserving map Λ. Hence, we canprobabilistically implement every such map by means of LOCC operationssupplemented by a (possibly entangled) PPT state.

Measurements: Let us finally discuss measurements, i.e., instruments where weare merely interested in the probability of the classical output. Measurements aredescribed by assigning to each outcome α from a device an “effect operator” Fαwhich satisfies 0 ≤ Fα ≤ 1. The probability for the outcome α is then given bytr [ρFα], such that the set of effect operators has to satisfy

∑α Fα = 1. The set

Fα is sometimes called a “positive operator valued measure” (POVM). A specialinstance of measurements are “von Neumann measurements”, where each Fα is aprojector. When considering these projectors as eigenspaces of a Hermitian operatorwe are led to the widespread believe that observables correspond to Hermitianoperators.

In the context of Bell inequalities we will discuss measurements with two possibleoutcomes ±1. The expectation value of the measurement with respect to a state ρ

25That is, ωA01is for instance a maximally entangled state acting on HA0

⊗ HA1, id1 is the

identity on B(HA1⊗HB1

) and tr01 is the partial trace over system 0 and 1.

25

Matrix reorderings

is then given by tr [ρE] with E = (F+ − F−). The set of admissible operators E isthe convex set of Hermitian operators satisfying −1 ≤ E ≤ 1 with extreme pointssatisfying E2 = 1. Hence, the extremal measurements correspond to Hermitian (andunitary) operators with eigenvalues ±1 and are thus von Neumann measurement.

2.5 Matrix reorderings

Obviously, the partial transposition is a reordering of matrix elements of the den-sity matrix, when written in terms of a product basis. This section will brieflydiscuss a more general set of matrix reorderings. These were already investi-gated in [OR85] (before quantum information) and utilized as separability crite-ria in [HHH02] motivated by [Rud02]. Let us first fix a product basis and writeρijkl = 〈ij|ρ|kl〉. The partial transposition with respect to the first tensor factoracts then as (ρT1)ijkl = ρkjil. Consider now a more general reordering of the form

(ρR)ijkl = ρτ(ijkl) (2.49)

where τ is a permutation of the four indices and therefore one out of 24 elementsof the permutation group S4. The action of the linear map ρ 7→ ρR on products ofrank one operators is given by

A = |ψ1〉〈ψ2| ⊗ |ψ3〉〈ψ4| 7→ AR = |ψτ(1)〉〈ψτ(2)| ⊗ |ψτ(3)〉〈ψτ(4)| ,(2.50)

where the tilde denotes complex conjugation if the respective vector was permutedfrom a bra to a ket or vice versa (e.g. τ(1) = 2), otherwise it is the identity.For every tensor product of rank one operators in Eq.(2.50) there are unitariesU, V such that AR = UAV . Since the trace norm is unitarily invariant we have||AR||1 = ||A||1. Due to convexity of the norm and linearity of the map ρ 7→ ρR wehave that ||ρR||1 ≤ 1 has to hold for every separable state ρ. Note that for ρR = ρT1

this separability criterion is indeed equivalent to ρT1 ≥ 0.Let us now come to permutations different from the partial transposition. In

fact, with respect to the trace norm criterium ||ρR||1 ≤ 1 there are only threedifferent permutations: the identity, which is of course not a useful separabilitycriterion, the partial transposition, and the map

(ρR)ijkl = ρljki . (2.51)

Concerning the other permutations note that the eight operators ρR, FρR, ρRF,FρRF, FρRTF, ρRT , FρRT and ρRTF all have the same trace norm. Moreover, theycorrespond to different permutations, such that we have 3 · 8 = 24 elements of theS4 which is the whole group.

In the context of separability criteria, the map in Eq.(2.51) appeared at first in aslightly different form in [Rud02]. As discussed below Prop.2, every density matrixcan be considered as an element of the Hilbert space of Hilbert-Schmidt operators.Hence, it has a Schmidt decomposition of the form

ρ =∑

j

√λj Ej ⊗ Fj , λj ≥ 0 , (2.52)

where Ej and Fj are orthonormal operator bases. It is then straight forward toverify that ||ρR||1 = ||(ρF)T1 ||1 =

∑j

√λj for the map in Eq.(2.51). It was shown

in [Rud02] that the separability criterion∑

j

√λj ≤ 1 is independent of the PPT

criterion. That is, it is in general neither weaker nor stronger than ρT1 ≥ 0.

26

Transposition as time reversal

2.6 Transposition as time reversal

As discussed in Sec.2.2 the transposition, as well as the partial transposition, doesnot correspond to a physically implementable operation. However, it can physicallybe interpreted as “time reversal” [STV97] and appears as such quite often as a(global) symmetry26, in particular in particle physics [Per82]. Although we will notmake explicit use of this interpretation or symmetry in the following, we will brieflydescribe this relation.

Since time is a parameter in quantum mechanics, there cannot be a generaltransformation in Hilbert space which directly affects this parameter and acts ast 7→ −t. In this sense the term “time reversal” is misleading. It characterizes areversal of momenta,27 including angular momenta and spins, rather than a reversalof the arrow of time. That is, the time reversal operator T acts on position andmomentum operators as

TQT−1 = Q and TPT−1 = −P, (2.53)

and for consistency also TLT−1 = −L and TST−1 = −S for angular momentaand spins. Left and right multiplication of the canonical commutation relations[Q,P ] = i1 leads with Eq.(2.53) to TiT−1 = −i 1. Thus T is an “anti-linear”operator, and, as it is supposed to be norm preserving, it is “anti-unitary”.28

The simplest example of an anti-unitary operator is the complex conjugation Γ(which is on Hermitian matrices equal to the transposition). In fact, in Schrodingerrepresentation we may identify the time reversal operator T with complex conjuga-

tion, since the latter leaves the position operator unchanged and maps −i ddxΓ7→ i ddx

and therefore changes the sign of momentum and angular momentum operators.The form of the time reversal operator depends on the representation (resp. basis)and therefore T cannot be identified with Γ in general. However, every anti-unitaryoperator can be written as the product of a unitary operator with complex conju-gation.

Let us now turn to spin angular momenta. Consider the standard representation,in which Sz is diagonal and S± = Sx ± iSy are both real. In this basis we have

ΓSxΓ−1 = Sx, ΓSyΓ

−1 = −Sy, ΓSzΓ−1 = Sz, (2.54)

hence we cannot identify T with complex conjugation. However, we may writeT = ΓV , where V has to invert the signs of Sx and Sz, while leaving Sy unchanged.Moreover, V is, as a product of two anti-unitaries, a linear unitary operator, andit is supposed to act only on the spin degrees of freedom. These requirements aresatisfied by V = exp−iπSy, which rotates the spin through the angle π about they axis. Hence, we have with respect to the considered representation29

T = Γe−iπSy , (2.55)

and applying the time reversal to a density matrix describing a finite dimensionalspin system gives then T−1ρT = V ∗ρTV .

26The time reversal symmetry, i.e., the invariance of a Hamiltonian under time reversal, doesnot lead to a conserved quantity (like parity for space inversion). However, it sometimes increasesthe degeneracy of the energy eigenstates, which is known as “Kramer’s theorem” (cf.[Bal98]).

27Note that we leave the case dimH <∞ here.28An anti-linear operator A satisfies Az = zA for any complex number z. The product of two

anti-linear operators is thus again a linear operator. If the inverse A−1 exists and ∀Ψ ∈ H :||AΨ|| = ||Ψ||, then we say that A is anti-unitary. We have then 〈AΦ|AΨ〉 = 〈Ψ|Φ〉.

29Note that time reversal squared is not the identity. The operator T 2 in Eq.(2.55) has eigen-values -1 for states with total spin n/2, with n odd.

27

Symmetric states

Let us now consider (finite dimensional) composite systems, where ρ ∈ B(H1 ⊗H2). The global time reversal operator is then T ⊗ T , which is a well defined anti-unitary operator again. Unfortunately, there exists no operator on Hilbert space,which corresponds to a “partial time reversal” of only one of the two subsystems.In fact, the action of a tensor product of the form 1 ⊗ T (linear⊗anti-linear) ona vector Ψ ∈ H1 ⊗ H2 is not well defined. However, since for product vectors(1⊗T−1)(|ψ〉⊗|φ〉) gives only rise to a phase ambiguity, the action on the respectiveprojectors

(1⊗ T−1

)(|ψ〉〈ψ| ⊗ |φ〉〈φ|

)(1⊗ T

)(2.56)

becomes well defined without any ambiguity, and Eq.(2.56) describes a linear op-erator again. Since the real linear hull of projectors onto product states equals theset of all Hermitian operators, we can in this way define the “partial time reversal”as a map on density matrices. Due to Eq.(2.55) this is equivalent to the partialtransposition up to a local unitary transformation. The latter is however irrelevant,since it does not change the eigenvalues, in which we are interested in quantuminformation theory.

Hence, we may indeed interpret the (partial) transposition as (partial) timereversal. In the case of Gaussian continuous variable systems (Ch.6) we will usethis interpretation as main characterization of the partial transposition: it reversesthe momenta of the respective subsystem.

2.7 Symmetric states

Symmetric states play an important paradigmatic role in quantum informationtheory [VR01]. Their symmetry often considerably reduces the complexity of aproblem while preserving some of the interesting features of the full structure. Inthe following we will discuss sets of states, which are invariant under a group of localunitaries. For simplicity, we will restrict the general discussion to bipartite systems(always finite dimensional). The generalization to multipartite systems is straightforward and it will be investigated in detail for the case of GHZ type symmetriesin Sec.2.7.3.

Let ρ ∈ B(H) with H = Cd⊗C

d be a bipartite density matrix, which is invariantunder a local unitary operation

ρ = (U1 ⊗ U2)ρ(U1 ⊗ U2)∗. (2.57)

We call then U = (U1 ⊗ U2) a local symmetry of ρ. The set of all local symmetriesof ρ forms a group G of unitaries acting on H. Since G is a closed subgroup of thegroup of all unitaries, it is equipped with a unique normalized measure, which isinvariant under left and right group translation — the Haar measure.

Conversely, we can start with any closed group of local unitaries and investigatethe states lying in the commutant G′. By enlarging the group, we reduce the set ofsymmetric states and vice versa. Moreover, we define a twirl operation

T (X) =

∫

G

dg gXg∗ , (2.58)

as an averaging over the group with respect to the Haar measure (and∫Gdg →

|G|−1∑g∈G for finite groups). The twirl operation has the following properties:

1. Every operator X ∈ B(H) is mapped onto a symmetric element T (X) ∈ G′.

2. T is a projection, i.e., T 2 = T . In particular, ∀X ∈ G′ : T (X) = X.

28

Symmetric states

3. T is a completely positive trace preserving and unital map.30 Moreover, itcorresponds to a LOCC operation. One possible implementation is, that oneparty chooses randomly an element g ∈ G, communicates its choice to thesecond party, and both apply the respective local unitaries.

4. Observables invariant under G are related to symmetric states via

tr [T (ρ)X] = tr [ρT (X)] = tr [T (ρ)T (X)] . (2.59)

The last property implies that a symmetric state ρ = T (ρ) is uniquely definedby its expectation values with all invariant observables X ∈ G′, where it sufficesto consider basis elements of the algebra G′. If G′ is an abelian algebra, then theset of symmetric states is a simplex, with extreme points corresponding to minimalprojectors.

If the local Hilbert spaces are tensor products by themselves, we may considera tensor product of two symmetry groups G = G1 ⊗ G2. The commutant is thengiven by G′ = G′1 ⊗ G′2, i.e., G

′ is the set of all linear combinations of elements ofthe form g1 ⊗ g2, with gi ∈ Gi. If G′1 and G′2 are abelian, then so is G′ and theminimal projectors of G′ are given by the tensor products of minimal projectors ofG′1 and G′2.

Applying the partial transposition to Eq.(2.57) leads to

ρT1 = (U1 ⊗ U2)ρT1(U1 ⊗ U2)∗. (2.60)

Hence, the operators invariant under the group G = U1 ⊗ U2 are the partialtransposes of those invariant under G.31 Unfortunately, an abelian commutant G′

does in general not imply that G′ is abelian, too.

2.7.1 Werner symmetry

Consider the group G = U⊗n of local unitaries acting on Cd⊗n. It is well known

from group representation theory [Wey46] that the commutant G′ is spanned bythe permutation operators of the tensor factors. Hence, for n > 2, G′ is not abelianand the state space is therefore not a simplex. For n = 2 the set of invariantstates is the interval of all positive and normalized linear combinations of identity1 and flip operator F. The respective minimal projectors are the projectors ontothe symmetric and anti-symmetric subspace P+ and P− respectively, where

P± =1± F

2, with d± = dimP± =

d2 ± d2

. (2.61)

States invariant under all local unitaries of the form U ⊗ U can thus be written as

ρ(p) = pP−d−

+ (1− p)P+d+

, 0 ≤ p ≤ 1. (2.62)

Equivalently, we may parameterize these states by their flip expectation valuetr [ρ(p)F] = 1 − 2p. States of the form in Eq.(2.62) are called “Werner states”,were introduced in [Wer89a], and play a crucial role in the theory of entanglement.

Due to Eq.(2.15) a Werner state is entangled if tr [ρ(p)F] < 0, i.e., p > 12 . On

the other hand, p ≤ 12 corresponds to separable states, since there are pure product

states, e.g. |Ψ〉 = 12

[(|0〉 + |1〉) ⊗ (|0〉 − |1〉)

], with 〈Ψ|F|Ψ〉 = 0: the twirled state

30A map is called unital if it takes the identity operator to itself. Trace preserving unital mapsare called doubly stochastic.

31Note that in order to make the map (U1 ⊗ U2) 7→ (U1 ⊗ U2) (i.e. G 7→ G) well defined it isnecessary to assume that G contains the group of all phases.

29

Symmetric states

T (|Ψ〉〈Ψ|) has then the same flip expectation value and is furthermore separable.Since every other Werner state with p < 1

2 can be written as a mixture of twoseparable states (ρ(0) and ρ( 12 )), we have indeed that ρ(p) is separable if and onlyif p ≤ 1

2 . Note that this is independent of the dimension of the system.As the flip operator is the entanglement witness corresponding to the partial

transposition (see Sec.2.2), ρ(p) is PPT if and only if it is separable. The symmetrygroup corresponding to the partial transposes is the group of all local unitaries ofthe form U ⊗ U .

2.7.2 Isotropic symmetry

States on Cd ⊗ C

d, which are symmetric with respect to the symmetry group G =U ⊗ U are called “isotropic states” [HH99b]. Since operators invariant underU ⊗ U and those invariant under U ⊗ U are related via partial transposition, wecan read off the commutant G′ from the previous paragraph. G′ is then spanned byFT1 = d|Ω〉〈Ω| (see Eq.(2.10)) and the identity operator. Isotropic states are thus

of the form

ρ(f) = f |Ω〉〈Ω|+ (1− f)d2 − 1

(1− |Ω〉〈Ω|

), 0 ≤ f ≤ 1 , (2.63)

where f is called the “maximally entangled fraction” or “fidelity” of ρ(f). We maythink of ρ(f) being generated by adding “white noise” to the maximally entangledstate Ω.

From Eq.(2.23) we know that ρ(f) is entangled if f > 1d . Conversely, since

|〈Ω|00〉|2 = 1d we have that f ≤ 1

d corresponds to separable states. As shown inSec.2.2, the operator

(1d1 − |Ω〉〈Ω|

)is the entanglement witness corresponding to

the reduction criterion in Eq.(2.22), which is in turn weaker than the PPT criterion.Therefore ρ(f) is PPT if and only if it is separable.

Note that for d = 2 and f = p isotropic states and Werner states only differby a local unitary operation, since in this case the antisymmetric subspace is one-dimensional and the respective projector corresponds to a maximally entangledstate.

2.7.3 GHZ and Bell type symmetries

We will now study a discrete symmetry group for multi-partite states acting onCd⊗n. The resulting symmetric states will be diagonal in a basis of vectors, which

are up to local unitaries of the form

|Ψ〉 = 1√d

d−1∑

k=0

|k〉⊗n. (2.64)

Since for (n, d) = (3, 2) the state in Eq.(2.64) is known as the “Greenberger-Horne-Zeilinger” (GHZ) state [GHZ89], we may as well refer to these states as GHZ statesfor n ≥ 3 and arbitrary dimension. For n = 2, Ψ is a maximally entangled stateand we will in analogy to the case d = 2 call states diagonal in such a basis “Belldiagonal states” [BDSW96].

The main ingredient for our discussion is the set of unitaries of the form

Ukl =d−1∑

r=0

ηrl|k + r〉〈r| , η = e2πid , (2.65)

where addition inside the ket is modulo d and k, l = 0, . . . , d − 1. This set ofunitaries has the following properties:

30

Symmetric states

1. It forms a unitary operator basis which is orthonormal with respect to theHilbert Schmidt scalar product, i.e., tr

[U∗ijUkl

]= dδikδjl. Since U00 = 1 we

have in particular that tr [Ukl] = 0 for all (k, l) 6= (0, 0).

2. It is a discrete Weyl system since UijUkl = ηjkUi+k,j+l.

3. The set is generated by U10 and U01 via Ukl = (U10)k(U01)

l.

4. If d is odd, then Ukl ∈ SU(d). For d even, this is the case iff (k + l) is even,i.e., detUkl = (−1)(k+l).

5. For d = 2 it reduces essentially to the set of Pauli matrices with identity, i.e.,(σx, σy, σz) = (U10, iU11, U01).

Let us now consider the group of local unitaries

G =

n⊗

k=1

Ui,jk

∣∣∣n∑

k=1

jk mod d = 0

, (2.66)

with i ∈ Zd, j ∈ Znd and one of the components of j, say j1, is fixed by the additional

constraint. The group properties of G can easily be checked by utilizing properties1. and 2. of the set Ukl together with the fact that U−1kl = ηklU−k,−l. Moreover,G is an abelian group with |G| = dn elements, and it spans its own commutant.32

The respective minimal projectors are one-dimensional and can be parameterizedby ω ∈ Z

nd , such that they correspond to vectors of the form

|Ψω〉 =1√d

d−1∑

l=0

ηlω1

n⊗

k=1

|ωk + l〉. (2.67)

These vectors form an orthonormal basis of the entire Hilbert space Cd⊗n and they

are all equal to the GHZ state in Eq.(2.64) up to local unitaries.For n = 2 Eq.(2.67) corresponds to an orthonormal basis of maximally entan-

gled states, which reduces to the “Bell basis” for d = 2. Moreover, every statewith isotropic symmetry with respect to any of these maximally entangled states isinvariant under G.

We can enlarge the group G, and therefore reduce its commutant, by droppingthe constraint

∑nk=1 jk mod d = 0 and multiplying with the group of roots of unity

ηl |l ∈ Zd:

G+ =

ηl

n⊗

k=1

Ui,jk

. (2.68)

Comparison of coefficients in the commutation equation then shows, that the com-mutant G′+ is spanned by the set of operators ⊗n

k=1 U0,jk |∑n

k=1 jk mod d = 0.The minimal projectors are now of dimension d and have the form

Pω =

d−1∑

l=0

(U0,l ⊗ 1)|Ψω〉〈Ψω|(U0,l ⊗ 1)∗ , (2.69)

where Pω does not depend on ω1.Twirl operations onto GHZ diagonal states for d = 2 will play an important role

in Sec.5.6. Separability and distillability properties of such states for the case ofn-qubit systems (d = 2) were studied extensively in [DC00].

32Constructing the commutant is straight forward, as Ukl is an orthonormal operator basis.We can therefore expand a general operator with respect to this basis, and have just to comparethe coefficients in the commutation equation.

31

Symmetric states

32

Chapter 3

Distillability

The main obstacle against the physical realization of applications in quantum com-putation and quantum communication is their sensitivity against noise. Fortunately,we can overcome decoherence and the interaction with a noisy environment withoutrequiring a complete isolation from the latter. This is made possible by “quantumerror correction” for quantum computation, and “entanglement distillation” in thecase of quantum communication. Entanglement distillation enables us to convert ahuge number of mixed and weakly entangled states into a smaller number of almostmaximally entangled states. This is in fact indispensable for many protocols inquantum communication, like entanglement based quantum cryptographic schemesor faithful teleportation — their advantage compared to their classical counterpartscrucially depends on the usage of highly entangled states.

The usual way of thinking about entanglement distillation is the following: Twoparties, Alice and Bob, situated at distant locations share n copies of a mixedentangled quantum state ρ, which they may have obtained by sending one part of apure maximally entangled state through a noisy quantum channel. We assume thatboth parties are able to perform any collective quantum operation, which merelyacts locally on their part of the n copies. Moreover, Alice and Bob are connectedvia a classical channel, such that they can perform arbitrary many rounds of localquantum operations, where each round may depend on the measurement outcomesof all the preceding operations on both sides. The set of operations accessible in thisway is the set of LOCC operations (local operations and classical communication,see Sec.2.4).

It was shown in one of the seminal works from the early years of quantuminformation theory [BBP+96] that under the above conditions Alice and Bob canfor certain two-qubit states “distill” (or “purify”) a smaller number m ≤ n ofstates ρ′, which are closer to the maximally entangled state than the initial ones.This can be done in such a way that in the limit n → ∞ the output states ρ′

become maximally entangled. The asymptotic ratio m/n is called the rate of thedistillation protocol and the maximally accessible rate under all LOCC protocols isan important measure of entanglement, the “distillable entanglement” D(ρ). Thelatter quantifies in some sense the amount of useful entanglement contained in thestate ρ. We will call a state “distillable” if D(ρ) > 0.

Despite its practical relevance and quite considerable effort in that direction,however, many of the basic questions concerning entanglement distillation are yetunanswered. Most notably the question whether a given quantum state is distillableor not. The present chapter is concerned with this question and provides somepartial results. We will in particular prove that the set of undistillable states is equalto the set of PPT states if the LOCC operation is supplemented by certain PPTentangled states, which contain in turn an arbitrarily small amount of entanglement.

33

Distillation protocols

We will start with a brief description of the most important distillation protocolsand then summarize known results about the distillability of mixed states.

3.1 Distillation protocols

The most important distillation protocols for bipartite systems are adapted to Belldiagonal states (see Sec.2.7.3). In order to apply them to other states, one has tofind a LOCC operation, which maps the given state onto such a Bell diagonal state.One may furthermore distinguish two kinds of protocols with respect to their mainpurpose:

1. Protocols, which increase the entanglement contained in a subset of the in-put states, without leading to a non-zero rate in the asymptotic limit. Suchprotocols are sometimes called “quasidistillation” protocols.

2. Protocols, which yield a non-zero rate in the asymptotic limit, and thereforelead to a non-trivial lower bound to the distillable entanglement.

The paradigms of these protocols are the “recurrence protocol” (1.) and the “hash-ing protocol” (2.), which we will briefly describe in the following. To date the mostefficient distillation protocols are “hybrid protocols”: we first do some preprocessingin order to obtain a Bell diagonal state, then we start a recurrence protocol until weexceed a certain threshold of purity/entanglement and finally we apply the hashingprotocol.

Recurrence protocol:1 This protocol works for any Bell diagonal state in arbi-trary dimension d if the largest eigenvalue of the density matrix satisfies λmax > 1/d.We will restrict to isotropic states here (λmax = f). The protocol is, however, thesame for all Bell diagonal states. In the first step of the protocol Alice and Bobuse two copies of the state ρ and each of them applies a local “controlled shift”operation, which acts as

C|i〉 ⊗ |j〉 = |i〉 ⊗ |j + i mod d〉 (3.1)

to his/her part of the two states. Then they measure the state which correspondsto the second tensor factor in Eq.(3.1) in computational basis. They keep theremaining state (corresponding to the first tensor factor) if their measurement out-comes coincide and discard it otherwise. The resulting state has then a maximallyentangled fraction [HH99a]

f ′ =1 + f

[df(d2 + d− 1)− 2

]

d3f2 − 2df + d2 + d− 1, (3.2)

where f is the initial fidelity. Since f ′ > f for any f > 1/d, one can increase theentanglement of every entangled isotropic state, and by iterating the protocol wecan indeed achieve f → 1. However, this requires an infinite number of rounds,and as we discard at least half of the states in every round, the distillation rate ofthis protocol is zero. Nevertheless, it is important as preprocessing for the hashingprotocol or for other tasks where we need only to have entanglement above a certainthreshold.

Breeding and hashing protocols:2 These protocols are more complicated than

1For the two-qubit case this kind of protocol was introduced and extensively discussed in[BBP+96, BDSW96, DEJ+96]. For arbitrary dimensions see [HH99a, VW02].

2The breeding protocol is the entanglement assisted version of the hashing protocol. Whereashashing is only proven to work in dimensions which are primes (or powers of primes), the breedingprotocol works for every finite dimension [VW02].

34

Distillable and undistillable states

the recurrence method, and we will thus only sketch the idea. They were intro-duced in [BBP+96, BDSW96] for two-qubit Bell diagonal states and generalized toarbitrary dimension in [VW02].

Assume that Alice and Bob share n copies of a Bell diagonal state ρ =∑

kl λklPkl,k, l = 1, . . . , d − 1 where Pkl are the projectors onto vectors of a maximally en-tangled basis. We have then

ρ⊗n =∑

k1...kn,l1...ln

λk1l1 · · ·λknln(Pk1l1 ⊗ · · · ⊗ Pknln

). (3.3)

An appropriate interpretation of Eq.(3.3) is to say that Alice and Bob share thestate Pk1l1 ⊗ · · · ⊗ Pknln with probability λk1l1 · · ·λknln . If they knew the sequencek1, . . . , kn, l1, . . . , ln, they would already have a maximally entangled state. There-fore the task is to identify this sequence by means of LOCC operations with-out destroying too many of the maximally entangled states. It was shown in[BBP+96, BDSW96, VW02] that this can be achieved in the limit n→∞ if the en-tropy of the state satisfies S(ρ) < log2 d, and the obtained rate is then log2 d−S(ρ).Together with the recurrence protocol, this implies that every entangled isotropicstate is distillable.

3.2 Distillable and undistillable states

This section summarizes known results about the distillability of mixed bipartitequantum states of finite dimension. These results were essentially obtained by theHorodecki family. Based on the previous sections, we will, however, present them ina rather self-contained way. The starting point for the implication chain relating thefollowing propositions is the fact that every entangled isotropic state is distillable.In general we will say that a state ρ is “distillable” if for every ε > 0 there exists anLOCC operation Λ and a number n such that 〈Ω|Λ(ρ⊗n)|Ω〉 > (1 − ε)tr [Λ(ρ⊗n)],where Ω is a maximally entangled two-qubit state.

Proposition 8 [HH99a] If a bipartite state ρ ∈ B(Cd ⊗ Cd) does not satisfy the

reduction criterion, i.e., if it violates either ρ1 ⊗ 1 ≥ ρ, or 1 ⊗ ρ2 ≥ ρ, then it isdistillable.

Proof: Assume without loss of generality that the first inequality is violated, i.e.,there is a vector Φ such that 〈Φ|ρ1 ⊗ 1 − ρ|Φ〉 < 0. By Eq.(2.5) there exists anoperator X ∈ B(Cd) such that |Φ〉 = (X ⊗ 1)|Ω〉 and we can, after a suitablerescaling of X, define a filtering operation leading to a state

ρ′ ∝ (X∗ ⊗ 1)ρ(X ⊗ 1). (3.4)

In terms of the state ρ′ the initial inequality reads then 〈Ω|ρ′1 ⊗ 1 − ρ′|Ω〉 < 0,which is in turn equivalent to 〈Ω|ρ′|Ω〉 > 1

d . Hence we can twirl the state ρ′ ontoan entangled (and thus distillable) isotropic state.

As shown in Sec.2.2, for 2 ⊗ 2 systems the reduction criterion is equivalent tothe positivity of the partial transpose, which is in turn equivalent to separability inthis case. This implies:

Proposition 9 [HHH97] Every entangled two-qubit state ρ ∈ B(C2 ⊗ C2) is dis-

tillable.

Due to Prop.9 we may call a state ρ distillable iff there is an LOCC protocol whichmaps a finite number n of copies ρ⊗n with non-zero probability onto an entangledtwo-qubit state. This leads us to the following criterion for distillability:

35


Proposition 10 [HHH98] A bipartite state ρ ∈ B(H), H = Cd1⊗C

d2 is distillableiff there exists a finite number n ∈ N and a pure state Ψ ∈ H⊗n with Schmidt ranktwo, i.e., |Ψ〉 = λ|e1〉 ⊗ |f1〉+

√1− λ2|e2〉 ⊗ |f2〉, such that

〈Ψ∣∣(ρT1)⊗n

∣∣Ψ〉 < 0. (3.5)

Proof: Suppose ρ satisfies Eq.(3.5), and let P and Q be two two-dimensional localprojectors defined by P = |e1〉〈e1|+ |e2〉〈e2| and Q = |f1〉〈f1|+ |f2〉〈f2| respectively.Then Eq.(3.5) reads

tr[ρ⊗n|Ψ〉〈Ψ|T1

]= tr

[(P ⊗Q)ρ⊗n(P ⊗Q)|Ψ〉〈Ψ|T1

]< 0. (3.6)

Hence, the 2⊗ 2 state (P ⊗Q)ρ⊗n(P ⊗Q) is entangled and by Prop.9 we have thatρ is distillable.

Now assume that there is a LOCC operation described by a separable superop-erator Λ with Kraus operators Ai⊗Bi such that Λ(ρ⊗n) is an entangled two-qubitstate. Then at least one term in the Kraus decomposition of Λ must satisfy

[(Ai ⊗Bi)ρ⊗n(Ai ⊗Bi)∗

]T1

6≥ 0, (3.7)

since mixtures of PPT states are again PPT. The Kraus operators satisfy (Ai⊗Bi) =(Ai ⊗ Bi)(P ⊗ Q) for appropriate two-dimensional projectors P and Q, and sinceseparable superoperators are PPT preserving we have that Eq.(3.7) is still true ifwe replace (Ai⊗Bi) by (P ⊗Q). This means that there is a vector Ψ in the supportof P ⊗Q such that Eq.(3.5) is satisfied.

Prop.10 suggests to distinguish distillable states with respect to the requirednumber n in Eq.(3.5):

Definition 11 A bipartite state ρ is called “m-distillable” if there exists a purestate Ψ with Schmidt rank two, such that Eq.(3.5) holds for n = m. It is called“m-undistillable” if it is not m-distillable.

In other words, a state is called m-distillable if there is an LOCC protocol whichmaps m copies of it onto an entangled two-qubit state. It is worth being mentionedthat so far we do not know any example, which is distillable but not 1-distillable.In general the question whether a given state is distillable or not is very difficult todecide on the basis of Prop.10, since it still corresponds to a variational problemwith an apparently unbounded number of parameters. However, Prop.10 indicatesa close relation between distillability and non-positivity of the partial transpose(NPPT):

Proposition 12 3 Every state having a positive partial transpose ρT1 ≥ 0 is undis-tillable. Conversely, if ρ ∈ B(C2 ⊗ C

d) is undistillable, then it is PPT.

Proof: This is an immediate corollary of Prop. 10. The first statement followsfrom the fact that separable superoperators are PPT preserving and the second isimplied by the fact that every vector in C

2⊗Cd has at most Schmidt rank two.

We have seen in Sec.2.3 that there exist entangled PPT states. Since we needsome entanglement for their preparation, which can however not be extracted againby entanglement distillation, these states are called “bound entangled”. The entan-glement contained in a bound entangled state is useless for most of the purposes inquantum information theory.

This leads us to the big open question concerning distillability: is the propertyof having a positive partial transpose equivalent to being undistillable? That is,

3The first statement in Prop.12 was proven in [HHH98]. The implication for 2⊗ d was shownin [DCLB00].

36


are there bound entangled states, which are not PPT? An affirmative answer wouldbe very surprising at least in the sense that the decision of these two propertiesis a priori of very different complexity: the one is a simple eigenvalue problemwhereas the other is by definition a highly non-trivial decision problem involvingan unbounded number of tensor factors.

However, the question whether or not these two properties coincide can be de-cided by looking at a one parameter family of states only:

Proposition 13 [HH99b] There exist undistillable states in Cd1 ⊗ C

d2 , d1 ≤ d2,which do not have a positive partial transpose, iff there exist undistillable entangledWerner states in C

d1 ⊗ Cd1 .

Proof: It is sufficient to show, that every state with ρT1 6≥ 0 can be mappedonto an entangled Werner state by means of LOCC operations. If the state is notPPT, then there exists a vector Φ ∈ C

d1 ⊗ Cd2 such that 〈Φ|ρT1 |Φ〉 < 0. Since the

maximal Schmidt rank of Φ is d1, we can project ρ locally onto a state ρ′ acting onCd1 ⊗ C

d1 for which 〈Φ|ρ′T1 |Φ〉 < 0 again. Due to the cyclicity of the maximallyentangled state (see Eq.(2.5)), we can write |Φ〉 = (1 ⊗ X)|Ω〉. Now we apply afilter operation to ρ′ such that

ρ′′ ∝ (1⊗X∗)ρ′(1⊗X), (3.8)

and we have

0 > 〈Φ|ρT1 |Φ〉 ∝ 〈Φ|ρ′T1 |Φ〉 ∝ tr [ρ′′F] . (3.9)

Since the proportionality constants are both strictly positive, ρ′′ has a negative flipexpectation value and can thus be twirled onto an entangled Werner state.

The distillability properties of entangled Werner states were extensively dis-cussed in [DCLB00] and [DSS+00]. It was in particular shown, that for every finitem there exists an interval of states for which m-undistillability can be proven. Un-fortunately, this interval, which is next to the separable boundary, vanishes form → ∞, so the matter is not decided yet. The following two extremal cases arestill possible:

1. Every distillable state is 1-distillable. Due to the results in [DCLB00, DSS+00],there would exist bound entangled states in this case which are not PPT.

2. Every state having a non-positive partial transpose is distillable. In thiscase Def.11 has some discriminating power in the sense that there are m-undistillable states which are (m+1)-distillable for some m.

37

Distillability via PPT preserving protocols

3.3 Distillability via PPT preserving protocols

As already mentioned in the previous section, it is to date not known whetherall states having a non-positive partial transpose (NPPT) are distillable. One ofthe main difficulties in deciding this question is that the set of LOCC protocolsis mathematically rather intractable, even when applied to symmetric states likeWerner states. The following letter answers the question when replacing LOCCprotocols by the larger set of PPT preserving protocols. The work was essentiallymotivated by a paper by Pavel Horodecki [Hor01], who tried to show that there existNPPT states which are undistillable with respect to PPT preserving protocols. Hisattempt failed, but it remained unclear whether his idea could be made work. Theletter proves the converse statement:

• Every NPPT state is distillable with respect to PPT preserving protocols.

38

VOLUME 87, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 17 DECEMBER 2001

Distillability via Protocols Respecting the Positivity of Partial Transpose

Tilo Eggeling,* Karl Gerd H. Vollbrecht,† Reinhard F. Werner,‡ and Michael M. Wolf §

Institut für Mathematische Physik, TU Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig, Germany(Received 15 May 2001; published 28 November 2001)

We show that all quantum states that do not have a positive partial transpose are distillable via channels,which preserve the positivity of the partial transpose. The question whether bound entangled stateswith nonpositive partial transpose exist is therefore closely related to the connection between the set ofseparable superoperators and positive partial transpose-preserving maps.

DOI: 10.1103/PhysRevLett.87.257902 PACS numbers: 03.67.–a, 03.65.Ca, 03.65.Ud

I. Introduction.—One of the main tasks of quantum in-formation theory is the systematic investigation of quan-tum entanglement, which is one of the key ingredients inquantum computation and quantum information process-ing. In spite of considerable research efforts, however,there are still many aspects of entanglement which are notfully understood. This is not only true for the quantita-tive theory (the explicit computation or at least the esti-mation of entanglement measures) but even for qualitativefeatures.

These qualitative features are best explained by lookingat the history of the problem. In 1989 [1] it was a newrealization that there is a proper gap between obviouslyentangled states (those violating some Bell inequality) andthe obviously nonentangled states, which are now calledseparable. The next step, made by Popescu in 1995 [2]was the striking result that this gap could be narrowed bydistillation: By local filtering and classical communica-tion one could sometimes get highly entangled states evenfrom states not violating any Bell inequality. For a whileit was everybody’s favorite conjecture that there shouldbe no more gap, i.e., that all nonseparable states shouldbe distillable. This folk conjecture was shattered in 1998by counterexamples [3], which are now called bound en-

tangled states. The way these examples were establishedwas by showing that the property of a density operator ofhaving a positive partial transpose, i.e., of being a PPT(positive partial transpose) state, does not change underdistillation. Therefore any nonseparable state with posi-tive partial transpose has to be “bound entangled.” Theobvious white spot on the entanglement map is then: Arethese all bound entangled states, or are there undistillablestates, whose partial transpose is not positive?

There were two recent papers [4–6] presenting someevidence for the existence of non-PPT-bound entangledstates. However, the matter is not decided [7], and in viewof the rapidly growing dimensions of the Hilbert spacesinvolved, numerical evidence can be treacherous in thisfield. The latest development was an attempt by Horodecki[9] at showing the existence of a gap between PPT statesand distillable states, using a stronger protocol [10] ofdistillation. The attempt failed due to an error in anotherpaper, but it remained unclear whether the idea could be

made to work. What we show in the present Letter is thatit cannot work: using the same distillation protocol [10],every non-PPT-state becomes distillable.

The rather subtle dependence of distillability on “proto-cols” requires some explanation. Typically a protocol fixesthe amount of classical communication allowed to Aliceand Bob in the process. Thus we may distinguish distil-lation with no communication allowed or with one-wayor two-way communication. Even stronger protocols thantwo-way communication protocols exist: these are definedby requiring only a subset of the properties which are truefor all two-way distillation procedures. One example isthe requirement that the overall operation can be writtenas a sum of tensor products of local operations (“separablesuperoperator”). Another such property, which is the onewe consider in this Letter following Rains [10], is that op-erators with positive partial transpose are again taken tosuch operators. An example of such a PPT-preserving pro-tocol is the case where Alice and Bob share a PPT-boundentangled state and use a protocol consisting of local op-erations and classical communication (LOCC) [11].

Obviously, the weaker the requirements on the admis-sible transformations, the larger the set of distillable states.However, when we do not care about rates of distillation,the dependence on the protocol is not as strong as onemight think. For example, distillability with two-waycommunication and with separable superoperators isknown to be equivalent [3]. Moreover, the stronger proto-cols have the virtue of being much more manageable andmore easily parametrized than two-way communicationprocesses, which may involve an arbitrarily large numberof exchanges of classical information.

Therefore it seemed quite reasonable to study the prob-lem of a proper gap between PPT states and distillablestates under this “PPT-preserving” protocol. Moreover,since a proper gap is the currently favored conjecture, itwas reasonable to expect a gap even with such a protocol.The main result of this Letter is, however, that the gap dis-appears, if we allow such a strong protocol. Unfortunately,this does not provide conclusive evidence about the gap forweaker protocols.

II. Distillation via PPT-preserving channels.—For thesake of completeness we begin by recapitulating the result

257902-1 0031-90070187(25)257902(3)$15.00 © 2001 The American Physical Society 257902-1


of Rains [10] for the fidelity of distillation via PPT-preserving channels. Let r be a density operator corre-sponding to a quantum state on d ≠ d and r T ra trace-preserving positive map, such that s $ 0 impliesT sT2 T2 $ 0.

Here the notion s $ 0 means that s has a non-negativespectrum and the superscript T2 denotes the partial trans-position. The latter is defined in terms of matrix elementswith respect to a given basis by ijjsT2 jkl iljsjkjhaving the property that it is Hermiticity but not positivitypreserving.

Let us further write Pm jcm cmj for the projectoronto the maximally entangled state in m 3 m dimensions,i.e., jcm

1pm

Pmi1 ji ≠ ji. Theorem 3.1 of [10] then

relates the maximal fidelity obtained by distillation viaPPT-preserving maps to the expectation value of certainHermitian operators. That is, it relates a problem fromchannel theory to the calculation of expectation values ofa special class of observables:

Lemma 1: The maximal fidelity of distillation via PPT

and trace-preserving positive maps with respect to the

m-dimensional maximally entangled state is given by

Fmr : maxT

trPmT r maxA

trrA , (1)

where the maximum on the right side is taken over all

Hermitian operators A satisfying

0 # A # 1 and 2 1 # mAT2 # 1 . (2)

Proof: First note that since every unitary of the formU ≠ U commutes with Pm it suffices to consider trace-preserving positive maps mapping into the set of isotropicstates, i.e., states which are obtained by averaging over allthese unitaries [12]:

Tr trrB 1 2 Pm 1 trrAPm . (3)

The coefficients in Eq. (3) have to be linear functionalsof r so that we are free to write them as traces, and T rbeing again a proper state requires that 0 # A, B # 1,and m2 2 1B 1 A 1. In order to obtain a PPT-preserving map we have additionally to demand thats $ 0 implies that

TsT2 T2 trsBT2

µ1 2

1

mF

∂1 trsAT2

1

mF $ 0 ,

(4)

where F denotes the flip operator, i.e., Fjf ≠ jc

jc ≠ jf. Inequality (4) is satisfied if and only if (iff)the absolute value of the coefficient of the flip operator isless than or equal the weight of the identity operator:

61

mtr

∑s

µAT2 2 1

m2 2 11 AT2

∂∏# tr

∑s

1 2 AT2

m2 2 1

∏. (5)

Since this inequality has to hold for all positive operatorss we can reformulate it as an operator inequality which isin turn equivalent to 21 # mAT2 # 1.

Hence, there is a one-to-one correspondence betweenPPT-preserving maps T of the form (3) and the respective

Hermitian operators A satisfying the constraints specifiedin Lemma 1 given by trPmTr trrA.

In fact, positive maps of the form (3) are even com-pletely positive; i.e., Lemma 1 holds also for PPT-preserving channels as can easily be seen by writing downa Kraus decomposition:

Tr trrB 1 2 Pm 1 trrAPm

1 2 Pm trp

B rp

B 1 2 Pm

1 Pm trp

A rp

A Pm .

This, however, is a special property of positive maps ofthe form (3). Indeed positivity and PPT preservation donot imply complete positivity in general, a counterexamplebeing the transposition.

Now we can utilize Lemma 1 in order to prove thefollowing:

Theorem 1: Any NPT state, i.e., state with nonposi-

tive partial transpose, is distillable via PPT-preserving

channels.

Proof: We recall that a state is known to be distillablevia standard LOCC distillation protocols if trrPm .

1m

[13]. The task is therefore to find an appropriate operatorA such that trrA . 1

m .Let Pneg be the projector onto the negative eigenspace

of rT2 . We choose A to be of the form

A

1

m1 2 ePT2

neg, 0 , e # min2, kPT2

negk21` ,

(6)

where k ? k` denotes the operator norm, which is in thiscase just the largest absolute value of an eigenvalue. Nowwe have to check whether A satisfies the constraints inLemma 1.

Positivity of the parameter e implies mAT2 # 1. To en-sure A # 1 it is sufficient that e # m 2 1 kPT2

negk21` , but

0 # A requires the even stronger condition e # kPT2negk21

` .

Moreover, mAT2 $ 21 is equivalent to e # 2, whichshows that Eq. (6) indeed defines an admissible operatorA. With the above A we obtain

trrA

1 1 eN r

m, (7)

where N r is the negativity [14], which is just the sumover the absolute values of the negative eigenvalues of rT2 .Since the state has at least one such negative eigenvalue byassumption, we end up with a fidelity larger than 1

m , whichcompletes our proof.

Of course one may further evaluate Eq. (7) for morespecific states. Let us, for instance, consider states com-muting with all unitaries of the form U ≠ U [1], whichcan be written as

rp 1 2 pP1

r1

1 pP2

r2

, 0 # p # 1 , (8)

where P1 (P2) is the projector onto the symmetric (anti-symmetric) subspace of d ≠ d and r6 trP6

d26d

2 are the respective dimensions. Evaluating Eq. (7)

257902-2 257902-2


for these states (e 2) then leads to

trArp

krpT2k1

m

d 2 2 1 4p

md, (9)

where k ? k1 denotes the trace norm, which is in the caseof Hermitian operators the sum over the absolute values ofthe eigenvalues.

In fact, this turns out to be already the maximal valuefor Fmrp. This can easily be seen by decomposing thepartial transpose of the state into its positive and negativeparts, i.e., rT2

r1 2 r2. Then

trrA trAT2rT2 trAT2 r1 2 r2

#1

mtrr1 1 r2

1

mkrT2k1 , (10)

where the estimate is due to the constraint mkAk1 # 1.This bound for the maximal fidelity can always be

reached for states with kPT2negk` #

12 .

III. Conclusion.—We have argued that enlarging theset of distillation protocols to PPT-preserving channelsimmediately implies that any NPT state can be distilled.Since we know that a state can be distilled via properLOCC operations iff trPmSr . 1

m for some separable

superoperator S [3,15], this raises the question about theconnection between the sets of separable superoperatorsand PPT-preserving channels. It is obvious that any sepa-rable superoperator is PPT preserving, but we do not knowyet any efficient method for deciding whether a givenoperator A from Lemma 1 corresponds to a separablesuperoperator.

There is a standard argument telling us that NPT boundentangled states exist iff there exist undistillable entangledstates of the form special U ≠ U-invariant form (8) [13].So the question about the existence of NPT bound en-

tangled states becomes to decide whether PPT-preservingchannels that distill U ≠ U-invariant states near the sepa-rable boundary can be realized as separable superoperatorsor not.

Moreover, the results in Ref. [16] show that any PPT-preserving channel can be implemented by a stochasticLOCC channel acting additionally on certain PPT-boundentangled resource states.

Another interesting feature of the distillation we dis-cussed is that we needed only a single copy of the given

bipartite state and not a tensor product of many identicallyprepared ones. This raises the question whether distillabil-ity under LOCC protocols can also be decided at the singlecopy level. All examples known to us would be consistentwith this.

The authors thank Barbara Terhal for interesting dis-cussions and for pointing out Ref. [11]. Funding by theEuropean Union project EQUIP (Contract No. IST-1999-11053) and financial support from the DFG (Bonn) aregratefully acknowledged.

*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]§Electronic address: [email protected]

[1] R. F. Werner, Phys. Rev. A 40, 4277 (1989).[2] S. Popescu, Phys. Rev. Lett. 74, 2619 (1995).[3] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.

Lett. 80, 5239 (1998).[4] D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal,

and A. V. Thapliyal, Phys. Rev. A 61, 062312 (2000).[5] W. Dür, J. I. Cirac, M. Lewenstein, and D. Bruss, Phys.

Rev. A 61, 062313 (2000).[6] See also problem page 2 at our open problem site

http://www.imaph.tu-bs.de/qi/problems/2.html[7] An indication to the contrary is the very recent result [8]

that the gap between PPT and distillability does not existfor Gaussian (continuous variable) states.

[8] G. Giedke, L.-M. Duan, P. Zoller, and J. I. Cirac, QuantumInf. Comput. 1, 79 (2001).

[9] P. Horodecki, quant-ph/0103091 v1.[10] E. M. Rains, IEEE Trans. Inf. Theory 47, 2921–2933

(2001).[11] P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett.

86, 2681 (2001).[12] K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64,

062307 (2001).[13] M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206

(1999); S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden,S. Popescu, and R. Schack, Phys. Rev. Lett. 83, 1054(1999).

[14] G. Vidal and R. F. Werner, quant-ph/0102117.[15] E. Rains, quant-ph/9707002.[16] J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein, Phys.

Rev. Lett. 86, 544 (2001).

257902-3 257902-3

Activating distillation with symmetric states

3.4 Activating distillation with symmetric states

We have seen in the last section that every state with non-positive partial transposeis distillable via PPT preserving protocols. By the operator-map dualism presentedin Prop.7 these protocols can however be implemented by LOCC protocols sup-plemented by an additional (entangled) PPT state. In this way the set of PPTpreserving protocols gets a physical meaning. For every m-undistillable entangledstate ρ there exists a PPT state σ, which is undistillable itself, such that σ ⊗ ρbecomes distillable. We will call this process “activation of distillation”, and thePPT state σ the “activator”.

The following letter investigates the requirements, which the supplementarystate σ has to fulfill. Of particular interest are the following results:

• An infinitesimal amount of (bound) entanglement contained in the state σ issufficient for the activation process.

• There are “universal activators”, capable of activating the distillation of everyentangled state. Hence σ does not have to depend on ρ. Moreover, there areinfinitesimally entangled universal activators.

43

Activating distillation with symmetric states

44

VOLUME 88, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 17 JUNE 2002

Activating Distillation with an Infinitesimal Amount of Bound Entanglement

Karl Gerd H. Vollbrecht* and Michael M. Wolf†

Institute for Mathematical Physics, TU Braunschweig, Germany(Received 25 January 2002; published 31 May 2002)

We show that bipartite quantum states of any dimension, which do not have a positive partial transpose(NPPT), become 1-distillable when one adds an infinitesimal amount of bound entanglement. To thisend we investigate the activation properties of a new class of symmetric bound entangled states of fullrank. It is shown that in this set there exist universal activator states capable of activating the distillationof any NPPT state. The result shows that even a small amount of bound entanglement can be useful forquantum information purposes.

DOI: 10.1103/PhysRevLett.88.247901 PACS numbers: 03.67.–a, 03.65.Ca, 03.65.Ud

Introduction.—The possibility of distillation plays acrucial role in quantum communication and quantum infor-mation processing (cf. [1]). Together with quantum errorcorrection it enables all the fascinating applications pro-vided by quantum information theory in the presence ofa noisy and interacting environment. Despite its practicalrelevance and quite considerable effort in that direction,however, many of the basic questions concerning distil-lation are yet unanswered. Most notable is the questionwhether a given quantum state is distillable or not, i.e.,whether it is possible to obtain pure maximally entangledstates from several copies of it by means of local opera-tions and classical communication (LOCC).

A necessary condition for the distillability of a state de-scribed by a density matrix r is the fact that its partialtranspose rTA , defined with respect to a given product ba-sis by ijjrTA jkl kjjrjil, has a negative eigenvalue[2]. Except for special cases like states on 2 ≠ n [3,4]and Gaussian states [5], it is, however, unclear whether thiscondition is sufficient as well. There is some evidence pre-sented in [4,6] that this may not be the case and that thereare indeed undistillable states, whose partial transpose isnot positive (NPPT). At least there exist n-undistillableNPPT states for every finite n, meaning that no LOCC op-eration on n copies leads even to a single entangled twoqubit state [4,6].

However, if we enlarge the class of allowed distillationprotocols from LOCC to channels respecting the positivityof the partial transpose, then every NPPT state becomes1-distillable [7,8] (which can be shown by using entangle-

ment witnesses [8]). Moreover, it is a result from [9] thatthese channels can always be stochastically implementedby an LOCC operation where the two parties are given anentangled state with positive partial transpose (PPT) as anadditional resource. The latter is known to be bound en-

tangled since the entanglement needed for the preparationof the state cannot be recovered by distillation [2]. Nev-ertheless, PPT bound entangled states can be useful in or-der to activate the distillability of bipartite NPPT states[10,12].

The aim of the present paper is to investigate the limitsand requirements of such an activation process. We will

show that there exist states with an arbitrary small amountof PPT bound entanglement, which are capable of activat-ing any NPPT state. The required additional resource istherefore universal as well as arbitrarily weakly entangled.

Preliminaries on symmetric states.—One of the keyideas in what follows will be the exploitation of the sym-metry properties of states commuting with certain localunitaries. Two well known one-parameter families of suchstates are the Werner states and isotropic states, both play-ing an important role in the sequel.

Werner states [16] acting on a Hilbert space H

HA ≠ HB with dimensions dimHA dimHB d

commute with all unitaries of the form U ≠ U and canbe written as

ra

µ1 2

a

d

∂d2 2 a, a [ 2d,d , (1)

with being the flip operator defined with respect to someproduct basis by jij jji. A Werner state is entanglediff a [ 1, d and 1-distillable iff a [ d2, d. More-over, it was shown in [14] that any NPPT state can bemapped onto an entangled Werner state by means of LOCCoperations. Therefore we can in the following restrict ourdiscussion to the activation of Werner states keeping inmind that the obtained results hold for any NPPT state.

Isotropic states [14,15] commuting with all unitaries ofthe form U ≠ U (where U is the complex conjugate of U)are combinations of the maximally mixed state 1d2 andthe projector jV Vj onto the maximally entangledstate jV 1

pd

Pdi1 jii:

vf f 11 2 f

d2 2 11 2 , f [ 0, 1 . (2)

An isotropic state v is known to be 1-distillable iff themaximally entangled fraction f VjvjV . 1d,which is a sufficient condition for any other state as well[14]. Hence, an activation protocol succeeded if thiscondition is fulfilled by the final state.

The symmetric states playing the central role inthe present paper act on a larger Hilbert space H

HA1≠ HA2

≠ HB1≠ HB2

of total dimension d4, whereA and B again label the two parts of the system situated at

247901-1 0031-90070288(24)247901(4)$20.00 © 2002 The American Physical Society 247901-1


different locations. The symmetry group under con-sideration is the group of all unitaries of the formW U ≠ V A ≠ U ≠ V B. States s commuting withall these unitaries can most easily be expressed in termsof the minimal projectors Pi spanning the commutant ofthe group [16]:

; W : s, W 0 , s

4Xi1

liPitrPi , (3)

with

P12

1

21 7 1 ≠ 2 , (4)

P34

1

21 7 1 ≠ 1 2 2 . (5)

Note that, as labeled by the indices, the tensor productscorrespond to a split 1j2 (and not AjB. Positivity and nor-

malization of s requires li $ 0 andP4

i1 li 1 suchthat any state of the considered symmetry can be charac-

terized by a vector l [ 3 lying in a tetrahedron, whichis given by these constraints. We note further that the setof symmetric states in (3) is Abelian; i.e., all symmetricstates commute with each other.

The activation protocol we use follows closely an ideaof Ref. [9]. Initially the two parties A and B are supposedto share a Werner state ra acting on H0 HA0

≠HB0

with dimH0 d2 and a symmetric state s on H1 ≠H2 given by Eq. (3). After a local filtering operation is ap-plied by projecting onto maximally entangled states A0,1

and B0,1(acting on HA0

≠ HA1and HB0

≠ HB1, re-

spectively) the maximally entangled fraction of the result-ing state on system 2 is given by

fra; s : trra ≠ s A0,1

≠ B0,1≠ 2

trra ≠ s A0,1≠ B0,1

≠ 12.

(6)

We know that s activates ra if fra; s . 1d.Since the output state of the protocol is itself isotropic,this condition is also necessary for the activation.

Of course, we are interested only in cases where a [

1, d2, i.e., ra is entangled but not 1-distillable, and sis in turn a PPT bound entangled state. The latter requiresthe classification of the symmetric states in (3), which isthe content of the next section.

Classification and Activation.—The following discus-sion will mainly take place in the three dimensional space

given by the expansion coefficients l l1, l2, l3

from Eq. (3). A vector l corresponds to a (positive and

normalized) symmetric state s l iff l [ S y [

3jyi $ 0,P

i yi # 1, where the state space S is atetrahedron.

The set P corresponding to normalized operators with apositive partial transpose can easily be obtained by observ-ing that the symmetry group of the partially transposed op-erators sTA in (3) is equal to the group of unitaries W when

interchanging the systems 1 $ 2. The respective minimalprojectors Qi can therefore be obtained from the pro-jectors Pi simply by relabeling the systems 1 $ 2, andthe kth coordinate of an extreme point pi of P is thusgiven by

pik trQ

TA

i PktrQi . (7)

Hence, P is again a tetrahedron and Eq. (7) leads to theextreme points:

p1

µd 2 1

2d,2d 2 1

2d,1 2 d2

2d

∂,

p2

µ1 2 d

2d,1 1 d

2d,2d 2 12

2d

∂,

p3

µ21

2d,21

2d,d 1 1

2d

∂,

p4

µ1

2d,

1

2d,d 2 1

2d

∂.

Straightforward linear algebra now allows us to com-pute the extreme points ti of the intersection S > Pcorresponding to the set of symmetric PPT states, which isshown in Fig. 1:

0.2

0.2

0.2

0.2

0.2

0.1

0.1

0.1

0.40.4

00

0

0

0

0.3

(5)

τ (5)

(0)

τ (1)

→

τ(2)

τ(3)

τ (4)

→

→

→

→

FIG. 1. The set of symmetric PPT states s (thick wired ob-ject) parametrized by the three coordinates li trsPi , plot-ted for d 3. The solid object inside corresponds to the setof separable states. The universal activators lie on the plane t3, t4, t5 and contain an arbitrary small amount of en-tanglement near the line t3, t4.

247901-2 247901-2


t1

µ0, 0,

1

2

∂, t2

0, 0, 0 ,

t3

µ1

2d,

1

2d,d 2 1

2d

∂, t4

µ0,

1

d, 0

∂,

t5

µ1

d 1 2, 0, 0

∂.

The PPT state space given by the convex hull of thesepoints can be divided into three parts: separable states,as well as activating and not activating bound entangledstates.

(i) Separable states: A vector l [ S > P correspondsto a separable symmetric state iff we can find any (notnecessarily symmetric) separable state rsep such that li

trPirsep (cf. [15]).A special case of a symmetric separable state is, of

course, a tensor product of a Werner and an isotropic states ra1 ≠ vf2, where both are separable. In fact, ifwe choose these two states lying on the separable bound-aries, i.e., a [ 2d, 1 and f [ 0, 1d, we retrieve theextreme points t1, . . . , t4.

Another point t0 1

2d , 0,d22

4d that will also turn outto be an extreme point of the set of separable symmetricstates is obtained from the product state:

rsep jF1 F1jA ≠ jC2 C2jB, (8)

where jF1 1p2j11 1 j22 and jC2

1p2j12 2

j21 are two-dimensional maximally entangled states.The convex hull of the points t0, . . . , t4 (see Fig. 1)

already covers the entire separable region as we are go-ing to show in the following that the complement of thispolytope within S > P corresponds to bound entangledstates.

(ii) Bound entangled and activating: The equation

fra; s l 1

d written out asP

i ciali 0, with

cia trra ≠ Pi A0,1

≠ B0,1≠ d 2 12

trPi

is linear in li and thus defines a plane separating sym-metric states activating ra from states apparently notactivating it. The task is now to construct this separatingplane depending on the parameter a.

As we have already used above, the points t3, t4

correspond to product states of the form s ra1 ≠v

1

d 2 for which fra; s 1

d obviously holds for any

Werner state ra. Thus t3, t4 are two fixed points ofthe separating plane with respect to a variation of a, andwe need to know only one more point. For this purpose we

consider the line lt t t5 1 1 2 tt1. Solving the

equation fra; s lt 1

d yields the required thirdpoint with

t 2 1 d

2a 1 d. (9)

This is obviously a strict monotone function in a, and itleads to the following properties of the separating plane.

(1) For a d

2 corresponding to the boundary Werner

state which is not 1-distillable, Eq. (9) leads to lt t0,showing that t0 indeed lies on the boundary of the set ofseparable states. That is, any PPT state in front of the plane t0, t3, t4 must be bound entangled since it activates

at least ra d

2 .(2) In the limit a ! 1, i.e., ra becoming less and

less entangled, lt approaches t5. However, for any

a 1 1 ´, ´ . 0 the polytope t0, t3, t4, lt has anonempty interior corresponding to PPT bound entangledstates capable of activating any ra with a . 1 1 e.

(3) Except for the line t3, t4 all PPT states on theplane t3, t4, t5 lie on the activating side of the sepa-rating plane for any a . 1. The corresponding symmetricstates can thus be considered to be universal activators inthe sense that they activate any entangled Werner state andtherefore any NPPT state.

The set of bound entangled universal activators con-tains states arbitrarily close to the line t3, t4 whichin turn corresponds to separable states. By the continu-ity properties of the entanglement measures entanglement

of formation [17] and relative entropy of entanglement

[18] this geometric vicinity, however, translates directly tothe proposition that these states contain an arbitrary smallamount of entanglement.

(iii) Bound entangled and not activating: In order tocomplete the classification of the symmetric states in-troduced in Eq. (3) we have still to determine the en-tanglement properties of the states corresponding to thetetrahedron t0, t2, t4, t5. The plane separating thisset from the separable states derived above is characterized

by a linear operator W via trWs l 0, where

W 1 2 1 ≠µ1 2

d

2

∂2

. (10)

However, this operator is an entanglement witness

(cf. [19]), meaning that trWr $ 0 holds for any sep-arable state r. In order to see this property we havejust to utilize the results from [4,6], where it was shown

that 1 2d

2 has a positive expectation value on anypure state of Schmidt rank two. Since the antisymmetric

projector1

2 1 2 is a sum of Schmidt rank two states,fA ≠ cBjWjfA ≠ cB is a sum of such positive expec-tations for any pure product state jfA ≠ cB. Hence,trWr $ 0 is indeed fulfilled by any separable state im-plying that the tetrahedron under discussion correspondsto bound entangled symmetric states which are, however,not activating (with respect to the considered protocol).

Conclusion.—We investigated the entanglement proper-ties of a new Abelian set of symmetric states with regardto the activation of NPPT distillation. The set containsPPT bound entangled states of full rank providing a uni-versal resource for the activation of any NPPT state (in

247901-3 247901-3


any finite dimension). Some of these universal activators

lie arbitrarily close to separable states. Hence, the acti-vation process turns out to require only an infinitesimalamount of entanglement. This indicates that the differencebetween 1-distillable and n-undistillable or even bound en-tangled NPPT states is very subtle. Moreover, the aboveresults show that even weakly entangled bound entangledPPT states can be useful for some quantum informationprocessing purposes [20].

As the problem discussed in this paper is primarily afeasibility problem, we have at this stage not asked forthe obtained rates. In fact, one could, for instance, easilyimprove the probability of success for the used activationprotocol by a factor of d2 by measuring in a basis of max-imally entangled states and retaining the state wheneverthe measurement outcomes coincide. An interesting ques-tion going one step further and requiring knowledge aboutrates is whether there is the possibility of self-activation

after some initial activation with a limited resource tookplace. In other words, is it possible to yield asymptoticallymore entanglement from distillation than is needed for thepreparation of the activator states?

The authors thank R. F. Werner and T. Eggeling forstimulating discussions and, in particular, M. Lewensteinfor bringing the witness property of W to their attention.Funding by the European Union project EQUIP (ContractNo. IST-1999-11053) and financial support from the DFG(Bonn) is gratefully acknowledged.

*Electronic address: [email protected]†Electronic address: [email protected]

[1] G. Alber et al., Quantum Information, Springer Tracts inModern Physics. Vol. 173 (Springer, Berlin, 2001).

[2] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.Lett. 80, 5239 (1998).

[3] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.Lett. 78, 574 (1997).

[4] W. Dür, J. I. Cirac, M. Lewenstein, and D. Bruss, Phys.Rev. A 61, 062313 (2000).

[5] G. Giedke, L.-M. Duan, J. I. Cirac, and P. Zoller, Quant.Inf. Comp. 1, 79 (2001).

[6] D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal,and A. V. Thapliyal, Phys. Rev. A 61, 062312 (2000).

[7] T. Eggeling, K. G. H. Vollbrecht, R. F. Werner, and M. M.Wolf, Phys. Rev. Lett. 87, 257902 (2001).

[8] B. Kraus, M. Lewenstein, and J. I. Cirac, e-print quant-ph/0110174, (2001).

[9] J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein, Phys.Rev. Lett. 86, 544 (2001).

[10] In fact, the first example of such an activation processwas given in [11] where it was also shown that if thereexist bound entangled NPPT states, then the distillable

entanglement is neither additive nor convex.[11] P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett.

86, 2681 (2001).[12] It was recently shown, however, that there exist three-

partite NPPT states, with the property that two copies canneither be distilled nor activated [8].

[13] R. F. Werner, Phys. Rev. A 40, 4277 (1989).[14] M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206

(1999).[15] K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64,

062307 (2001).[16] By the Commutation Theorem for von Neumann algebras,

the commutant of the tensor product of the consideredgroups is the tensor product of the commutants. Hence, wehave just to tensor the minimal projectors known from thediscussion of Werner and isotropic states. For more detailsabout symmetric states commuting with local unitaries seeRef. [15].

[17] M. A. Nielsen, Phys. Rev. A 61, 064301 (2000).[18] M. J. Donald and M. Horodecki, Phys. Lett. A 264, 257

(1999).[19] M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki,

Phys. Rev. A 62, 052310 (2000).[20] The first example showing that bound entangled PPT states

can indeed be useful as an additional resource for telepor-tation protocols was given in Ref. [21].

[21] P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev.Lett. 82, 1056 (1999).

247901-4 247901-4

Chapter 4

Conditional entropies

Consider two correlated classical random variables x ∈ X and y ∈ Y with a jointprobability distribution p(x, y). The “marginals” of this distribution are given byp(x) =

∑y p(x, y) and p(y) =

∑x p(x, y). If we compare the entropy of the joint

distribution with the entropy of one of its marginals, we obtain

S(X,Y )− S(X) = −∑

x,y

p(x, y) log2 p(x, y) +∑

x

p(x) log2 p(x) (4.1)

=∑

x,y

p(x, y) log2p(x)

p(x, y)≥ 0 . (4.2)

Hence, a composite classical system can never have a smaller entropy than any ofits parts. This is of course a well known fact in classical information theory andstatistical mechanics, and the non-negative difference of the two entropies is knownas the “conditional entropy” S(Y |X) = S(X,Y )− S(X).

However, for entangled quantum systems the conditional entropy can take onnegative values. In the quantum case the Shannon entropy has to be replaced bythe von Neumann entropy of the considered density matrix S(ρ) = −tr [ρ log2 ρ]which is in turn nothing but the Shannon entropy of the spectrum of ρ. The bestexample for a bipartite quantum state having a negative conditional entropy is themaximally entangled state. As a pure state it has zero entropy S(ρ) = 0, whereasthe respective reduced states are maximally chaotic, i.e., S(ρ1) = S(ρ2) = log2 d,with d being the dimension of each subsystem.

The following paper considers the whole set of “Renyi entropies”, which includethe Shannon resp. von Neumann entropy as a special case. The main results are:

• Every conditional Renyi entropy is non-negative if the considered state satis-fies the reduction criterion. This implies in particular:

• The conditional entropies of every PPT state are non-negative.

• If any conditional entropy is negative, then the state is distillable.

49

Conditional entropies and their relationto entanglement criteria

Karl Gerd H. Vollbrechta) and Michael M. Wolfb)

Institute for Mathematical Physics, TU Braunschweig, Germany

~Received 12 February 2002; accepted for publication 16 May 2002!

We discuss conditional Renyi and Tsallis entropies for bipartite quantum systems offinite dimension. We investigate the relation between the positivity of conditionalentropies and entanglement properties. It is in particular shown that any state hav-ing a negative conditional entropy with respect to any value of the entropic param-eter is distillable since it violates the reduction criterion. Moreover, we show thatthe entanglement of Werner states in odd dimensions can neither be detected byentropic criteria nor by any other spectral criterion. © 2002 American Institute ofPhysics. @DOI: 10.1063/1.1498490#

I. INTRODUCTION

Entanglement has always been a key issue in the ongoing debate about the foundations andinterpretation of quantum mechanics since Einstein1 and Schrodinger2 expressed their deep dis-satisfaction about this astonishing part of quantum theory. Whereas for the long period from 1935to 1964, until Bell3 published his famous work, discussions about entanglement were purelymeta-theoretical, nowadays quantum information theory has established entanglement as a physi-cal resource and key ingredient for quantum computation and quantum information processing.This led to a dramatic increase of general structural knowledge about entanglement in the last fewyears, and the resource point of view often led to results that are reminiscent of those known fromthermodynamics: free entanglement is distinguished from bound entanglement,4 irreversibility canbe observed in the process of preparing and distilling entangled states5 and entanglement itself isdefined in a way that it must not increase by means of local operations and classical communi-cation ~LOCC!. Moreover, there is recent effort in order to quantify quantum correlations throughheat engines.6

Entropies lay at the heart of both theories, thermodynamics and entanglement theory. Con-cerning the latter it was shown that a few reasonable assumptions lead to a unique measure ofentanglement7 for pure bipartite quantum states, which is just the von Neumann entropy of thereduced state. Hence, it is obvious that the two subsystems of a pure entangled state exhibit moredisorder than the system as a whole, so that the respective conditional entropy is negative. This isa remarkable property of entangled states, which is impossible for classical systems ~i.e., classicalrandom variables!.

The present article is primarily devoted to settling the relationship between the negativity ofconditional Renyi and Tsallis entropies and other entanglement properties. We will in particularshow how the property of having a positive conditional entropy enters into the known implicationchain of entanglement, resp., separability criteria.

In the second part we will then follow the result of Nielsen and Kempe8 and give examples ofentangled states having the property that their entanglement can neither be detected by entropiccriteria nor by any other spectral criterion. In Sec. IV we show that this is indeed the case forsymmetric Werner states ~in odd dimensions!, which play a crucial role in entanglement theory.

a!Electronic mail: [email protected]!Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 43, NUMBER 9 SEPTEMBER 2002

42990022-2488/2002/43(9)/4299/8/$19.00 © 2002 American Institute of Physics

II. PRELIMINARIES ON SEPARABILITY CRITERIA

To fix ideas we will start by recalling some of the basic notions and previous results concern-ing separability, resp., entanglement criteria.

A bipartite quantum state described by its density matrix r acting on a Hilbert space H

5H(A)

^ H(B) is said to be separable, unentangled or classically correlated if it can be written as

a convex combination of tensor product states9

r5(j

p jr j(A)

^ r j(B) , ~1!

where the positive weights p j sum up to one and r (A) (r (B)) describes a state on H(A) (H (B)).

This means in particular that pure states are separable if and only if they are product states.Moreover, all entanglement properties of pure states, which can always be written in their Schmidtform ~cf. Ref. 10! as uC&5( iAl iui& ^ ui& , are completely determined by the eigenvalues $l i% ofthe reduced state rA5trBuC&^Cu. The unique measure of entanglement for pure states is thengiven by the von Neumann entropy of the reduced state:

S1~rA!52tr~rA log rA!. ~2!

For mixed quantum states, however, the situation is much more difficult and deciding whethera state is entangled or separable is not yet feasible in general. Currently, the most efficientnecessary criterion for separability is the positivity of the partial transpose ~PPT!, i.e., the condi-tion that rTA has to be a positive semi-definite operator.11 The partial transpose of the state isthereby defined in terms of its matrix elements with respect to some basis by ^klurTAumn&5^mlurukn&. For the smallest nontrivial systems with 232, resp., 233, dimensional Hilbertspaces and a few other special cases the PPT criterion also turned out to be sufficient.12 In higherdimensional systems, however, so-called bound entangled states exist, which satisfy the PPTcondition without being separable.4

Another well known condition is given by the reduction criterion13,14

rA ^ 12r>0 and 1^ rB2r>0, ~3!

which is implied by the PPT criterion but is nevertheless an important condition since its violationimplies the possibility of recovering entanglement by distillation ~which is yet unclear for PPTviolating states!. For the case of two qubits ~and 233! the reduction criterion is also known to besufficient for separability.13,14 Moreover, it was shown in Ref. 15 that Eq. ~3! implies that the rankof the reduced state has to be smaller than or equal to the rank of r. The general line of implicationis then

r separable⇓

rTA>0⇓

r undistillable⇓

rA ^ 12r>0`1^ rB2r>0⇓

max@rank~rA!,rank~rB!#<rank~r !.

~4!

The last condition we want to mention was recently derived by Nielsen and Kempe8 and is basedon majorization. However, it is yet not known how the majorization criterion enters into the aboveimplication chain. Since it is closely related to conditional entropies we will discuss it in moredetail in the following section.

4300 J. Math. Phys., Vol. 43, No. 9, September 2002 K. G. H. Vollbrecht and M. M. Wolf

III. CONDITIONAL ENTROPIES

The idea to use entropic inequalities as separability, resp., entanglement, criteria for mixedstates goes back to the mid-1990s when Cerf and Adami16 and the Horodecki family17 recognizedthat certain conditional Renyi entropies are non-negative for separable states, and it was recentlyresurrected by several groups18–23 in the form of conditional Tsallis entropies.

The quantum Renyi entropy depending on the entropic parameter aPR is given by

Sa~r !5log tr~ra!

12a, ~5!

where S0 ,S1 ,S` reduce to the logarithm of the rank, the von Neumann entropy and the negativelogarithm of the operator norm, respectively. For the case of separable states it was shown in Refs.15–17 that the conditional entropy24

Sa~BuA;r !ªSa~r !2Sa~rA! ~6!

is non-negative for a50, ` and aP@1,2# .In Refs. 18 and 20 essentially the same criterion was expressed in terms of the Tsallis entropy

Ta~r !512tr~ra!

a21, ~7!

which is non-negative, concave ~convex! for a.0 (a,0) and becomes the von Neumann en-tropy in the limit a→1. The conditional Tsallis entropy defined in Ref. 18 reads

Ta~BuA;r !ªtr~rA

a!2tr~ra!

~a21 !tr~rAa!

. ~8!

Concerning positivity, however, the two conditional entropies are equivalent, i.e.,

Ta~BuA;r !>0⇔Sa~BuA;r !>0, ~9!

which is in turn equivalent to tr(rAa)>tr(ra) for a.1, tr(rA

a)<tr(ra) for 0<a,1, and thepositivity of the conditional von Neumann entropy for a51.

Obviously, for pure states the conditional entropies are negative if and only if the state isentangled.

A. Monotonicity counterexample

It was conjectured in Ref. 20 that Ta(BuA;r) is monotonically decreasing in a, such that itwould be sufficient to calculate T`(BuA;r) in order to decide positivity. However, monotonicitydoes not hold in general and can most easily be ruled out by low rank examples like

r512

~ uF1&^F1u1u01&^01u!, uF1&51

&~ u00&1u11&),

for which the reduced state has eigenvalues 14, 3

4 and therefore T05T`50ÞT2515. We note that

similar counterexamples can be found for the monotonicity of the conditional Renyi entropy aswell. Fortunately, however, monotonicity is not necessary for proving the positivity of the condi-tional Tsallis/Renyi entropies for separable states for other values than a50, `, aP@1,2# .25

B. Majorization and convex functions

Majorization turned out to be a powerful tool in the discussion of quantum state transforma-tions by means of LOCC operations ~cf. Ref. 26!, and it was recently proven to yield the strongest

4301J. Math. Phys., Vol. 43, No. 9, September 2002 Conditional entropies and entanglement criteria

separability criterion, which is based on the spectra of a state and one of its reductions. It wasproven in Ref. 8 that any separable state r acting on C

d^C

d is majorized by its reduced state rA :

rAsr , i.e., ;k<d:(i51

k

l i(A)>(

i51

k

l i , ~10!

where $l i% and $l i(A)% are the decreasingly ordered eigenvalues of r, resp., rA .

It is a well known result in the theory of majorization that xsy iff tr( f (x))>tr( f (y)) for allconvex functions f :R→R.27 Since f (x)5xa is convex for a>1 and concave on R

1 for 0<a<1 and the von Neumann entropy is concave ~needed for a51!, this immediately implies ~cf.Ref. 28! the following.

Theorem 1: Let r be a bipartite quantum state, which is majorized by its reduction rAsr .Then for every a>0 the conditional Tsallis/Renyi entropies of r are non-negative, i.e.,

Sa~BuA;r !>0 and Ta~BuA;r !>0. ~11!

The result of Nielsen and Kempe implies that this holds in particular for any separable state.It is yet not known how the majorization criterion ~10! is related to other separability criteria

like PPT, undistillability and the reduction criterion. However, we will show in the next subsectionhow the positivity of conditional entropies is related to these properties.

C. Conditional entropies and the reduction criterion

Positivity of the conditional entropies for a50 reduces to the rank criterion in the implicationchain ~4!. The following theorem will show, however, that all the other properties stated in ~4! inturn imply positivity of the conditional entropies for every value of the entropic parameter a.

Theorem 2: Let r be a bipartite quantum state satisfying the reduction criterion rA ^ 1>r .Then for every a>0 the conditional Tsallis/Renyi entropies are non-negative:

Sa~BuA;r !>0 and Ta~BuA;r !>0. ~12!

We note that Theorem 2 implies, in particular, that states with negative conditional entropiesare distillable.

Proof: We will divide the proof into three steps depending on the value of the entropicparameter.

For a.1 the proof is essentially based on the Golden–Thompson inequality ~cf. Ref. 29!stating that

tr~eAeB!>tr~eA1B! ~13!

for Hermitian matrices A ,B . Utilizing the definition of the reduced state, i.e.,

;P>0:tr~r~P ^ 1!![tr~rAP !, ~14!

this leads to

tr~rAa!5tr@r~rA

a21^ 1!#5tr@exp~ ln r !exp~~a21 !ln~rA ^ 1!!#

>tr@exp~ ln~r !1~a21 !ln~rA ^ 1!!# . ~15!

At this point we need two monotonicity properties in order to exploit the validity of the reductioncriterion. First of all we use the fact that the logarithm is operator monotone,30 i.e.,

A>B⇒ln A>ln B . ~16!

Thus, for a.1 the reduction criterion rA ^ 1>r implies


ln~r !1~a21 !ln~rA ^ 1!>ln~r !1~a21 !ln~r !5a ln~r !. ~17!

In the second step we utilize the fact that the exponential function is monotone under the trace.This can be seen by noting that for any A Hermitian, P>0 and B5(A1eP) with e>0:

]

]etr~eB!5tr~eBP !>0. ~18!

Hence tr(eB)>tr(eA) is implied by B>A . Combining ~15! and ~17!, this leads to

tr~rAa!>tr@exp~a ln r !#5tr~ra!. ~19!

For 0<a,1 the reduction criterion can immediately be applied since f (A)5Ar is an operatordecreasing function for 21<r<0, A>0 ~cf. Ref. 31! and thus

tr~rAa!5tr@r~rA

a21^ 1!#<tr~ra!. ~20!

For the case a51 we have to look at the conditional von Neumann entropy S1(r)2S1(rA), for which positivity is directly implied by the reduction criterion and the operatormonotonicity of the logarithm:

S1~rA!52trrA log rA ~21!

52trr log rA ^ 1 ~22!

<2trr log r ~23!

5S1~r !, ~24!

which completes the proof. h

D. Negative entropic parameters

So far we have only discussed conditional entropies for non-negative values of the entropicparameter a. For these cases we know that they can become negative for entangled states, thesimplest examples being pure entangled states. However, for a,0 ~and states of full rank! thesign of the conditional entropy contains no information

Theorem 3: Let r be a bipartite quantum state of full rank. Then for every a,0 the condi-tional Tsallis/Renyi entropies are non-negative:

;a,0:Sa~BuA;r !>0 and Ta~BuA;r !>0. ~25!

Proof: Let $ua&% be an eigenbasis of rA . Then

tr~rAa!5(

aâurAua&a ~26!

5(a

F(i

â ^ iurua ^ i&Ga

~27!

<(a ,i

â ^ iurua ^ i&a

<tr~ra!, ~28!


where Eqs. ~27! and ~28! use that (( ib i)a<( ib i

a holds for b i>0, a<0, and the last inequality isimplied by the convexity of negative powers on R

1.

IV. ISOSPECTRAL STATES

The fact that positivity of conditional entropies is implied by the reduction criterion ~Theorem2! shows already that such an entropic criterion cannot be sufficient for separability. In fact, it wasshown in Ref. 8 that no spectral property is capable of distinguishing any entangled state fromseparable ones.

We will in this section follow the idea of Ref. 8 and construct particular examples of states,such that their entanglement cannot be detected by any spectral criterion, since there exist sepa-rable states having the same spectrum and the same reductions.

Werner states9 have always played an important and paradigmatic role in quantum informa-tion theory. Their characteristic property is that they commute with all unitaries of the form U^ U and they can be expressed as

r~p !5~12p !P1

r1

1pP2

r2

, 0<p<1, ~29!

where P1 (P2) is the projector onto the symmetric ~antisymmetric! subspace of Cd^C

d and r6

5tr@P6#5 (d26d)/2 are the respective dimensions. Werner showed that these states are en-

tangled iff p.12 independent of the dimension d . The following shows, however, that none of

these entangled states for odd dimension d can be detected by any separability criterion, which isbased on the spectrum of the state and its reductions.

Theorem 4: Any entangled state in Cd

^Cd with maximal chaotic reductions and eigenvalues

having multiplicities, which are multiples of d , has a separable isospectral counterpart, which islocally undistinguishable as it has the same reductions.

Proof: Let us consider a special basis of maximally entangled states in Cd

^Cd:32

uC jk&51

Ad(n51

d

expS 2pid

jn D un ,n % k&, ~30!

where j ,k51,.. . ,d and % means addition modulo d . Any equal weight combination of all states ofthe form ~30!, which belong to the same value of k , is then a projector onto a separable state since

Pk5(j51

d

uC jk&^C jku

51d (

j ,n ,m51

d

expF2pid

j~n2m !G un ,n % k&^m ,m % ku

5 (n51

d

un&^nu ^ un % k&^n % ku ~31!

is an equal weight combination of product states. Here we have used that(1/d) ( j51

d exp@(2pi/d) j(n2m)#5dn,m . Moreover, the reductions of the respective states Pk /d aremaximally chaotic, i.e., rA51/d , just as the reductions of any maximally entangled state.

If we now have a state with multiplicities being multiples of d , we can replace the projectorsonto its eigenspaces with sufficiently many projectors of the form Pk . The resulting state will thenbe again a convex combination of product states, i.e., separable, having the same spectrum andmaximal chaotic reductions. h


For the case of Werner states we note that the unitary invariance of the state r(p) in Eq. ~29!implies that its reductions are rA51/d . Moreover, r(p) has two eigenvalues (12p)/r1 and p/r2

with multiplicities r1 , r2 which are indeed multiples of d in odd dimensions.Following Proposition 2 we can now construct a state

r8~p !5~12p !

r1

(k51

r1 /d

Pk1p

r2

(l51

r2 /d

P l1r1

/d , ~32!

which has then both, the same spectrum and the same reductions as r(p). However, as a convexcombination of separable states, it is itself separable for any 0<p<1.

V. CONCLUSION

We discussed conditional Renyi and Tsallis entropies and the relation between their positivityand other separability properties. We showed in particular that states having a negative conditionalentropy are distillable since they violate the reduction criterion.

Conditional entropies are a special instance of criteria using just the spectra of a state and itsreductions. Concerning the detection of entanglement, it was shown in Ref. 8 that majorization isthe strongest spectral criterion, which uses the spectra of a state and just one of its reductions. Itsrelation to other separability criteria is yet not known. The present result and numerical evidencemay indicate that majorization is also implied by the reduction criterion. However, the proofpresented in Sec. III C does not work for arbitrary convex functions and, in fact, majorization isnot implied by the conditional entropy criteria.

Concerning separability, the most efficient criterion is still the PPT criterion, which is also aspectral criterion, however, for the partially transposed state. One interesting question in thiscontext would therefore be: how can other ~easy calculable! invariants provide information aboutthe separability of a state, which is not yet encoded in the smallest eigenvalue of its partialtranspose?

ACKNOWLEDGMENTS

The authors would like to thank R. F. Werner for useful discussions and M. A. Nielsen forbringing the relation to majorization to their attention. Funding by the European Union projectEQUIP ~Contract No. IST-1999-11053! and financial support from the DFG ~Bonn! are gratefullyacknowledged.

1 A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 ~1935!.2 E. Schrodinger, Naturwissenschaften 23, 807 ~1935!; 23, 823 ~1935!; 23, 844 ~1935!.3 J. S. Bell, Physics ~Long Island City, N.Y.! 1, 195 ~1964!.4 M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 ~1998!.5 G. Vidal and J. I. Cirac, Phys. Rev. Lett. 86, 5803 ~2001!.6 J. Oppenheim, M. Horodecki, P. Horodecki, and R. Horodecki, quant-ph/02020044.7 M. J. Donald, M. Horodecki, and O. Rudolph, quant-ph/0105017.8 M. A. Nielsen and J. Kempe, Phys. Rev. Lett. 86, 5184 ~2001!.9 R. F. Werner, Phys. Rev. A 40, 4277 ~1989!.

10 A. Peres, Quantum Theory: Concepts and Methods ~Kluwer, Dordrecht, 1995!.11 A. Peres, Phys. Rev. Lett. 77, 1413 ~1996!.12 M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1 ~1996!.13 M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 ~1999!.14 N. J. Cerf, C. Adami, and R. M. Gingrich, Phys. Rev. A 60, 898 ~1999!.15 P. Horodecki, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, quant-ph/9910122; B. M. Terhal, quant-ph/0101032.16 N. Cerf and C. Adami, Phys. Rev. Lett. 79, 5194 ~1997!.17 R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 210, 377 ~1996!; R. Horodecki and M. Horodecki, Phys.

Rev. A 54, 1838 ~1996!.18 S. Abe and A. K. Rajagopal, Physica A 289, 157 ~2001!.19 S. Abe and A. K. Rajagopal, Phys. Rev. A 60, 3461 ~1999!.20 C. Tsallis, S. Lloyd, and M. Baranger, Phys. Rev. A 63, 042104 ~2001!.21 A. Vidiella-Barranco, Phys. Lett. A 260, 335 ~1999!.22 A. K. Rajagopal and R. W. Rendell, quant-ph/0106050.


23 F. C. Alcaraz and C. Tsallis, quant-ph/0110067.24 Note that throughout this article we omit equations corresponding to the reduction with respect to the second subsystem

rB5trA(r). However, all the equations hold in a symmetric form for both reductions and the presented ones should beread as representatives for either of the two.

25 Abe and Rajagopal already tried to prove this proposition for separable states in Ref. 18. However, they failed since theywrongly supposed that all the states in the decomposition in Eq. ~1! mutually commute.

26 M. A. Nielsen and G. Vidal, Quant. Inf. Comp. 1~1!, 76 ~2001!.27 R. Bhatia, Matrix Analysis, Springer Graduate Texts in Mathematics ~Springer, New York, 1991!.28 A. Wehrl, Rev. Mod. Phys. 50, 221 ~1978!.29 M. Ohya and D. Petz, Quantum Entropy and Its Use ~Springer, New York, 1993!.30 K. Lowner, Math. Z. 38, 177 ~1934!.31 A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications ~Academic, New York, 1979!.32 R. F. Werner, J. Phys. A 34, 7081 ~2001!.


Chapter 5

Bell inequalities

The theory of entanglement was born in the year 1935 when Einstein together withPodolsky and Rosen recognized that quantum theory allows very particular corre-lations — entanglement — between two distant parts of a system [EPR35]. Theyargued that if such correlations allow the prediction of the result of a measurementon one part of a system by looking at the very distant part, then, in a completeand local theory, the predicted quantity has to have a definite value even beforethe measurement. In their example, however, this value could not be obtained fromquantum mechanics. The presence of entanglement led them in this way to theconclusion that quantum theory is an apparently incomplete theory.

For the following three decades the debate about the “EPR dilemma” was philo-sophical in nature and for many physicists it was nothing more than that. Thissituation, however, changed dramatically in the year 1964, when John Bell showedthat the matter could be decided by an experiment [Bel64]. He derived correlationinequalities, which can be violated in quantum mechanics, but have to be satisfiedwithin every model that is local and complete in a particular sense — models, whichwe nowadays call “local hidden variable models”.

Experiments showing the first reliable violations of “Bell’s inequalities” andthereby confirming quantum mechanics and demonstrating the presence of entan-glement in nature were then made from the early eighties on [AGR81]. Remarkably,the demonstrated violation of Bell’s inequalities does not only rule out a single the-ory, but the very way scientific theories had been formulated for centuries.

We will begin the discussion of Bell inequalities studying the general assump-tions required for their derivation. In the sections 5.4, 5.5 and 5.6 we will then focuson correlation experiments, where each of n parties has two two-valued observablesto be measured. We will in particular investigate the relation between the viola-tion of Bell inequalities, non-positivity of the partial transpose and distillability ofentangled multipartite quantum systems.

5.1 Hidden variables and joint distributions

A main part of the work in the derivation of “Bell’s Theorem” is conceptual ratherthan mathematical: if we want to draw far-reaching conclusions ruling out wholeclasses of theories, or ways of formulating natural laws, we have to analyze theorieson a very general and abstract level in order to even state the essential assumptions.Naturally, there are many ways to say what the really essential assumptions are,depending on philosophical taste and scientific background.

However, in all derivation of Bell inequalities two types of elements can beidentified:

59

Hidden variables and joint distributions

locality classicalityno-signalling hidden variables

non-contextuality classical logicjoint distributions

“realism”

Since Bell’s inequalities are found to be violated in nature [AGR81], one of these twoassumptions needs to be dropped. Quantum mechanics (in statistical interpreta-tion) drops the possibility of a hidden variable description, whereas hidden variabletheories drop locality in order to retain a description by classical parameters. Ineither case, however, fundamental features of the pre-quantum way of describingthe world are lost. In the following we will show how these two assumptions –locality and the existence of hidden variables – lead to Bell’s inequalities and theirgeneralizations.

To clarify matters we restrict the general discussion to the case of correlationmeasurements involving only two observers – Albert and Boris. Once the basicconcepts are clear, the generalization to multipartite systems (Albert, Boris, . . . ,Nathan) is straight forward.

Basic notation: Assume that Albert and Boris have some measurement devicesA1, . . . , AmA

and B1, . . . , BmBrespectively. The numbers of possible outcomes

of these observables may be different (but countable for simplicity) and we willrefer to them by v(Ai) and v(Bj). Assume further that Albert and Boris get oneparticle each from a common source and choose one of their measurement apparatusto perform a measurement. The basic objects of the theory are then the jointprobability distributions obtained by repeating this correlation experiment. We willdenote a typical measured probability by PAi,Bj (a, b), where a and b are possiblemeasurement outcomes of the devices Ai and Bj respectively. The collection of allthese probabilities are the basic raw data, we might call the“correlation table”.

Of course, these data have to satisfy some constraints which follow alreadyfrom the probability interpretation: PAi,Bj (a, b) ≥ 0, and all the probabilities in aparticular setup (Ai, Bj) have to add up to 1:

∑

a,b

PAi,Bj (a, b) = 1 . (5.1)

An interesting role is played by the marginals∑

b

PAi,Bj (a, b) . (5.2)

These are the probabilities measured by Albert in a given setup (Ai, Bj). Forgeneral correlation tables such marginals might depend on the whole setup and, inparticular, on the device Bj chosen by Boris. For example, the device Bj mightbe a transmitter with a particular input fed into it, and Ai might be a receiver.Then this dependence on Bj would be precisely what is required for Albert to‘get the message’. Note, however, that this usually requires some signal-carryingphysical system to go from Boris to Albert, contrary to the basic description of thecorrelation setup (“particles are coming from a common source”). What we expectin a general correlation experiment (without communication between the parties)is the following “no-signalling condition”:

∑

b

PAi,Bj (a, b) ≡ PAi(a) is independent of Bj (5.3)

and similarly for the other site and observables.

60


Non-local hidden variable models: “Hidden variables” have a bad name inthe physics community. Yet the question whether we can understand the observedquantum randomness as arising from our ignorance of some underlying classicalvariables (this is what the term means) is a fundamental question, which must beaddressed seriously if we want to understand quantum mechanics at all.

It is a widespread belief that constructing non-local hidden variable theoriesis a non-trivial task. However, this is not at all the case. The simplest classicalstructure from which all measured results can be obtained was already mentioned:it is the collection of correlation data itself. In this “theory”, which is remarkableonly for having explanatory power exactly equal to zero, the hidden variables arethe data to be measured, and they are hidden in the same way that the future is.The hidden variable thus contains a complete description of the experimental setup,i.e., of the devices (Ai, Bj) chosen by the two parties1. This feature marks so-called“contextual” or, more simply, “non-local” hidden variable theories.

Contextuality is, of course, not always as blatant, especially in hidden vari-able theories focusing on dynamical laws (such as the Bohm/Nelson theory[Boh52a,Boh52b, Nel67] and its generalizations[Wer86]), where the “setups” are not appar-ent, but enter via a description of the measuring devices inside the theory. By farthe most wide-spread hidden variable theory is the “individual state” interpreta-tion of quantum mechanics, according to which some wave function is somehowattached to each individual system, and constitutes a “catalog of all expectations”to be measured on the system. Technically, this is indeed nothing but the descrip-tion of a hidden variable theory, although such statements can also be found by theCopenhagen Masters, who are not usually associated with hidden variable views.

Local hidden variable models: To formalize the idea of a local hidden variabletheory, let us explicitly introduce a hidden variable λ which takes values in a spaceΛ. We assume that the systems sent to Albert and Boris are described by λ inall details necessary to compute their response to any measurement, or at least todetermine the probabilities of all such responses. Thus for any measuring device Ai

of Albert, and any possible outcome a of this device we get a response probabilityfunction λ 7→ χAi(a, λ). The source of the correlation experiment is characterizedby the probabilities with which the different λ occur, i.e., by a probability measureM on Λ. With these data we can thus compute all the correlations

PAi,Bj (a, b) =

∫M(dλ) χAi(a, λ)χBj (b, λ) . (5.4)

We will say that a correlation table “allows a local classical model”, if it can berepresented in this form. The locality assumption in Eq.(5.4) is expressed in the factthat the response functions factorize, such that χAi(a, λ) does not depend on Bj andχBj (b, λ) is independent of Ai. The probabilities obtained in this way satisfy thusthe non-signalling condition in Eq.(5.3). In fact, the locality of the hidden variablemodel is precisely the condition that the no-signalling property persists even whenwe know the value of λ, or the source has been upgraded to produce only one λ.Conversely, given a local hidden variable model, the no-signalling condition for theexperimental correlation data is merely “locality property on average”.

An important motivation for the search for hidden variables was to restore thedeterminism of classical physics or, using Einstein’s famous metaphor, to allowGod to quit gambling. We can easily formulate determinism as a requirement fora local classical model: the knowledge of λ should not only allow us to predictthe probabilities of outcomes, but the outcomes themselves with certainty. Thus

1We could also say that there is a separate probability space to be chosen for each experimentalsetup.

61


we call a local hidden variable model “deterministic” (as opposed to “stochastic”),if the response functions take on only the values 0 and 1. It turns out, however,that this seemingly much stronger constraint on the model does not lead to sharperconditions on the correlation data (cf.[Fin82a]):

Proposition 14 For every non-deterministic local hidden variable model, there ex-ists a deterministic one, which produces the same correlation table.

Proof: Let(Λ, M , χAi, χBj

)be the probability space, measure and response

functions for the stochastic model. The idea is to incorporate the randomness inthe measuring devices into the hidden variable. To this end we replace the hiddenvariable λ ∈ Λ by λ = (λ, ξ), where ξ = (ξA1

, . . . , ξAmA , ξB1, . . . , ξBmB ) is a set

of uniformly distributed random variables on [0, 1], which are independent of each

other and of λ. In addition, we define M(dλ) = dξ M(dλ) and

χAi(a, λ) ≡

1 ξAi ≤ χAi(a, λ)0 otherwise,

(5.5)

and similarly for all the other response functions. This is then clearly a deterministicmodel and since

∫dξAi χAi(a, λ) = χAi(a, λ) we have

∫M(dλ) χAi(a, λ)χBj (b, λ) =

∫M(dλ) χAi(a, λ)χBj (b, λ) . (5.6)

Joint distributions: An important point in the above list of “classicality”assumptions is the existence of joint probability distributions. Given a set of proba-bility distributions PX1

, . . . , PXk, there always exists joint distribution that returns

them as marginals, namely the product distribution∏ki=1 PXi

. However, if wefix in addition the joint distributions PXi,Xj

for a certain subset of pairs (i, j) in anon-trivial way, a joint distribution with these marginals in general no longer exists.The constraints imposed by the existence of a joint distribution will precisely be theBell inequalities. Before we come to their derivation, we will, however, have a closerlook at the relation between local hidden variable models and joint distributions(cf.[Fin82a, Fin82b]):

Proposition 15 There exists a local hidden variable model iff there exists a jointprobability distribution for all the considered observables, which is compatible withthe given correlation table.

Proof: Let A = (A1, . . . , AmA), a = (a1, . . . , amA

) be vectors of observables andtheir respective outcomes, and similarly for B, b. If PA,B is the joint distributionfor all pair distributions PAi,Bj , then

PAi,Bj (α, β) =∑

a,b

PA,B(a, b) δα,ai δβ,bj (5.7)

is an admissible (deterministic) local hidden variable model.Conversely, if

(Λ,M , χAi, χBj

)are probability space, measure and response

functions for a local hidden variable model, then

PA,B(a, b) ≡∫M(dλ)

mA∏

i=1

χAi(ai, λ)

mB∏

j=1

χBj (bj , λ) (5.8)

is a joint distribution, which returns all pair distributions PAi,Bj .

The fact that there exists a joint distribution for all mA +mB observables canbe expressed by looking only at the respective 1 +mB marginals:

62

Bell inequalities and Bell polytopes

Proposition 16 [Fin82b] Consider a set of fixed pair distributions PAi,Bj withi = 1, . . . ,mA and j = 1, . . . ,mB. There exists a joint probability distribution forall mA+mB observables which is compatible with the given joints, iff there are jointdistributions PAi,B1,... ,BmB

for all i, which have in turn equal marginals PB1,... ,BmBand are compatible with all pair distributions.

Proof: We use the notation of the proof of Prop.15. Obviously, if there is jointdistribution PA,B for all mA +mB observables, the statement holds. Conversely,we can define a joint distribution

PA,B(a, b) ≡[PB1,... ,BmB

(b)]1−mA

mA∏

i=1

PAi,B1,... ,BmB(ai, b) (5.9)

which indeed has the right marginals for all PAi,B1,... ,BmBand therefore is compat-

ible with all pair distributions as well.

Finally, we will utilize Prop.16 to formalize the intuition that commuting ob-servables correspond to a classical system (cf.[Fin82b]):

Proposition 17 Consider a set of pair distributions PAi,Bj obtained from lo-cal POVM measurements on a given bipartite density matrix. If the measurementdevices on one site, say Bj, correspond to commuting observables, then there ex-ists a joint distribution for all observables, which is compatible with the given pairdistributions. The considered correlation table thus admits a local hidden variabledescription.

Remark: Note that the existence of the local hidden variable model is independentof the type and number of observables Ai.Proof: Let Fi(ai), Gj(bj) be the effect operators corresponding to the mea-surement devices Ai and Bj with respective measurement outcomes ai and bj . Then

PAi,B1,... ,BmB(ai, b) ≡ tr

ρ(Fi(ai)⊗

mB∏

j=1

Gj(bj)) (5.10)

is a well-defined probability distribution, since∏j Gj(bj) is a psd operator due

to [Gj(bj), Gk(bk)] = 0. Moreover, Eq.(5.10) is compatible with the given pairdistributions and the marginal PB1,... ,BmB

is independent of Ai. Hence, we canapply Prop.16, which proves the statement.

5.2 Bell inequalities and Bell polytopes

We will now briefly discuss, how Bell inequalities arise from the assumption thatthe considered correlations admit a description within a local classical model. Tothis end we will at first discuss the role of classical configurations for local hiddenvariable models.

Classical configurations: For a fixed value of the hidden variable λ the re-sponse function χA(a, λ) in a deterministic model takes on the value one for oneoutcome a and vanishes for all the others. If we now consider an n-partite system,where each of the parties has the choice of m v-valued observables to be measured,any of the nm observables thus divides the hidden variable space Λ into v pieces(which do not necessarily have to be connected). In this way Λ is build up of vnm

regions Λc, such that every region is characterized by a single “classical configura-tion” c, i.e., an assignment of one of the v outcomes to each of the nm observables.

63

Bell inequalities and Bell polytopes

This enables us to rewrite Eq.(5.4) in form of a sum (and analogous for more thantwo sites)

PAi,Bj (a, b) =∑

c

∫

Λc

M(dλ) χAi(a, λ)χBj (b, λ)

=∑

c

pc χAi(a, c)χBj (b, c), (5.11)

where pc =∫ΛcM(dλ) is the probability corresponding to the classical configuration

c. Locality is here expressed in the fact that the assignment of a value to anobservable at one site does not depend on the observables chosen at other sites.

All the Bell inequalities There is not only one Bell inequality but an infinitehierarchy of such inequalities, which can basically be classified by specifying the typeof correlation experiments they deal with. As mentioned previously, the essentialassumption leading to any Bell inequality is the existence of a local hidden variabletheory, which describes the outcomes of a certain class of correlation measurements.The modus operandi for the derivation of a class of Bell inequalities would thereforebe the following: We first fix the type of correlation measurements we want to dealwith – say we consider n-partite systems, where each of the parties has the choiceof m v-valued observables to be measured.2 Then we consider the space spanned bythe entire set of the raw experimental data, i.e., the (mv)n probabilities, and ask forthe inequalities, which bound the region that is accessible within the framework ofa local realistic model. Whatever this underlying model looks like, if only it is localand classical, the accessible region will be contained in a convex polytope, whoseextreme points are by Eq.(5.11) the classical configurations. The classical regionis thus bounded by a finite albeit huge number of linear inequalities. These arethe natural generalizations of the original inequality John Bell published in 1964[Bel64]. We may therefore call the polytope containing the classical region the “Bellpolytope”.3

Hence we are faced with a whole hierarchy of inequalities. The task of findinga minimal set of these inequalities, which is complete in the sense that they aresatisfied if and only if the correlations considered allow a local hidden variablemodel, is however closely related to some known hard problems in computationalcomplexity[Pit89, Pit91]. So it is not surprising that complete solutions only existeither in case where additional symmetries can be exploited[WW01, ZB02], or forsmall values of (n,m, v), where we can utilize today’s computing power for a bruteforce approach.

The CHSH inequality The Clauser-Horne-Shimony-Holt (CHSH) inequality isby far the best studied case of Bell inequalities [CHSH69]. It was proven by Finethat every non-trivial face of the Bell polytope for (n,m, v) = (2, 2, 2) correspondsto a CHSH inequality [Fin82a]. In the following lines we will prove the inequality,however, without showing Fine’s completeness result.

The CHSH inequality refers to correlation experiments with two ±1 valued ob-servables on two sites. From the response probabilities χA(a, λ), a = ±1 we formthe mean value of the random variable a (given the hidden variable λ):

a(λ) = χA(+1, λ)− χA(−1, λ) , (5.12)

2Of course we are free to require further restrictions, e.g. we might just want to look at a subsetof all possible correlations or restrict to a special class of observables or systems.

3The first to consider the construction of a complete set of Bell inequalities as a problem inconvex geometry apparently was M. Froissart [Fro84].

64

Quantum states admitting a local classical description

and from these the correlation function

B(λ) = 1

2

[a1(λ)

(b1(λ) + b2(λ)

)+ a2(λ)

(b1(λ)− b2(λ)

)]. (5.13)

We claim that B satisfies the pointwise inequality |B(λ)| ≤ 1: indeed, the extreme

values are attained, when each ai(λ), bi(λ) is extremal, i.e., ±1. But then 2B(λ) is aneven integer, and since 2B(λ) = 4 requires a1(λ) = b1(λ) = b2(λ) = a2(λ) = −b2(λ),a contradiction, we must have 2B(λ) ≤ 2.

Since B is pointwise bounded, its expectation, the so-called Bell correlation,

β =

∫M(dλ)B(λ) (5.14)

is also bounded by unity. But β can be expressed directly in terms of the mea-sured correlation table. To this end we introduce the expectation values E(A,B) =∑

a,b=±1 a b PA,B(a, b). The inequality |β| ≤ 1 then becomes the CHSH inequality:

1

2

∣∣∣E(A1, B1) + E(A1, B2) + E(A2, B1)− E(A2, B2)∣∣∣ ≤ 1. (5.15)

In the case where the expectation values come from quantum mechanics, we maydefine a “Bell operator” B from Eq.(5.13) such that Eq.(5.15) reads |tr [ρB] | ≤ 1.

A more detailed discussion of the CHSH inequality, together with its general-izations to more than two parties, will be given in Sec.5.4 and Sec.5.5.

5.3 Quantum states admitting a local classical de-

scription

The violation of Bell’s inequality was the first mathematically sharp criterion forentanglement. For quite a long time (’64-’89) the “violation of Bell inequalities”had become synonymous with “non-classical correlations”, i.e., entanglement.

One of the first papers in which finer distinctions were made was consideredwith the construction of states (Werner states [Wer89a]) with the property thatthey satisfy all the usual assumptions leading to the Bell inequalities, but can stillnot be generated by a purely classical mechanism (are not “separable” in modernterminology). This example pointed out a gap between the obviously entangledstates (violating a Bell inequality) and the obviously non-entangled ones, which aremerely classical correlated (separable).

In this section we discuss some first examples of bipartite quantum states forwhich a description within a local hidden variable model is possible. The case ofgeneral PPT states will then be investigated in detail in the subsequent sections.

1. Separable states: Separable states admit a local hidden variable descrip-

tion by definition. A separable density matrix is of the form ρ =∑

k pk ρ(1)k ⊗

ρ(2)k . If Fi(ai) and Gj(bj) are the effect operators corresponding to the mea-

surement devices Ai and Bj with respective outcomes ai and bj then

PAi,Bj (ai, bj) = tr[ρFi(ai)⊗Gj(bj)

]=∑

k

pktr[ρ(1)k Fi(ai)]tr[ρ

(2)k Gj(bj)],

which is evidently a local hidden variable description of the form in Eq.(5.4).

2. Werner states: For the case of von Neumann measurements it was proven in[Wer89a], that the states we now call “Werner states” (see Sec.2.7.1, Eq.(2.62))

65

Quantum states admitting a local classical description

admit a local classical description for

p = 1− d+ 1

2d2, (5.16)

which is in any nontrivial case larger than one half and therefore correspondsto an entangled state. For increasing dimension Eq.(5.16) even approaches 1,which is (within the family of U⊗U invariant states) as far from the classicallycorrelated state (p = 1

2 ) as possible. Recently, the local hidden variable modelfor entangled Werner states was, however for a smaller interval, generalizedto POVM measurements.

3. States with quasi-extensions: Consider a bipartite density matrix ρ ∈B(H1 ⊗ H2), and assume there is a “quasi-extension” ρ′ ∈ B(H1 ⊗ H⊗mB

2 ),which is such that tracing out any mB − 1 subsystems of H⊗mB

2 always gives

back ρ, and tr[ρ′P0 ⊗

⊗mB

j=1 Pj

]≥ 0 for all psd operators Pj . Then

PAi,B1,... ,BmB(ai, b1, . . . , bmB

) = tr

ρ′Fi(ai)⊗

mB⊗

j=1

Gj(bj)

(5.17)

is an admissible joint probability distribution compatible with all pair distri-butions PAi,Bj (ai, bj) = tr [ρFi(ai)⊗Gj(bj)]. By applying Prop.16 we havethen that there exists a local hidden variable model for every correlation ta-ble with an arbitrary number of observables acting on H1 and mB arbitrarymeasurement devices on H2. This result was obtained in [TDS02], and itwas shown that PPT entangled states constructed from unextendable prod-uct bases, like in Eq.(2.35) always have such an extension for mB = 2.

66

Bell inequalities for states with positive partial transpose

5.4 Bell inequalities for states with positive partial

transpose

States having a positive partial transpose are either separable or contain some boundentanglement, in the sense that no entanglement can be extracted from them viaentanglement distillation. This led Peres to the conjecture that every PPT stateadmits a description within a local hidden variable model and therefore satisfies allBell inequalities [Per99]. In its general form this conjecture is neither proven, nordoes there exist a counterexample yet. The following paper is, however, supportingevidence for Peres’ conjecture, since it answers the question for the largest classof Bell inequalities, that was explicitly known at that time.4 These inequalities,known as “Mermin-Klyshko” inequalities, deal with experimental setups where nparties have two two-valued observables each. Hence, they are generalizations ofthe CHSH inequality and include this for the case n = 2.

For such multipartite settings we have to specify what PPT means, since thepartial transposition is a bipartite concept. One may for instance look at the partialtransposes with respect to all partitions of the whole set into two groups and thencount non-positive and positive cases. This would lead to a finer distinction thanjust deciding PPT (w.r.t. all partitions) versus non-PPT. In fact, we will doesomething similar and first group the set of n parties into p ≤ n nonempty anddisjoint sets, which we consider thenceforward as individual systems. Then we askwhether or not this p−partite state has positive partial transposes with respect toall bipartite partitions. By varying p we obtain again the possibility of making afiner distinction.

Apart from giving a new simplified geometric proof for all Mermin-Klyshkoinequalities, the paper provides the following results:

• Multipartite states having a positive partial transpose w.r.t. every bipartitepartition do not violate any Bell inequality of the Mermin-Klyshko type.

• Assume that the p−partite system has positive partial transpose w.r.t. everybipartite partition. Then the largest possible violation decreases with growingp as 2(n−p)/2.

4The result will be extended in the following section.

67

Bell’s inequalities for states with positive partial transpose

R. F. Werner* and M. M. WolfInstitut fur Mathematische Physik, TU Braunschweig, Mendelssohnstraße 3, 38106 Braunschweig, Germany

~Received 15 October 1999; published 15 May 2000!

We study violations of n-particle Bell inequalities ~as developed by Mermin and Klyshko! under the as-sumption that suitable partial transposes of the density operator are positive. If all transposes with respect to apartition of the system into p subsystems are positive, the best upper bound on the violation is 2 (n2p)/2. Inparticular, if the partial transposes with respect to all subsystems are positive, the inequalities are satisfied. Thisis supporting evidence for a recent conjecture by Peres that positivity of partial transposes could be equivalentto the existence of local classical models.

PACS number~s!: 03.65.Bz, 03.67.2a

I. INTRODUCTION

One of the basic questions asked early in the developmentof quantum information theory was about the nature of en-tanglement. Extreme cases were always clear enough: a twoqubit singlet state was the paradigm of the entangled state@1#, whereas product states and mixtures thereof were obvi-ously not, but merely ‘‘classically correlated’’ @2#. But in thewide range between it was hardly clear where a meaningfulboundary between the entangled and the nonentangled couldbe drawn. Still today, some boundaries are not completelyknown, although, of course, general structural knowledgeabout entanglement has increased dramatically in the lastfew years. The present paper is devoted to settling the rela-tionship between two entanglement properties discussed inthe literature.

To fix ideas we will start by recalling some properties onemight identify with ‘‘entanglement’’ and the known relationsbetween them. For simplicity in this introduction, we willchoose the setting of bipartite quantum systems, i.e., quan-tum systems whose Hilbert space is written as a tensor prod-uct H5H1 ^ H2. Moreover, we consider finite dimensionalspaces only, leaving the appropriate extensions to infinitedimensions to the reader. All properties listed refer to a den-sity matrix r on this space. It turns out to be simpler todefine the entanglement properties in terms of there nega-tions, i.e., the various degrees of ‘‘classicalness.’’

(S) A state is called separable or ‘‘classically corre-lated,’’ if it can be written as a convex combination of tensorproduct states. Otherwise, it is simply called ‘‘entangled.’’

(B) Before 1990 perhaps the only mathematically sharpcriteria for entanglement were the Bell inequalities in theirClauser-Horn-Shimony-Holt ~CHSH! form @3#. A state issaid to satisfy these Bell inequalities if, for any choice ofoperators A i ,A i8 on Hi (i51,2) with 21<A i ,A i8<1, wehave

tr r@A1 ^ ~A21A28!1A18^ ~A22A28!#<2. ~1!

It is easy to see that (S)⇒(B).

(M ) Bell’s inequalities are traditionally derived from anassumption about the existence of local hidden variables.The same assumptions lead to an infinite hierarchy of corre-lation inequalities @4#, and it seems natural to base a notionof entanglement not on an arbitrary choice of inequality~e.g., CHSH! from this hierarchy. So we say that r admits alocal classical model, if it satisfies all inequalities from thishierarchy. Then (S)⇒(M )⇒(B). It was shown in @2# that(M )⇒” (S), and this was perhaps the first indication that dif-ferent types of entanglement might have to be distinguished.

(U) A key step for the development of entanglementtheory was a paper by Popescu @5#, showing that by suitablelocal filtering operations applied to maybe several copies of agiven r , one could sometimes obtain a new state r8 violatinga Bell inequality, even though r admitted a local hiddenvariable model, and hence satisfied the full hierarchy of Bellinequalities. Let us call a state undistillable, if it is impos-sible to obtain from it a two-qubit state violating the CHSHinequality, by any process of local quantum operations ~i.e.,operations acting only on one subsystem!, perhaps allowingclassical communication and several copies of the state as aninput. What Popescu showed was that (M )⇒” (U).

(P) The idea of distillation was later taken to muchgreater sophistication @6#, and for a while the natural conjec-ture seemed to be not only that (S)⇒(U) ~which is trivial tosee!, but that these two should be equivalent. The counter-example was provided in @7#. These authors used a property(P), which had been proposed by Peres @8# as a necessarycondition for separability @i.e., (S)⇒(P)], which turned outalso to be sufficient in the qubit case @9#. This condition ~P!is that r has positive partial transpose, i.e., rT1 is a positivesemidefinite operator. Here the partial transpose AT1 of anoperator A on H5H1 ^ H2 is defined in terms of matrixelements with respect to some basis by

^kluAT1umn&5^mluAukn&. ~2!

Equivalently,

S (a

Aa ^ BaD T1

5(a

AaT

^ Ba , ~3!

where the superscript T stands for transposition in the givenbasis. It was shown that (P)⇒(U), and the counterexample*Electronic address: [email protected], [email protected]

PHYSICAL REVIEW A, VOLUME 61, 062102

1050-2947/2000/61~6!/062102~4!/$15.00 ©2000 The American Physical Society61 062102-1

in @7# worked by establishing ~U! and not-(S) in this ex-ample. States of this kind are now called bound entangled.

There are further interesting properties, like usefulness forteleportation @10#, but the above are sufficient for explainingthe problem addressed in this paper. To summarize, it isknown that (S)⇒(P)⇒(U) and (S)⇒(M )⇒(B). For purestates all conditions are equivalent, and for systems of twoqubits (U)⇒(S), but (M )⇒” (S).

For multipartite systems, i.e., systems with Hilbert spaceH1 ^ H2 ^ ••• ^ Hn , the properties (S),(M ),(U) immedi-ately make sense. For ~B! there may be several choices ofinequalities following from (M ). The inequalities we use inthis paper are discussed in detail in the next section. Partialtransposition ~P! is an intrinsically bipartite concept. Thestrongest version of ~P! in multipartite systems is the one weuse below: the positivity of partial transposes with respect toevery subsystem.

Then the implication chains (S)⇒(P)⇒(U) and(S)⇒(M )⇒(B) hold as in the bipartite case. However, nodirect relations are known so far between these chains, evenin the bipartite case. It seems likely that the violation of ~B!is a fairly strong property, perhaps implying distillability.This certainly seems to be the intuition of Peres in @11#, whoconjectures that

~M !⇔~P !. ~4!

We will refer to this statement as Peres’ conjecture. It shouldbe noted, however, that neither we nor Peres have given asharp mathematical formulation, particularly of the way themodel is required to cover not only one pair but also tensorproducts and distillation processes. Some such condition iscertainly needed ~and implicitly assumed by Peres!, becauseotherwise the implication (M )⇒(P) would already fail fortwo qubits @2#. It is not entirely clear from @11# how stronglyPeres is committed to Eq. ~4!. We are not completely con-vinced. However, we do follow Peres’ lead in seeing here aninteresting line of inquiry. Indeed, the present paper is de-voted to proving one special instance of the conjecture,namely, the implication (P)⇒(B), for general multipartitesystems, where ~P! is taken as the positivity of every partialtranspose, and ~B! is taken as the n particle generalization ofthe CHSH inequality proposed by Mermin @12#, and furtherdeveloped by Ardehali @13#, Belinskii and Klyshko @15#, andothers @14,16#.

II. MERMIN’S GENERALIZATIONOF THE CHSH INEQUALITIES

Like the CHSH inequalities, Mermin’s n-party generali-zation refers to correlation experiments, in which each of theparties is given one subsystem of a larger system and has thechoice of two 61-valued observables to be measured on it.The expectations of such an observable are given in quantummechanics by a Hermitian operator A with spectrum in@21,1# , and with a choice of Ak ,Ak8 at site k the raw experi-mental data are the 2n expectation values of the formtr(rA1 ^ A28^ •••An) with all possible choices Ak vs Ak8 atall the sites.

If we look only at a single site, the possible pairs of ex-pectation values ~with fixed A ,A8 but variable r) lie in asquare. It will be very useful for the construction of the in-equalities and the proof of our result to consider this squareas a set in the complex plane: after a suitable linear transfor-mation ~a p/4 rotation and a dilation! we can take it as thesquare S with the corners 61 and 6i . The pair of expecta-tion values of A and A8 is thus replaced by the single com-plex number tr(ra), where

a512 @~A1A8!1i~A82A !# ~5!

5e2ip/4~A1iA8!/A2. ~6!

The idea of this transformation is that the square S has aspecial property: products of complex numbers zkPS lieagain in S. This is evident for the corners ~they form agroup! and follows for the full square by convex combina-tion. Suppose now that r5 ^ k51

n rk is a product state. Thenthe operator b5 ^ k51

n ak has expectation tr(r b)5) k51

n tr(rkak)PS. Since the expectation is linear in r , thesame follows for any separable state, i.e., any convex com-bination of product states. The statement ‘‘tr(r b)PS’’ isessentially Mermin’s inequality, although not yet written asan inequality. Note that the argument given here implies alsothat this statement ~written out in correlation expressions in-volving Ak ,Ak8) holds in any local classical model, becausein a classical theory every pure state of a composite system isautomatically a product, and hence every state is separable.Thus Mermin’s inequality indeed belongs to the broad cat-egory of Bell’s inequalities.

To write ‘‘tr(r b)PS’’ as a bona fide set of inequalities,we just have to undo the transformation ~5!, i.e., we intro-duce operators B ,B8 such that Eq. ~5! is satisfied with(b ,B ,B8) substituted for (a ,A ,A8). The operators B ,B8 areusually called Bell operators, and Mermin’s inequality sim-ply becomes

utr~r B !u<1 or utr~r B8!u<1. ~7!

Writing out B and B8 explicitly in terms of tensor productsof Ak ,Ak8 gives the usual CHSH inequality ~1! for n52, andbecomes arbitrarily cumbersome for large n. It is also nothelpful for our purpose. The above derivation also gets rid ofthe case distinction ‘‘n odd/even,’’ which has troubled theearly derivations. In fact, Mermin @12# first missed a factorA2 for even n, which was later obtained by Ardehali @13#,who in turn missed the same factor for odd n. Inequalitiesequally sharp for even and odd n were established in @14#and @15#.

III. VIOLATIONS OF MERMIN’S INEQUALITYIN QUANTUM MECHANICS

The idea of combining A ,A8 in the non-Hermitian opera-tor a has a long tradition for the CHSH case @17#. Its poweris not only in organizing the inequalities ~only linear trans-formations among operators are needed for that purpose!, butin the possibility of bringing in the noncommutative alge-

R. F. WERNER AND M. M. WOLF PHYSICAL REVIEW A 61 062102

062102-2

braic structure of quantum mechanics to analyze the possi-bility of violations in the quantum case. In this section wediscuss these violations, at the same time building up themachinery needed in the proof of our result. We will needthe following expressions:

a*a512 ~A2

1A82!1

i2

@A ,A8# , ~8!

aa*512 ~A2

1A82!2

i2

@A ,A8# , ~9!

a22a*2

5i~A822A2!. ~10!

It is clear from the first line that although tr(r a) lies in S,and hence in the unit circle for all r , the operator normiai5ia*ai1/2 may be greater than 1. Therefore, the tensorproduct operator b may have a norm increasing exponen-tially with n. This is the key to the quantum violations ofMermin’s inequality.

The largest possible commutators, i.e., operators saturat-ing the obvious bound i@A ,A8#i<2iAiiA8i are just Paulimatrices. A good choice is Ak5(sx1sy)/A2 and Ak85(sx

2sy)/A2 for all k. Then ak5A2v , where v5( 10

00 ). It is

readily verified that v^ n acts in the two-dimensional space

spanned by e1^ n and e2

^ n exactly as v acts in the spacespanned by the two basis vectors e1 ,e2PC

2. With the sameidentification of two-dimensional subspaces b52n/2

v^ n acts

like 2 (n21)/2a , so the possible expectations tr(r b) with rsupported in this subspace span the exponentially enlargedsquare 2 (n21)/2

S.In order to show that 2 (n21)/2 is the maximal possible

violation ~in analogy with Cirel’son’s bound @18# for theCHSH inequality!, but also in preparation for the proof ofour main result, it is useful to consider the following generaltechnique for getting upper bounds on tr(r b). It has beenused in the CHSH case by Landau @19#, among others. Notefirst that tr(r B) and tr(r B8) are affine functionals of eachAk or Ak8 . Hence, if we maximize the expectations of Belloperators by varying some Ak or Ak8 , keeping r fixed, wemay as well take Ak extremal in the convex set of Hermitianoperators with 21<Ak<1. That is to say, we may assumeAk

25Ak8

251 for all k. Taking tensor products of Eq. ~8! and

expanding the product we find

b*b5 ^

k51

n S 11

i2

@Ak ,Ak8# D

5(b

^kPb

i2

@Ak ,Ak8# , ~11!

where the sum is over all subsets b,$1, . . . ,n%, and onlyfactors different from 1 are written in the tensor product. Inparticular, the term for b5B is 1. For bb* we get a similarsum with an additional factor (21) ubu, where ubu denotes thecardinality of the set b . From Eq. ~10! we find ak

25ak*

2 andb2

5b*2 by taking tensor products. Again by applying Eq.

~10!, to (b ,B ,B8) this time, we find that B25B82. In fact, by

adding Eqs. ~8! and ~9! and inserting Eq. ~11!, we get

B25B82

512 ~b*b1bb*!

5 (b even

^kPb

i2

@Ak ,Ak8# .

~12!

By the variance inequality utr(r B)u2<tr(r B2), the expecta-tion of the right hand side is an upper bound on the square oflargest violation of Mermin’s inequality. There are two im-mediate applications: since each term in the sum has norm atmost 1, the norm of the sum is bounded by the number ofterms, i.e., 2n21. This shows the analog of Cirel’son’s in-equality, i.e., that the violation discussed above is indeedmaximal. The second application is to the case that all com-mutators vanish. Then only the term for b5B survives, andthere is no violation of the inequality. Our result to be statedand proved in the next section is a refinement of this idea.

IV. POSITIVE PARTIAL TRANSPOSESAND MAIN RESULT

We now apply the technique of the previous section to thepartial transpose. More specifically, for any density operatorr and any subset a,$1, . . . ,n%, let rTa denote the partialtranspose of all sites belonging to a . Suppose now that rTa

is positive semidefinite and hence again a density matrix.Then we can apply the variance inequality to rTa and BTa,obtaining

~ tr rB !25~ tr rTaBTa!2<tr@rTa~BTa!2#

<tr$r@~BTa!2#Ta%. ~13!

We note that @AT,A8T#T52@A ,A8# and thus

@~BTa!2#Ta5 (b even

~21 ! uaùbu ^kPb

i2

@Ak ,Ak8# . ~14!

Note that it does not matter whether we transpose a or itscomplement.

Now consider a partition of $1, . . . ,n% into p nonemptyand disjoint subsets a1 , . . . ,ap . Let us denote by P thecollection of all unions of these basic sets together with theempty set, so that P has 2p elements. We assume that rTa

>0 for all aPP. For p51 this is no constraint at all, be-cause the full transpose of r is always positive. At the otherextreme, for p5n , this assumption means the positivity ofevery partial transpose.

We now take the expectation value of Eq. ~14! and aver-age over the 2p resulting terms. The coefficient of the bthterm then becomes

22p (aPP

~21 ! uaùbu522p )

m51

p

@11~21 ! uamùbu# , ~15!

which is proved by writing the sum over P as a sum over ptwo-valued variables, labeling the alternative ‘‘am,a or

BELL’S INEQUALITIES FOR STATES WITH . . . PHYSICAL REVIEW A 61 062102

062102-3

am,” a ,’’ and using that the parity (21) uaùbu is the productof the parities corresponding to the am . Clearly, the expres-sion ~15! is 1 if and only if uamùbu is even for all m andzero otherwise. Let us call such sets b ‘‘P even.’’ There are

)m

2 uamu2152n2p ~16!

such sets. Hence we get the bound

~ tr rB !2< (b P even

trS r ^kPb

i2

@Ak ,Ak8# D<2n2p. ~17!

That this bound is optimal is evident by evaluating it on atensor product of pure states maximally violating Mermin’s

inequality for each partition element am , i.e., states as dis-cussed in Sec. III.

To summarize, we have established the best bound

utr~rB !u<2 (n2p)/2 ~18!

on violations of Mermin’s inequalities, under the assumptionthat the partial transposes rTa are positive for alla,$1, . . . ,n% subordinated to a partition into p subsets. Thisincludes three special cases: For p51 it is the analog ofCirel’son’s inequality, for p5n it proves our claim that theinequalities are satisfied if, all partial transposes are positive,and for partitions of the form $1%, . . . ,$m%,$m11, . . . n%,we obtain the result of Gisin and Bechmann-Pasquinucci@16# using Mermin’s inequalities to test for the number m ofindependent qubits.

@1# D. Bohm, Quantum Theory ~Prentice-Hall, Englewood Cliffs,NJ, 1951!.

@2# R.F. Werner, Phys. Rev. A 40, 4277 ~1989!.@3# J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt, Phys.

Rev. Lett. 23, 880 ~1969!.@4# I. Pitowsky, Quantum Probability—Quantum Logic ~Springer,

Berlin, 1989!.@5# S. Popescu, Phys. Rev. Lett. 74, 2619 ~1995!.@6# C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A.

Smolin, and W.K. Wootters, Phys. Rev. Lett. 76, 722 ~1996!.@7# M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.

Lett. 80, 5239 ~1998!.@8# A. Peres, Phys. Rev. Lett. 77, 1413 ~1996!.@9# M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A

223, 1 ~1996!.@10# S. Popescu, Phys. Rev. Lett. 72, 797 ~1994!.@11# A. Peres, Found. Phys. 29, 589 ~1999!.@12# N.D. Mermin, Phys. Rev. Lett. 65, 1838 ~1990!.@13# M. Ardehali, Phys. Rev. A 46, 5375 ~1992!.@14# S.M. Roy and V. Singh, Phys. Rev. Lett. 67, 2761 ~1991!.@15# A.V. Belinskii and D.N. Klyshko, Usp. Fiz. Nauk 163-165, 1

~1993! @Phys. Usp. 36, 653 ~1993!#.@16# N. Gisin and H. Bechmann-Pasquinucci, Phys. Lett. A 246, 1

~1998!.@17# S.J. Summers and R.F. Werner, J. Math. Phys. 28, 2440

~1987!.@18# B.S. Cirel’son, Lett. Math. Phys. 4, 93 ~1980!.@19# L.J. Landau, Phys. Lett. A 120, 54 ~1987!.


062102-4

A complete set of Bell inequalities for multipartite systems

5.5 A complete set of Bell inequalities for multi-

partite systems

In the preceding section it was shown that Bell inequalities of the Mermin-Klyshkotype are satisfied for states of n parties, which have positive partial transpose withrespect to all bipartite partitions. Except for the case n = 2 this does, however, notmean that there exists a local hidden variable model for the considered correlations,since the set of inequalities is not complete. The following paper will complete thisset of inequalities and extensively discuss their properties. Apart from deriving thecomplete set of Bell inequalities explicitely, the most important results are:

• The complete set of 22n

(linear) Bell inequalities can be expressed by a singlealbeit non-linear inequality.

• Multipartite GHZ states violate all these inequalities maximally.

• States having positive partial transpose with respect to every bipartite parti-tion satisfy all these inequalities. Hence, the considered correlations admit alocal hidden variable description.

• All the inequalities can be obtained by nesting CHSH inequalities.

• For every n the respective Mermin-Klyshko inequality can be characterized asthat inequality that can be violated by the widest margin in quantum theory.

73

All-multipartite Bell-correlation inequalities for two dichotomic observables per site

R. F. Werner* and M. M. Wolf†

Institut fur Mathematische Physik, TU Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig, Germany~Received 9 February 2001; published 17 August 2001!

We construct a set of 22nindependent Bell-correlation inequalities for n-partite systems with two dichotomic

observables each, which is complete in the sense that the inequalities are satisfied if and only if the correlationsconsidered allow a local classical model. All these inequalities can be summarized in a single, albeit nonlinearinequality. We show that quantum correlations satisfy this condition provided the state has positive partialtranspose with respect to any grouping of the n systems into two subsystems. We also provide an efficientalgorithm for finding the maximal quantum-mechanical violation of each inequality, and show that the maxi-mum is always attained for the generalized GHZ state.

DOI: 10.1103/PhysRevA.64.032112 PACS number~s!: 03.65.Ud, 03.67.2a

I. INTRODUCTION

Entanglement has not only been a key issue in the ongo-ing debate about the foundations of quantum mechanics,started by Einstein, Podolsky, and Rosen in 1935 @1#, it alsoplays a crucial role in the young field of quantum informa-tion theory. Here entangled states are one of the basic ingre-dients of quantum information processing, due to their roleas a resource in quantum key distribution, super dense cod-ing, quantum teleportation, and quantum error correction ~cf.@2#!. Although general structural knowledge about entangle-ment has improved dramatically in the last few years, thereare still many open problems. For example, there is still noefficient general method to decide whether a given state isentangled or not.

The first, and for a long time also the only, mathemati-cally sharp criteria for entanglement were the Bell inequali-ties @3#. They provided the first possibility to distinguish ex-perimentally between quantum-mechanical predictions andthose of local realistic models. But although Bell inequalitieshave been known for more than 30 years @4#, our knowledgeabout the precise border between the classical and quantum-mechanical accessible region is still mainly restricted to thesimplest nontrivial cases. Best known is the case of two sites,at each of which two dichotomic observables are chosen.This is characterized completely by the Clauser-Horne-Shimony-Holt ~CHSH! version of Bell’s inequalities @5#, inthe sense that the inequalities are satisfied if and only if alocal classical model exists @6#. Finding a complete set oflinear inequalities in more complicated situations ~moresites, more observables, more outcomes! turns out to be avery difficult problem in the sense of computational com-plexity @7#. There is only very little knowledge about Bell-type inequalities beyond the CHSH case @8–12#. Though nu-merical studies yield a large number of inequalities @13#, formost of them it is neither known by how much they can beviolated in quantum theory nor is there a general character-ization admitting further investigations.

We were therefore quite surprised ourselves at finding an

infinite sequence of multipartite correlation settings forwhich we could develop the theory to be as explicit andcomplete as in the CHSH case. Our setting generalizes theCHSH setting to an arbitrary number n rather than two dif-ferent sites, but retains the constraints of just two observ-ables per site with just two outcomes each. Thus each of then participants has the choice of two observables, each ofwhich can take the values 11 or 21. For any choice ofobservables we then consider the expectation value of theproduct of all n signs ~a ‘‘full’’ correlation function!. A Bellinequality is a linear constraint on the set of all such expec-tations, which is valid whenever the correlations can be ob-tained from a local classical model, and which cannot bewritten as a convex combination of other such constraints.Examples are the CHSH inequality @5# for n52 and theirgeneralizations going back to Mermin and others @8–11#leading to a single inequality for arbitrary n.

We remark that this problem setting could be generalizedto include the expectations not only of the product of all nsigns, but also the products of subsets of signs ~ ‘‘restricted’’correlation functions!. These data would be sufficient to re-construct the full joint probability distributions of signs forall choices of observables. However, most of the derivationsin this paper do not generalize to this setting, and it is not yetclear which statements would still be valid ~maybe with adifferent proof!. When we talk of the existence of a classicalmodel, however, it is understood that such a model wouldalso determine all restricted correlation functions. The omis-sion of restricted correlation functions from our setting onlymeans that we do not consider constraints depending onthem.

For this class of multipartite correlations we obtained thefollowing results.

~i! We construct a set of 22nBell inequalities, and show its

completeness: the correlations considered allow a local clas-sical model if and only if all these inequalities are satisfied~Sec. III!.

~ii! The convex set of collections of classical correlationfunctions is a 2n-dimensional hyperoctahedron, which can bedescribed alternatively by a single nonlinear inequality ~Sec.III!.

~iii! We discuss the symmetries connecting different in-*Electronic address: [email protected]†Electronic address: [email protected]

PHYSICAL REVIEW A, VOLUME 64, 032112

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equalities and develop a construction scheme, which yieldsall 22n

equalities by successive substitutions into the CHSHinequality ~Sec. IV!.

~iv! We reduce the computation of the maximal quantumviolations of each Bell inequality to a simple variationalproblem with just one free variable per site. The maxima arealready attained in qubit systems, more specifically for then-party generalization of the GHZ state @15#, with a choice ofobservables depending on the inequality under consideration~Sec. V!.

~v! We extend this method to a characterization of theconvex body of quantum-mechanically attainable correlationfunctions in terms of its extreme points, which are also foundin the generalized GHZ state. These results are analogous tothose of Tsirelson @16,17# for the bipartite case.

~vi! We characterize the Mermin inequality as that Bellinequality, which can be violated by the widest margin inquantum theory.

~vii! Section VI settles the relationship between the cor-relation Bell inequalities and another important entanglementproperty. We show that for states having positive partialtransposes with respect to all their subsystems, all 22n

in-equalities are satisfied, so the correlations in such quantumstates can be explained in the context of a local realisticmodel. This extends our earlier result @18# for Mermin’s in-equalities, and is further supporting evidence for a recentconjecture by Peres @19#, namely that positivity of partialtransposes should generally imply the existence of local re-alistic models.

In the Appendix we will discuss some of the general re-sults obtained in Secs. III, IV, and V in more detail for thespecial cases n53,4.

II. BELL’S INEQUALITIES AND CONVEX GEOMETRY

Before entering the discussion of Bell inequalities in ourspecial context it is useful to recall some geometric struc-tures of the general problem and basic facts concerning theduality of convex polytopes. Consider a system decomposedinto n independent subsystems. Suppose further that on eachof these subsystems one out of m v-valued observables ismeasured. Thus each of the mn different experimental setupsmay lead to v

n different outcomes, so that the raw experi-mental data are made up of (mv)n probabilities. These num-bers form a vector j lying in a space of dimension (mv)n

~minus a few for normalization constraints!. Classically, in alocal realistic model, j would be generated by specifyingprobabilities for each classical configuration, i.e., for everyassignment of one of the v values to each of the nm observ-ables. Here the ‘‘local’’ character of the theory is expressedby the property that the assignment of a value to an observ-able at site k does not depend on the observables chosen atother sites. Every configuration c also represents a possibleclassical ~ideally prepared! state, and hence a vector ec ofprobabilities. The classical accessible region, which we willdenote by V , is thus the convex hull of v

(nm) explicitlyknown extreme points. Even though the number of configu-rations is large, it is finite, hence V is a polytope.

Like every compact convex set, V is the intersection of

all half spaces containing it. A half space is completely char-acterized by a linear inequality, so we must look for vectorsb such that ^b ,j&<1 for all jPV . Since this property canbe checked on the extreme points ec we must look at theconvex set

B5$bu; c:^b ,ec&<1%, ~1!

also known as the polar of $ec%. For each bPB the inequal-ity ^b ,j&<1 is thus a necessary condition for jPV . More-over, the bipolar theorem @20# says that the collection of allthese inequalities is also sufficient.

Luckily, the inequalities are not all independent, since theinequality for a convex combination b5(l ib i , with b iPB already follows from the inequalities for the b i . It there-fore suffices to take only the extreme points of B. For apolytope this has a very intuitive geometrical interpretation:the half spaces determined by extreme points touch V in aface of maximal dimension. Moreover, there are only finitelymany such maximal faces, which is to say that B is also apolytope.

The task of finding all Bell inequalities is therefore a spe-cial instance of a standard problem in convex geometry,known as the hull problem: given the extreme points $ec% ofa polytope V , find its maximal faces or, equivalently, theextreme points of its polar.

The duality between B and V is a generalization of theduality between regular platonic solids, under which dodeca-hedron and the icosahedron, as well as the octahedron andthe cube are polars of each other. A generalized(d-dimensional! octahedron is the unit sphere in a sequencespace l1($1, . . . ,d%). Its polar is the unit sphere in the dualBanach space l`($1, . . . ,d%), i.e., a d-dimensional hyper-cube. This is precisely the situation we will find for the clas-sically accessible region considered in this paper, where d52n.

The first to consider the construction of a complete set ofBell-type inequalities as a problem in convex geometry ap-parently was M. Froissart @21#. Unfortunately, however, ageneral solution for all (n ,m ,v) is highly unlikely to exist.To find some extreme points of Eq. ~1! is not so difficult, butalgorithms providing the complete set are likely to run intoserious growth problems already for very small (n ,m ,v). Infact, there is a theorem by Pitowsky @7# to the effect that, ina closely related problem, finding all inequalities would alsosolve some known hard problems in computational complex-ity ~this is in fact strongly connected with the notoriousNP5P , respectively NP5coNP questions!. Pitowsky andSvozil @13# have recently performed an extensive numericalsearch for n53, and published their result, the coefficients of53 856 inequalities on their website. Unfortunately, there isnot much generalizable insight coming out of this kind ofwork, but it is nice to see what can be done in this hardnumerical problem. Moreover, a numerical approach basedon linear optimization methods for investigating the case(n ,m ,v)5(2,2,v) without explicitly constructing extremalinequalities was proposed in @14#. For further problems andpartial results in this genre we refer to the problem page @12#on our own website. In what follows we will restrict to the


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case (n ,m ,v)5(n ,2,2) and ‘‘full’’ correlation functions inthe sense described in the Introduction.

III. ALL BELL-CORRELATION INEQUALITIES

A. Basic notation

Talking about Bell inequalities one usually has in mindinequalities of the CHSH form @5#. These inequalities refer tocorrelation experiments, in which each of two parties has thechoice of two 61 valued observables to be measured, i.e.,(n ,m ,v)5(2,2,2). Focusing only on full correlation func-tions for multiparticle generalizations of such systems@(n ,m ,v)5(n ,2,2), n fixed arbitrarily# the raw experimentaldata are 2n expectation values, each corresponding to a dif-ferent experimental setup. Each setup is labeled by thechoice of observables at each site. We parameterize thesechoices by binary variables skP$0,1% so that sk indicates thechoice of the 61-valued observable Ak(sk) at site k. Eachfull correlation function is thus the expectation of a product) kAk(sk), and is labeled by a bit string s5(s1 , . . . ,sn).

We will consider these expectations as the componentsj(s) of a vector j in a 2n-dimensional space. Then any Bellinequality is of the form

(s

b~s !j~s !<1, ~2!

where we have normalized the coefficients b so that themaximal classical value is 1, in accordance with the defini-tion of polars in Sec. II. The linear combination in Eq. ~2!can also be computed under the expectation value, so thatthis inequality can be stated as an upper bound on the expec-tation of

B5(s

b~s !)k51

n

Ak~sk!. ~3!

We call such expressions Bell polynomials. They can be useddirectly in the quantum case, where all variables Ak(sk) aresubstituted by operators with 21<Ak(sk)<1, acting in theHilbert space of the kth site, and the product is taken as thetensor product. It is often useful to consider these polynomi-als rather than the set of coefficients, because often manycoefficients are zero, and we can sometimes simplify a poly-nomial algebraically ~e.g., by factorization!, even though thismay not be apparent from the coefficients.

Two convex sets in the real 2n-dimensional vector spaceare the subject of our investigation: firstly, the polytope V ofcorrelation vectors j coming from local classical models, andsecondly the set Q.V of such vectors arising from quantummodels. V will be characterized in terms of Bell inequalitiesin this section, Q will be considered in Sec. V.

B. Construction and completeness

In a local classical model every observable Ak(sk) is arandom variable in its own right, i.e., it is a function of the‘‘hidden variable’’ which does not depend on the choices s lof observables at other sites lÞk . A model must assign prob-abilities to any collection of values for these observables,

i.e., to each classical configuration. Since the extremalchoices of such probabilities just assign probability 1 to oneconfiguration and zero probability to all others, the extremepoints of V are simply labeled by the configurations.

One configuration c is the choice of ck(sk)P$21,1% forall k and sk . Clearly, there are 22n such configurations. Thecorresponding correlation vector j[ec has components

ec~s !5)k51

n

ck~sk!. ~4!

Since we only consider full correlation functions ~and notrestricted ones, see the Introduction!, different classical con-figurations may give the same extreme point ec . For ex-ample, we may choose two different sites, and change thevalues of all ck(sk) at these sites simultaneously. Then in Eq.~4! the sign changes cancel for all s. This is also apparentfrom the factorization

ec~s !5S )k51

n

ck~0 !D)l51

n

c l~0 !c l~s l!, ~5!

in which the first factor is just an s-independent sign, and inthe second factor it suffices to choose configurations withck(0)51. Thus we can write ck(sk)5(21)skrk with rkP$0,1%. Then

ec~s !56~21 !^r ,s&, ^r ,s&5 (k51

n

rksk , ~6!

where the extreme points are now labeled uniquely by the bitstring r5(r1 , . . . ,rn) and the overall sign. This leaves uswith exactly 2n11 extreme points of V .

Our task is now to find the extremal linear inequalities b ,characterizing this set, i.e., the extreme points of B from Eq.~1!. The bipartite case was indeed completely analyzed byFine @6#, who showed that there are only two classes of in-equalities: one is trivial in the sense that it just requires cor-relations to be in @21,11# , and the second consists of theCHSH-type inequalities, for which the prototype is b

5( 12 , 1

2 , 12 ,2 1

2 ). A construction of some Bell-type inequalitiesfor arbitrary n was first proposed by Mermin @8# and furtherdeveloped by Ardehali @9#, Belinskii and Klyshko @10#, andGisin and Bechmann-Pasquinucci @11#.

We will now find all extremal solutions b to the set ofinequalities

21<(s

b~s !~21 !^r ,s&<1, ~7!

where rP$0,1%n runs over all bit strings characterizing theconfigurations. Suppose that of these 2n inequalities p,2n

fixed ones are ‘‘tight’’ in the sense that the sum takes one ofthe extreme values 61. This will be consistent with a plane~affine manifold! of vectors b of dimension at least 2n

2p .We can now construct convex decompositions of b in anopen neighborhood of b in this plane, since each one of theremaining sums is continuous in b , and there is a finite mar-gin before another inequality becomes violated. This contra-dicts extremality, so we conclude that the inequality must betight for all r. Thus we have 2n signs f (r)P$11,21% with

ALL-MULTIPARTITE BELL-CORRELATION . . . PHYSICAL REVIEW A 64 032112

032112-3

(s

b~s !~21 !^r ,s&5 f ~r !. ~8!

Now we can read Eq. ~8! as a Fourier transform with respectto the group of n-tuples of $0,1% with addition modulo 2.Therefore, we easily obtain the entire set of extremal b byapplying the inverse transformation to the set of vectors fP$21,1%2n

:

b~s !522n(r

f ~r !~21 !^r ,s&. ~9!

These are the coefficients of the complete set of 22nextremal

Bell inequalities specifying the range of expectations of fullcorrelation functions for any local realistic model.

The inequalities constructed in this way have a naturalnumbering, defined by the following procedure: For anynumber between 0 to 22n

21, write the binary expansionwith ‘‘digits’’ 61 to get f, and perform the inverse Fouriertransform ~9!. From b compute the polynomial ~3!, which isoften the best form of writing the inequality, because one canapply algebraic simplifications. For examples of this number-ing, see the Appendix. The converse procedure is similar. Forexample, the MATHEMATICA package available from our web-site @12# finds that Mermin’s inequality for n56 has thenumber 1 692 930 046 964 590 721.

C. Structure of the classical region

From the preceding section it is clear that the classicalregion V is a polytope in d52n dimensions with 2d extremepoints and 2d maximal faces. This suggests that V should bea hyperoctahedron, whose polar B is a hypercube. Indeedfrom the parametrization of the inequalities by d valuesf (r)561, the latter statement is rather obvious. That V isan octahedron is not so apparent in the coordinates labeledby s as above. However, we can choose a basis transforma-tion making this geometric identification of V more obvious.The necessary transformation is, of course, just the Fouriertransform. With the notation

j~r !522n(s

~21 !^r ,s&j~s !, ~10!

we can summarize the findings of the preceding section bysaying that jPV if and only if

; f P$21,1%2n:(

rf ~r !j~r !<1. ~11!

The expression in Eq. ~11! reaches its maximum with respectto f if f (r) is just the sign of j(r). Therefore, the whole setof 22n

linear inequalities ~or the statement jPV) is equiva-lent to the single nonlinear inequality

(r

u j~r !u<1. ~12!

Obviously, this nonlinear inequality is nothing but the char-acterization of the hyperoctahedron in 2n dimensions as theunit sphere of the Banach space l1.

From this simple characterization of V it might seem thatour problem is essentially trivial. However, the vast symme-try group of V , which includes among other transformationsthe set of (2n)! permutations of the coordinates is mislead-ing, because these are not really symmetries of the underly-ing problem of finding all correlations within a classicalmodel. This is apparent from the observation that the Bellpolynomials associated with the extreme points may lookquite different algebraically. That is, the 2n dimensions arenot really equivalent, but carry some structure coming fromthe division of the system into n sites. This is even moreobvious when looking at the set of quantum correlations,which has a much lower symmetry. Nevertheless, the under-lying problem has a large symmetry group, which will bestudied in the next section.

IV. SYMMETRIES AND SUBSTITUTIONS

Browsing through the complete set of linear correlationinequalities one quickly gets the feeling that there are manyrather similar ones, and also some inequalities which can beobtained in a rather trivial way ~e.g., as a product! fromlower-order ones. In this section we will describe the group-ing of the inequalities into ‘‘essentially different ones,’’ andalso how they can be obtained by an efficient constructionfor composing higher-order inequalities from lower-orderones. Both ways of structuring the set of inequalities makesense for more general cases (n ,m ,v) ~see Sec. II!, but forthe moment we only apply them to our restricted class.

A. Symmetry group

Some symmetries acting on Bell inequalities are obviousand, in fact, present in any problem of this type, involvingany number of outcomes and observables. The basic symme-tries leading to equivalent inequalities are as follows.

~i! Changing the labeling of the observables at each site.~ii! Changing the names of the outcomes of each observ-

able.~iii! Permuting subsystems.Since we have two observables per site, there are 2n ways

of swapping the labels of observables at each site. Swappingthe 61 outcomes of an observable Ak(sk) at site k results ina sign in all correlation functions involving this observable.We have already utilized the fact that swapping both Ak(0)and Ak(1) only results in an overall sign, so it is enough toconsider sign changes for Ak(1) only. Clearly, there are 2n

such sign changes. Expressed in terms of the function f thesetransformations amount to

f ~r !°~21 !^s0 ,r& f ~r1r0!, ~13!

where r ,s0 ,r0 all lie in $0,1%n, and r0 and s0 are the param-eters describing the sign changes and observable swaps, re-spectively. Together with the global sign change and the n!permutations we thus find the group G of symmetry transfor-mations in our case to have the order


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uGu5n! 22n11. ~14!

The orbit of a given inequality is defined as the set of all theinequalities generated from it by symmetry transformations.The number of elements in an orbit is uGu, divided by theorder of the group of symmetries leaving an element of theorbit invariant. The number of different orbits is the numberof ‘‘essentially different’’ inequalities. Obviously, Eq. ~14! isan upper bound on the number of elements in each orbit.Since the union of all orbits is the set of all inequalities, thisleads to a lower bound on the number of essentially differentinequalities.

Note that uGu increases much more slowly than 22n, the

total number of extremal inequalities. Therefore, for large nthe classification up to symmetry hardly reduces the numberof cases. Explicitly, we find

n Inequalities uGu Orbits

2 16 64 23 256 768 54 65 536 12 288 395 4 294 967 296 245 760 >17 476

For n up to 4, the number of orbits was obtained explicitly.However, for n>5 the lower bound on the number of orbitsmakes it clear that listing all essentially different inequalitiesis not going to be useful. More detailed results up to n54will be shown in the Appendix.

B. Generating new inequalities by substitution

A simple way of generating inequalities for higher n is topartition the n sites into two subsets of sizes n1 and n25n2n1 and to take arbitrary Bell polynomials for n1 and n2sites, appropriately rename the variables, and to multiply thetwo expressions. For example, the polynomial

12

~a1b11a1b21a2b12b1b2!c1 ~15!

is obtained by multiplying a CHSH polynomial for the firsttwo sites with the trivial polynomial ‘‘c1’’ on the third @notethat for the sake of clarity we have substituted A1(0),A2(1)with a1 ,b2, etc.#. It is clear that this gives an extremal Bellinequality for three sites.

This procedure can be generalized considerably by notingthat the product operation corresponds to the trivial two-siteBell polynomial ‘‘a1b1,’’ but nothing restricts us to using atrivial expression here. So in general, consider a partition ofthe sites into K subsets of sizes nk , (k51

K nk5n . Then pickan extremal Bell polynomial for K sites, written out in vari-ables A1(0),A1(1), . . . ,AK(1). Now substitute for eachAk(sk) an extremal Bell polynomial for nk sites. We claimthat the resulting polynomial in n variables is an extremalBell polynomial.

Indeed, if we substitute for each of the variables either11 or 21, we will get Ak(sk)561 for each k ,sk because

we substituted extremal Bell polynomials. But then the sameargument on the level of K sites shows that the value will be61.

We will say that a Bell polynomial is elementary if itcannot be obtained by substitution from lower order polyno-mials. Obviously, if an inequality is elementary, so is itsentire orbit. Clearly, the CHSH inequality is elementary.Moreover, it is known that it is a good tool for generatinghigher-order inequalities by substitution: one of the construc-tions @10,11# of the Mermin’s inequalities is based on thisidea. But in view of the rapid increase of the double expo-nential one might think that there must be many more el-ementary inequalities. However, we have the following re-sult.

Proposition. The CHSH inequality is the only elementaryBell inequality in the class we consider, i.e., all these in-equalities for n.2 can be constructed by successive substi-tutions into the CHSH inequality.

It is an interesting open problem, whether this statementholds for other families of Bell inequalities, e.g., the onetabulated in @13#.

We start the proof on the level of vectors f P$21,1%2n

parametrizing an arbitrary extremal Bell inequality for nsites. We decompose the system into a partition of K52subsets of size n21, 1 and rewrite

~16!

The respective coefficients b(s) of the n-site inequality arethen obtained via Fourier transformation according to Eq.~9!, and we get

b~s !522n(r

f ~r !~21 !^r ,s&

512

b0~ s !(rn

~21 !snrndrn,0112

b1~ s !(rn

~21 !snrndrn,1

512

@b0~ s !1~21 !snb1~ s !# , ~17!

where bk( s) are coefficients for extremal Bell inequalitiesfor n21 sites. If we now add the respective observablesAk(sk) and write out the corresponding Bell polynomial

B5(s

b~s !)k51

n

Ak~sk!

512

B0@An~0 !1An~1 !#112

B1@An~0 !2An~1 !# ,

~18!

we immediately see that this is just a CHSH polynomial,where the observables of one site have been substituted byBell polynomials B0 and B1 for n21 sites.

V. QUANTUM VIOLATIONS

Provided with a huge number of Bell-type inequalities wenow go beyond the classical accessible region. The first


032112-5

question to arise is of course whether or not and to whatextent quantum systems can violate these inequalities. Toanswer this question we will first provide an effective varia-tional method for computing the maximal quantum viola-tions and show, that they are bounded by those obtained forMermin’s inequalities. In Secs. V A and V B we will thenbriefly discuss the structure of the underlying quantum do-main, and prove that the generalized GHZ state maximallyviolates any of the correlation inequalities.

A. Obtaining the maximal violations

In order to compute the maximal quantum violation ofany correlation inequality we have to vary over one densityoperator r on a tensor product of n factors, and two operatorsin each factor. Assuming all tensor factors to have dimensiond, this means d2n parameters for the density operator and2nd2 for the observables. Hence the numerical solution ofthis variational problem is not feasible, except for the mosttrivial cases ~and even impossible, because d is, in principle,a free parameter!. Fortunately, however, it turns out thatcomputing the overall maximum is much easier than com-puting the maximal violation for a fixed state: we will reducethe computation to a variational formula in just n variables.

First we have to recall some basic notions. In quantummechanics expectations of 61-valued observables are de-scribed by Hermitian operators Ak(sk) with spectrum in@21,11# . Since we are only interested in maximal correla-tions, we may as well take the observables extremal in theconvex set of Hermitian operators with 21<A<1, i.e., wemay assume the observables to be unitary and thus A2

51.The general form of a Bell inequality for an n-partite

quantum system, which is characterized by a density opera-tor r , is then

tr~rB !ªtrFr(s

b~s ! ^ k51n Ak~sk!G<1, ~19!

where we will refer to B as the Bell operator, which is justthe quantum counterpart of the Bell polynomial defined inEq. ~3!. Of course, every expectation value ~19! larger than 1is called a violation of Bell’s inequality.

In order to derive the maximal quantum violation, whichis nothing but the operator norm of the Bell operator, we firstdefine another operator C by

CªB ^ k51n Ak~0 !5(

sb~s ! ^ k51

n Cksk , ~20!

where we have set Ck5Ak(1)Ak(0) and Ck051. Since the Ck

are commuting unitary operators, all summands of C can bediagonalized simultaneously, and the eigenvectors of C aretensor products of eigenvectors of the Ck . Every eigenvalueg of C is therefore of the form

g5(s

b~s !)k51

n

gksk , ~21!

where gk is an eigenvalue of Ck . It is clear from the aboveremarks that C commutes with its adjoint, so iCi is just the

modulus of the largest eigenvalue. Now we utilize the factthat iB*Bi5iC*Ci , i.e., iBi5iCi and obtain

iBi5sup$gk%

U(s

b~s !)k51

n

gkskU , ~22!

where each gk runs over the eigenvalues of Ck5Ak(1)Ak(0). This formula allows us to compute the larg-est expectation tr(rB) for fixed real coefficients b ~comingfrom a Bell inequality or not! and a fixed choice of observ-ables Ak(sk), but with r chosen without further constraints tomaximize the expectation.

What we are now interested in is the maximum also withrespect to the Ak(sk). Since formula ~22! depends only onthe eigenvalues gk this will be given by the same expression,but with gk running not just over the eigenvalues of a par-ticular operator Ck but over all gk which can be eigenvaluesof products of unitary and Hermitian operators. Since such aproduct is again unitary, we have ugku51. Moreover, as iseasily seen in 232 examples, this is the only constraint inany Hilbert space dimension ~see also Sec. V D!. Hence forany choice of real coefficients b(s) and observables 21

<Ak(sk)<1, we have, as the best possible bound,

trFr(s

b~s ! ^ k51n Ak~sk!G<sup

$gk%U(

sb~s !)

k51

n

gkskU ,

~23!

where the supremum runs over all $g1 , . . . ,gn% with ugku51. Moreover, the bound does not change with Hilbertspace dimension, as long as all factors are nontrivial. A moredetailed discussion of quantum violations utilizing Eq. ~22!for the special cases n53,4 can be found in the Appendix.

B. Mermin’s inequalities and the overall maximum

Asking for the overall maximal quantum violation wemay additionally vary over the set of inequalities. In utilizingthe result obtained in the preceding section, we are able toexpress the norm of a Bell operator in terms of lower-orderBell operators. Moreover, it suffices to consider qubit sys-tems and we may therefore set Ak(sk)5aW k(sk)sW , where sW isthe vector of Pauli matrices and aW k(sk) is a normalized vec-tor in R

3.Squaring Eq. ~18! this leads to

B25

B02

2^ @11aW n~0 !aW n~1 !#1

B12

2^ @12aW n~0 !aW n~1 !#

1@B0 ,B1# ^

i2

@aW n~0 !3aW n~1 !#sW . ~24!

Without loss of generality we now assume that iB0i<iB1iand estimate by induction

iBi25iB2i<2iB1i2<2n21. ~25!

This bound is indeed saturated by the set of inequalitiesgoing back to Mermin @8–11#, which thus provides the over-all maximal quantum violation. In fact, we will show that theconverse is also true, so that we have the following.


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Proposition. The orbit corresponding to Mermin’s in-equality is the only one for which the maximal violation2 (n21)/2 is attained.

Before we continue proving the claimed uniqueness, weemphasize, however, that this does not, in general, imply thatfor a fixed quantum state Mermin’s inequality is morestrongly violated than any other.

We begin our proof with noting that the maximal norm ofthe Bell operator in Eq. ~24! requires orthogonality of theobservables, such that the respective phases in Eq. ~22! haveto be 6i . Without loss of generality we can thereby restrictto the case 1i since the remaining sings just correspond to atransformation between two inequalities of the same orbitaccording to Eq. ~13!. Hence, Eq. ~22! leads to

iBmaxi522nU(r ,s

f max~r !)k51

n

~21 !skrkiskU522nU(

rf max~r !)

k51

n

@11i~21 !rk#U522n/2U(

rf max~r !)

k51

n

@e i(p/4)(122rk)#U522n/2U(

rf max~r !~2i !(

krkU . ~26!

If we now want Bmax to saturate the bound in Eq. ~25!, thenfollowing Eq. ~26! we are left with four possible choice forf max , like f max(r)51 for (2i)(krk51,i and f max(r)521 oth-erwise. Since these four inequalities again belong to thesame orbit, the correlation inequality leading to the overallmaximal quantum violation is indeed uniquely determined~up to equivalence transformations within one orbit!.

C. Structure of the quantum domain

In the same manner as we did for the classical case wemay ask for the structure of the region in the space of corre-lations, which is accessible within the framework of quantummechanics. One of the first to investigate this question inmore detail apparently was Tsirelson @16,17#, while studyingquantum generalizations of Bell’s inequalities.

Let us begin with defining the quantum counterpart of theclassical accessible region V , introduced in Sec. II,

Qª$jujs5tr@r ^ k51n Ak~sk!#%,R

2n, ~27!

where $Ak% are suitable observables and r is a quantum statein arbitrary dimension. The structure of Q is much morecomplicated than that of V,Q. In particular, it is not a poly-tope. Nevertheless, we can explicitly parametrize its extremepoints. For the sake of completeness we will first prove con-vexity of Q, although this follows closely the work ofTsirel’son @17#.

Consider a convex combination of vectors in Q

(a

l (a)j (a), j (a)PQ ~28!

and an associated Hilbert space

H5 ^ k51n

Hk5 ^ k % aH k(a)> % a1 . . . an

^ kH k(ak) . ~29!

Then with r5 % al (a)r (a), which is a density operator actingon the ‘‘diagonal subspace’’

% a ^ kH k(a)

,H, ~30!

and Ak(sk)5 % aAk(a)(sk) we are given a state and observ-

ables such, that the convex combination in Eq. ~28! is indeeda proper element of Q. Hence, Q is convex.

Now let us return to the result obtained in Sec. V B. Fol-lowing Eq. ~22! we can write the maximal quantum violationof an arbitrary inequality b as

supg0 . . . gn

(s

b~s !ReS g0)k51

n

gkskD ~31!

5 supw0 . . . wn

(s

b~s !js~w0 , . . . ,wn!, ~32!

where we have set js(w0 , . . . ,wn)5cos(w01(kwksk). Nowby the bipolar theorem @20# the convex set Q is just given bythe convex hull of these vectors:

Q5co$j~w0 , . . . ,wn!%. ~33!

D. Generalized GHZ states

It is a well-known fact that the generalized GHZ statedefined by

uCGHZ&51

A2~ u00¯0&1u11¯1&) ~34!

maximally violates Mermin’s inequalities @10#. Astonish-ingly this is also true for any other of the 22n

correlationinequalities.

Proposition. Any extreme point of the convex set of quan-tum correlation functions as defined in Eq. ~27! is alreadyobtained for the generalized GHZ state. In particular, thisimplies that GHZ states maximally violate any of the pre-sented correlation inequalities.

We have to show that for any set of angles $w0 , . . . ,wn%there are suitable observables such, that

^CGHZu ^ k51n Ak~sk!uCGHZ&5cosS w01(

kwkskD .

~35!

Therefore we choose observables Ak(sk)5aW k(sk)sW with

aW k~0 !5~cos a ,sin a ,0 !

aW k~1 !5@cos~wk1a !,sin~wk1a !,0# . ~36!

These observables simply swap the basis vectors providingthem with an additional phase factor, i.e.,

aW k~sk!sW u j&5exp@ i~21 ! j~a1wksk!#u j % 1& , ~37!

where j50,1 and % means addition modulo 2. Hence, forthe left-hand side of Eq. ~35! two terms occur, which are justcomplex conjugates of each other, and we get


032112-7

^CGHZu ^ k51n Ak~sk!uCGHZ&5ReH e iane i(

kwkskJ , ~38!

so that it just remains to set a5w0 /n .

VI. STATES WITH POSITIVE PARTIAL TRANSPOSES

The violation of one of the inequalities, which can bederived from Eq. ~9!, is a rather physical entanglement cri-terion, since we can at least in principal decide it experimen-tally by measuring the respective correlations. However, thedifficulty in doing so is the choice of the observables, andoptimizing them for a fixed state leads in general to a veryhigh dimensional variational problem. An entanglement cri-terion, which is in contrast easy to compute, is the partialtranspose proposed by Peres in @22#. Before we settle therelationship between these two entanglement criteria, we willbriefly recall some basic notions.

The partial transpose of an operator on a twofold tensorproduct of Hilbert spaces H1 ^ H2 is defined by

S (j

C j ^ D j D T1

5(j

C jT

^ D j , ~39!

where C jT on the right-hand side is the ordinary transposition

of matrices with respect to a fixed basis. The generalizationof this definition to an n-fold tensor product is straight for-ward, and we will denote the transposition of all sites be-longing to a set t,$1, . . . ,n% by the superscript Tt .

Recall further that a state is called separable or classicallycorrelated if it can be written as a convex combination oftensor product states—otherwise, it is called entangled. Anecessary condition for separability, which also turned out tobe sufficient in the case of two qubits @23#, but not in general~cf. @24#!, is the positivity of all partial transposes with re-spect to all subsystems. Moreover, there is a conjecture byPeres @19# that this might even imply the existence of a localrealistic model. In @18# we showed that the set of inequalitiesgoing back to Mermin @8# is indeed fulfilled for states satis-fying this positive partial transpose condition. In the follow-ing we will show that this implication is not due to a specialproperty of these inequalities, but holds for any Bell-typeinequality in Eq. ~19!, as long as we consider expectations offull n-site correlations. This leads to the main result of thissection.

Proposition. Consider an n-partite quantum system, whereeach of the parties has the choice of two dichotomic observ-ables to be measured. Assume further, that the partial trans-poses with respect to all subsystems of the correspondingdensity operator are again positive semidefinite operators.Then all the 2n correlations can be described in the contextof a local realistic model.

In particular, this implies that if a state is biseparable withrespect to all partitions, all the inequalities are satisfied evenif there exists no convex decomposition into n-fold productstates. In order to prove this proposition and to derive anupper bound for the expectation of the Bell operator, we firstapply the variance inequality to rTt and BTt,

~ tr rB !25~ tr rTtBTt!2<tr@rTt~BTt!2#<tr$r@~BTt!2#Tt%.

~40!

Since we suppose that rTt>0;t this holds for any partialtransposition, and we may take the average with respect toall subsets t , and have therefore to estimate the expectationof the operator12n (

t(s ,s8

b~s !b~s8! ^ kPt Ak~sk!Ak~sk8! ^ k¹t Ak~sk8!Ak~sk!

5(s ,s8

b~s !b~s8! ^ k51n 1

2$Ak~sk!,Ak~sk8!%1 , ~41!

where $• ,•%1 denotes the anticommutator. Note that in thefirst line of Eq. ~41! we have rearranged the tensor productand made use of

@Ak~sk8!TAk~sk!T#T5Ak~sk!Ak~sk8!. ~42!

Since A251 and sk ,sk8P$0,1% only two different operators

can arise in every tensor factor in Eq. ~41!: either

2$Ak(0),Ak(1)%1 or the identity operator. These two obvi-ously commute, and we can therefore simultaneously diago-nalize all the summands. What remains to be done is to sub-stantiate our intuition that ‘‘if everything commutes, then weare in the classical regime.’’ For this purpose note that eigen-values of the operator ~41! are of the form

(s ,s8

b~s !b~s8!)k51

n H xk , skÞsk8

1, sk5sk8J , ~43!

for suitable 21<xk<1. But since we can always find clas-sical observables C with correlations

^Ck~0 !Ck~1 !&5xk , ~44!

we are able to construct a system which is classical in thesense that it may be described in the context of classicalprobability theory, such that Eq. ~43! is the expectation of thesquare of the respective Bell polynomial. However, due tothe defining properties of the Bell inequalities, this is indeedbounded by unity, which proves our claim that all the con-sidered Bell inequalities are satisfied for states having posi-tive partial transposes with respect to all their subsystems.

VII. CONCLUSION

We provided two approaches for constructing the entireset of multipartite correlation Bell inequalities for two di-chotomic observables per site: the Fourier transformation ofa 2n-digit binary number and nesting CHSH inequalities.This set of inequalities led us to a single nonlinear inequality,which detects the existence of a local classical model withrespect to the considered correlations. We were able to sim-plify the variational problem of obtaining the maximal quan-tum violation of the linear correlation inequalities, in particu-lar showing, that these are attained for generalized GHZstates, and proved, that ‘‘ppt states’’ satisfy all these 22n

in-equalities.

One crucial assumption was that each site has the onlychoice of two dichotomic observables to be measured. Per-mitting more observables per site, more outcomes per ob-servable, or even the choice of ‘‘not measuring,’’ i.e., includ-


032112-8

ing restricted correlation functions, would lead tononcommuting terms, and most of the arguments would fail.So this is obviously a starting point for further investigations.In particular, one may think of applying the mechanism ofsubstitution ~Sec. IV! in order to derive new classes of Bellinequalities.

Another open question concerns the hierarchy of the in-equalities with respect to their quantum violations. That is, ifa given inequality is violated for a fixed quantum state, isthere a set of inequivalent inequalities that have to be vio-lated as well?

Finally, we want to mention that there is recent work byScarani and Gisin @25# pointing out that there might be aclose relation between the quantum violation of multipartiteBell inequalities and the security of n-partner quantum com-munication.

Note added. Recently a closely related paper @30# wasposted by Zukowski and Brukner. They also study con-straints on correlation functions and obtain the inequalitieswe presented in Sec. III.

ACKNOWLEDGMENTS

Funding by the European Union project EQUIP ~ContractNo. IST-1999-11053! and financial support from the DFG~Bonn! are gratefully acknowledged.

APPENDIX

Recently more and more attention has turned to tri- andfour-partite states, especially to symmetric states as labora-tories for multipartite entanglement ~cf. @26–28#!. Thereforewe will provide the complete set of Bell inequalities forthese cases in a more explicit form and additionally give themaximal quantum violations, which we have numerically@29# obtained utilizing the method presented in Sec. V.

1. Inequalities for three sites

For n53 Eq. ~9! leads to the five essentially different Bellpolynomials @for the sake of legibility we again substituteA1(0),A2(1) with a1 ,b2 etc.#,

a1b1c1 , ~A1!

14 (

k ,l ,makb lcm2a1b1c1 , ~A2!

12

@a1~b11b2!1a2~b12b2!#c1 , ~A3!

12

@a1b1~c11c2!2a2b2~c12c2!# , ~A4!

12

~a1b1c21a1b2c11a2b1c12a2b2c2!. ~A5!

Polynomials ~A1! and ~A3! are just trivial extensions oflower-order inequalities, and polynomial ~A5! belongs to theset developed by Mermin @8#. The maximal quantum viola-tions, the number of the first inequality of each of the fiveorbits, and the sizes of the respective orbits are stated in thefollowing table:

Inequality uOrbituQuantumviolation

Polynomial

0 16 1 ~A1!

1 128 5/3 ~A2!

3 48 A2 ~A3!

6 48 A2 ~A4!

23 16 2 ~A5!

2. Inequalities for four sites

For n54 we just give the number of the first inequality ofeach of the 39 orbits, its size, and the respective maximalquantum violations. The index p labels orbits, including anelement which is invariant under permutations of the sub-systems, and f indicates factorizing Bell polynomials @likepolynomials ~A1! and ~A3! for tripartite systems#.

Inequality uOrbituQuantumviolation

0p , f 32 11p 512 1.8433 f 1024 5/36 1536 5/37 3072 1.93215f 192 A222 2048 1.93223 1024 A524 1024 225 6144 A327 3072 A330 3072 A360f 384 A2105 128 A2278p 256 A5279p 512 2.556280 3072 2.139281 1536 1.819282 3072 1.819283 6144 2.078286 1536 2.078287 1536 2.326300 3072 2301 6144 5/3303 3072 1.819317 3072 2318 1536 2319 2048 2.139360 1024 2.326363 1536 A3367 1536 A3383p 256 2831f 128 2854f 96 2857 384 A2874 384 21632 96 A21647 192 26014p 32 2A2

The number of the inequality representing the orbit of Mer-min’s inequality is 6014.


032112-9

@1# A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777~1935!.

@2# E. Knill and R. Laflamme, Phys. Rev. A 55, 900 ~1997!; N.Gisin, G. Ribordy, W. Tittel, and H. Zbinden,quant-ph/0101098; R.F. Werner, quant-ph/0101061.

@3# J. S. Bell, Physics ~Long Island City, N.Y.! 1, 195 ~1965!.@4# Of course mathematicians started to investigate the possible

range of correlations in the form of inequalities long beforephysicists payed attention to it due to the work of Bell @3#. Fora better review cf. Ref. @7#.

@5# J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys.Rev. Lett. 23, 880 ~1969!.

@6# A. Fine, Phys. Rev. Lett. 48, 291 ~1982!.@7# I. Pitowsky, Quantum Probability—Quantum Logic ~Springer,

Berlin, 1989!.@8# N. D. Mermin, Phys. Rev. Lett. 65, 1838 ~1990!.@9# M. Ardehali, Phys. Rev. A 46, 5375 ~1992!.

@10# A.V. Belinskii and D. N. Klyshko, Phys. Usp. 36, 653 ~1993!.@11# N. Gisin and H. Bechmann-Pasquinucci, Phys. Lett. A 246, 1

~1998!.@12# http://www.imaph.tu-bs.de/qi/problems/1.html@13# I. Pitowsky and K. Svozil, e-print quant-ph/0011060.@14# D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklas-

zewski, and A. Zeilinger, Phys. Rev. Lett. 85, 4418 ~2000!.@15# D. M. Greenberger, M. A. Horne, A. Shimony, and A.

Zeilinger, Am. J. Phys. 58, 1131 ~1990!.@16# B. S. Cirel’son, Lett. Math. Phys. 4, 93 ~1980!.@17# B. S. Tsirel’son, J. Sov. Math. 36, 557 ~1987!.@18# R. F. Werner and M. M. Wolf, Phys. Rev. A 61, 062102 ~2000!.@19# A. Peres, Found. Phys. 29, 589 ~1999!.@20# H. H. Schaefer, Topological Vector Spaces ~Springer, Berlin,

1980!.@21# M. Froissart, Nuovo Cimento Soc. Ital. Fis., B 64, 241 ~1981!.@22# A. Peres, Phys. Rev. Lett. 77, 1413 ~1996!.@23# M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A

223, 1 ~1996!.@24# M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.

Lett. 80, 5239 ~1998!.@25# V. Scarani and N. Gisin, e-print quant-ph/0101110.@26# W. Dur, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314

~2000!.@27# T. Eggeling and R. F. Werner, e-print quant-ph/0010096.@28# K. G. H. Vollbrecht and R. F. Werner, e-print

quant-ph/0010095.@29# Especially for the case n54 we emphasize that the maximal

violations in the table are results of a numerical maximization.Although they seem very stable with respect to variation of theinitial conditions, we have no proof that these are the absolutemaxima.

@30# M. Zukowski and C. Brukner, e-print quant-ph/0102039.


032112-10

Bell inequalities and distillability in n-quantum-bit systems

5.6 Bell inequalities and distillability in n-quantum-

bit systems

In this section we will briefly discuss an “application” of the two previous papers.The results here are mainly due to A. Acin and were obtained within a collaborationduring this thesis. They are stated just for completeness and without detailedproofs. The latter can be found in [ASW02].

Starting point is the question whether there is a connection between the viola-tion of Bell inequalities and distillability for systems composed out of n qubits. Inanalogy to the bipartite case discussed in Ch.3 a multipartite state is called dis-tillable, if one can extract a highly entangled state between any pair of parties bymeans of LOCC operations. This is in turn equivalent to demand that an n-partiteGHZ state can be obtained by LOCC operations. It is not surprising that multipar-tite states exhibit a richer structure than bipartite states concerning qualitativelydifferent entanglement properties. For instance, an undistillable state of n partiesmay become distillable if two of the n parties come together and are treated likea single system. Very much like in Sec.5.4 this leads to a finer distinction thandistillable versus undistillable.

We have proven in [ASW01, ASW02] that the violation of any of the Bell in-equalities introduced in Sec.5.5 implies that entanglement can be distilled from thestate. However, the corresponding distillation protocol may require that some ofthe parties join into several groups. We have shown that there exists the followinglink between the amount of the Bell inequality violation and the size of the groupsone has to form for distillation:

Proposition 18 Consider an n-qubit state violating one of the Bell inequalitiesintroduced in Sec.5.5 by an amount β such that

β > 2n−p

2 . (5.18)

Then entanglement can be distilled between any two groups, if the n parties can joininto groups of at most p− 1 qubits.

The proof of this statement is rather lengthy and can be found in several steps in[ASW02]. The main ingredients of the proof are:

1. The complete set of inequalities presented in Sec.5.5 can be obtained by nest-ing CHSH inequalities.

2. States having a positive partial transpose with respect to every bipartite par-tition do not violate any of these inequalities.

3. Every Bell operator B of the considered set of inequalities is diagonal in aGHZ type basis. In particular

B =∑

k

λk(P+k − P−k

), λk ≥ 0 , (5.19)

where P±k are the projectors onto |ω±〉 = (eiφω |ω〉 ± |1 − ω〉)/√2, where

ω ∈ 0, 1n, and the phases φω depend on the type of the inequality andthe chosen observables.

4. There is a twirl onto a (different) GHZ basis as presented in Sec.2.7.3.

Prop.18 shows that Bell inequalities can provide information about the usefulnessof a state for quantum information applications.

85

Bell inequalities and distillability in n-quantum-bit systems

86

Chapter 6

Gaussian states

Gaussian continuous variable states, i.e., states having a Gaussian Wigner distribu-tion, play an important role in quantum optics. In particular, coherent, thermal andsqueezed states of light field modes are Gaussians. From a mathematical perspec-tive they are distinguished from other states acting on infinite dimensional Hilbertspaces, since they are completely specified by finite dimensional objects — a vectorof means and a covariance matrix. Hence, many of the typical questions of quantuminformation theory are of the same complexity for Gaussian states as for systemson finite dimensional Hilbert spaces. Each mode of a Gaussian state correspondsto one canonical degree of freedom described by a pair of canonical operators —one position and one momentum operator. In fact, the underlying mathematicalstructure for describing Gaussian states is precisely the same as in classical phasespace mechanics.

We will begin this chapter again with introducing the basic notions. Start-ing from canonical commutation relations we will introduce Weyl operators andphase space functions in Sec.6.1 and 6.2. Sec.6.3 will then discuss transformations,which preserve the canonical commutation relations and therefore leave the basickinematic rules unchanged. The letter in Sec.6.4 discusses the relation between en-tanglement and the positivity of the partial transpose for Gaussian states. Finally,we investigate in Sec.6.5 the possibility of generating entangled Gaussian states withnon-positive partial transpose by means of passive optical elements.

6.1 Phase space

A system of n canonical degrees of freedom is described classically in a phase space,which is a 2n-dimensional real vector space X ' R

2n. The canonical structureis given by a non-degenerate and antisymmetric bilinear form1 σ : X × X → R.The form σ is called “symplectic form” and the pair (X,σ) a “symplectic vectorspace”. We will write the “symplectic scalar product” as σ(ξ, η) =

∑ij ξiηjσij by

introducing the symplectic matrix defined as σij = σ(ei, ej), where ei is a basisin X.2

With a suitable choice of the basis (“canonical coordinates”), the symplecticmatrix can be brought into a standard form: The 2n variables are then grouped inton canonical pairs (e.g., position and momentum), for each of which the symplectic

1That is, σ has to satisfy σ(ξ, η) = −σ(η, ξ) and ∀ξ : σ(ξ, η) = 0⇒ η = 0.2To avoid complicated notation we will in the following formally not distinguish between the

symplectic form and the symplectic matrix.

87

Phase space

matrix is antisymmetric, and all other matrix elements vanish:

σ =

(0 1n−1n 0

). (6.1)

The block structure in Eq.(6.1) reflects the direct sum of configuration and momen-tum space.

In order to recognize the familiar phase space of classical mechanics as a sym-plectic vector space (cf.[Arn89]), let us write down Hamilton’s canonical equationsas

pi = −∂H

∂qi, qi =

∂H

∂pi⇔ ξi =

∑

j

σij∂H

∂ξj, (6.2)

where we have set ξ = (q1, . . . , qn, p1, . . . , pn). Moreover, if ξ(t) is a solution ofHamilton’s canonical equations, then an observable f evolves in time as

d

dtf(ξ(t)

)= f,H =

∑

ij

σij∂f

∂ξi

∂H

∂ξj, (6.3)

where ·, · are the “Poisson brackets”.

Canonical commutation relations: Let us now turn to the quantum world. Bythe “quantization transcription” q, p → −i[Q,P ], the Poisson brackets are for-mally related to the “canonical commutation relations” (CCR).3 The CCR are thengoverned by the symplectic matrix: if Rk, k = 1, . . . , 2n are canonical operators4,i.e. generalized position and momentum operators withR = (Q1, . . . , Qn, P1, . . . , Pn),the commutation relations read

[Rk, Rl] = iσkl1 . (6.4)

Equivalently the quantum system may be described using “annihilation” and“creation” operators aj , a

∗j , j = 1, . . . , n, which obey the standard bosonic commu-

tation relations

[aj , a∗k] = δjk 1, [aj , ak] = [a∗j , a

∗k] = 0. (6.5)

The Eqs.(6.4) and (6.5) are then related via the unitary matrix

Ω =1√2

(1n i1n1n −i1n

), (6.6)

which is such that (a1, . . . , an, a∗1, . . . , a

∗n)T = Ω(R1, . . . , R2n)

T . Thus, the matrixΩ translates between a real and a complex representation.

The standard quantum mechanical representation of the CCR in Eq.(6.4) is ofcourse the “Schrodinger representation”, where for each degree of freedom the rep-resenting Hilbert space is H = L2(R) and P and Q are i−1d/dx and multiplicationwith x respectively.

3Note that we use dimensionless position and momentum operators with ~ = 1.4One should think of the operator Rk as acting on a tensor product Hilbert space H = H⊗n

0 ,such that Rk is in turn a tensor product of n− 1 identity operators with the respective canonicaloperator acting on the k-th (resp. (k − n)-th) tensor factor.

88

Phase space functions and Gaussian states

Weyl systems: The representation of the CCR in Eq.(6.4) is not unique, noteven up to unitary equivalence. One way to remove the remaining ambiguities andto express the CCR with bounded rather than unbounded operators is to use a“Weyl system”, i.e., family of unitaries

Wξ = eiξ·σR, ξ ∈ R2n (6.7)

which satisfy the “Weyl relations”

WξWη = e−i2ξ·ση Wξ+η. (6.8)

The generators R satisfy then the CCR and by the “Stone-von Neumann theorem”[vN31] the representation of the Weyl relations is unique up to unitary equivalenceif the Weyl system is irreducible, strongly continuous and corresponds to a systemwith finitely many canonical degrees of freedom.5 Every Weyl system appearingin the following is assumed to be irreducible and strongly continuous. Moreover,the underlying phase space will always be finite dimensional. Weyl operators corre-sponding to systems with many degrees of freedom can be identified with a tensorproduct of Weyl operators, such that Wξ =

⊗nk=1W(ξk,ξk+n).

By differentiating Eq.(6.7) with respect to ξk and utilizing the Weyl relationsone can verify that

WξRkW∗ξ = Rk + ξk1. (6.9)

Hence, the Weyl operators correspond to translations in phase space, and we em-phasize for later use that translations are always local unitary operations.

6.2 Phase space functions and Gaussian states

It is often useful to deal with complex (or real) valued functions on phase spacerather than with operators in Hilbert space. In the present section we will brieflydiscuss how such a correspondence can be formulated using “characteristic func-tions” or “Wigner functions”. We will introduce Gaussian states and discuss howcertain properties of the density operator, like positivity and purity, translate intoproperties of their covariance matrix.

Characteristic functions: One possible starting point of such an “operator –phase space function” correspondence is the “Fourier-Weyl transform”

g 7→ g = (2π)−n∫g(ξ)W−ξ d

2nξ, (6.10)

where g(ξ) is a complex valued Lebesgue integrable function on phase space. Theinverse relation of Eq.(6.10) for trace class operators g ∈ T 1(H) reads6

g(ξ) = tr [Wξ g] , (6.11)

and has the property |g(ξ)| ≤ ||g||1 with equality for ξ = 0. The one-to-one cor-respondence between g and g is in fact analogous to the one in classical Fourieranalysis.

5Two Weyl systems W (1) and W (2) are called “unitarily equivalent” if there exists a unitary

operator U such that ∀ξ ∈ R2n :W

(1)ξ

= UW(2)ξ

U∗. A Weyl system is called “strongly continuous”

if ∀ψ ∈ H : limξ→0 ||ψ −Wξψ|| = 0, and “irreducible” if ∀ξ ∈ R2n : [Wξ, A] = 0⇒ A ∝ 1.

6An operator A is a trace class operator A ∈ T 1(H) if tr [|A|] = tr[√

A∗A]<∞.

89


Applied to density matrices the function

χ(ξ) = tr [Wξρ] (6.12)

is called the “characteristic function” corresponding to the density operator ρ.7 Thenormalization tr [ρ] = 1 translates to χ(0) = 1 and by utilizing the Parseval theorem[Hol82] we obtain that χ(ξ) corresponds to a pure state, i.e., ρ2 = ρ, iff

∫|χ(ξ)|2 d2nξ = (2π)n. (6.13)

The property of χ corresponding to ρ ≥ 0 will be discussed separately for Gaussianstates below.

Wigner functions: The map ρ 7→ χ(ξ) = tr [ρWξ] is reminiscent of the classicalFourier transform. If we take this analogy literally, i.e., if we interpret χ(ξ) as theclassical Fourier transform of a function W(ξ) (with the symplectic form insteadof a scalar product between the vectors), we are led to “Wigner’s phase spacedistribution function” [Wig32]

W(ξ) = (2π)−2n∫eiξ·ση χ(η) d2nη. (6.14)

If we chose ρ =∑

i λi|ψi〉〈ψi| with λi ≥ 0 and∑

i λi||ψi||2 = 1 to be a decompositionof ρ and define an integral kernel D(x; y) =

∑i λiψi(x)ψi(y) in the Schrodinger

representation, then we can write Eq.(6.14) as

W(ξ) = π−n∫D(q − x; q + x) e−i2x·p dnx, (6.15)

where we have set ξ = (q, p). Up to the sign in the exponent, the momentumrepresentation is given by the same equation.

The Wigner function plays the role of a quasi probability distribution on phasespace. In fact, it is normalized (

∫W(ξ)d2nξ = tr [ρ] = 1) and its marginal distri-

butions with respect to q and p return the usual position and momentum proba-bility distributions. However, the Wigner function may take on negative values.Whereas for pure states W(ξ) is everywhere positive iff the state is a Gaussian (seebelow)[Hud74], a characterization of mixed states with positive Wigner function isstill an open problem [BW95].

For a detailed discussion of this and other phase space distribution functions see[Lee95].

Displacement vectors and covariance matrices: In classical probabilitytheory the moments of a probability distribution are easily expressed through thederivatives of its characteristic function. An analogous relation holds for the quan-tum characteristic function χ(ξ). In particular for R′ = σR

1

i

∂

∂ξkχ(ξ)

∣∣∣ξ=0

= tr [ρR′k] , (6.16)

− ∂2

∂ξk∂ξlχ(ξ)

∣∣∣ξ=0

=1

2σkl + tr [ρR′lR

′k] =

1

2tr [ρR′k, R′l+] . (6.17)

7The name “characteristic function” is due to the analogy with classical probability theory.Classically, the characteristic function is the Fourier transform of the probability distribution, andits derivatives return the moments of the distribution. Due to non-commutativity the general-ization to the quantum world is not unique. Depending on the operator ordering (normal, anti-normal, symmetric) one obtains different characteristic functions as symplectic Fourier transformsof the respective phase space distribution functions (P-function, Q-function, Wigner function)(cf.[GZ00]). We will in the following only use the characteristic function defined in Eq.(6.12),which corresponds to a symmetric operator ordering (see Eq.(6.17)).

90


We will call the vector d with dk := tr [ρRk] the “displacement vector” and definethe “covariance matrix” γ as8

γkl := tr [ρRk − dk1, Rl − dl1+] (6.18)

= 2tr [ρ(Rk − dk1)(Rl − dl1)] + iσkl. (6.19)

Note that γ− iσ (and by complex conjugation also γ+ iσ) is a positive semi-definitematrix, since ξ · (γ − iσ)ξ = 2tr [ρA∗A] with A =

∑k ξk(Rk − dk1).

Gaussian states: Gaussian states can be defined as quantum states having aGaussian Wigner distribution

W(ξ) = c e−(ξ−d)·γ−1(ξ−d), c = π−n|γ|− 1

2 , (6.20)

where γ and d are covariance matrix and displacement vector respectively, defined asin Eq.(6.18). Hence, the first and second moments determine the state and thereforeall its higher moments. The characteristic function corresponding to a Gaussianstate is the (symplectic) Fourier transform of the Wigner function (Eq.(6.14)):

χ(ξ) = exp[iξ · d′ − ξ · γ′ξ/4

], (6.21)

where γ′ = σT γσ and d′ = σd are covariance matrix and displacement vector withrespect to the set of canonical operators R′ = σR.9 According to Eq.(6.10) thedensity matrix of the Gaussian state is then given by ρ = (2π)−n

∫χ(ξ)W−ξ d

2nξ.The following relations between the density operator of a Gaussian state and its

covariance matrix hold:

Proposition 19 Let ρ be a density operator corresponding to a Gaussian state withcovariance matrix γ. Then:

1. Positivity: A necessary and sufficient condition for ρ being a positive semi-definite operator is given by γ + iσ ≥ 0.

2. Purity: The state ρ is pure iff |γ| = 1.

3. Entanglement: The covariance matrix contains all information about the en-tanglement properties of a Gaussian state.

4. Normal mode decomposition: There exists a canonical basis in phase spacewith respective number operators nk = a∗kak, k = 1, . . . , n, such that ρ takeson the form

ρ =n⊗

k=1

eβknk

tr [eβknk ], (6.22)

where the inverse temperature βk of mode k is related to the mean number ofquanta in that mode by (eβk − 1)−1 = Nk, which is in turn equal to

12 (sk − 1)

with sk being the symplectic spectrum of γ (see the following section).Remarks : For a proof of condition 1 see [Hol82]. In fact, the inequality γ+iσ ≥ 0 isnothing but Heisenberg’s uncertainty relation. Condition 2 can easily be obtainedfrom Eq. (6.13). Concerning 3, note that displacement vectors do not contain anyinformation about entanglement since they can be changed arbitrarily by phasespace translations, which are in turn local unitary operations. Statement 4 followsfrom Williamson’s theorem (Prop.21).

8Note that the definition in Eq.(6.18) differs by a factor 2 from the one used in classicalprobability theory.

9Note that γ, d and γ′, d′ differ merely by local transformations. For the discussion of entan-glement properties of the Gaussian state one may therefore often disregard this distinction.

91

Symplectic Transformations

6.3 Symplectic Transformations

Of particular physical interest are transformations of the canonical operators, whichpreserve the canonical commutation relations and therefore leave the basic kine-matic rules unchanged. Let S : R 7→ R′ = SR be a real linear transformation, then[R′k, R

′l] = iσkl1 holds iff SσST = σ. The set of real 2n × 2n matrices satisfying

this condition form the n(2n + 1) dimensional real “symplectic group” Sp(2n,R)of “symplectic” resp. “canonical transformations”. Together with the phase spacetranslations these form the “affine symplectic group”. If S ∈ Sp(2n,R) then alsoST , S−1,−S ∈ Sp(2n,R). Moreover, the inverse of S is given by S−1 = σSTσ−1

and the determinant of every symplectic matrix is |S| = +1.It is a direct consequence of the Stone-von Neumann theorem [vN31] that every

symplectic transformation corresponds to a unitary operation acting on the repre-sentation space of the Weyl operators. This representation of the symplectic groupis sometimes called the “metaplectic representation”.

Generators and Hamiltonians: Let us now turn to the generators of thegroup, which means in physical terms to have a look at the Hamiltonians corre-sponding to symplectic transformations. The infinitesimal generators of the defin-ing representation of the symplectic group can easily be found by applying thesymplectic condition (SσST = σ) to an element of Sp(2n,R) close to the identity:S = eεJ ' 1+ εJ with |ε| << 1. This leads to (σJ)T = σJ , which means that theinfinitesimal generators are in one-to-one correspondence to real symmetric 2n×2nmatrices: we just have to multiply the latter with the symplectic form σ.

The generators of the metaplectic representation have the form 12Rk, Rl+.

Hence, every Hamiltonian which is a hermitian quadratic expression in the canon-ical operators (and thus in the annihilation/creation operators) corresponds to asymplectic transformation and vice versa. For mathematical as well as for physi-cal reasons we may express these generators in terms of the annihilation/creationoperators and divide them into two groups:

compact generators : G(1)kl = i(a∗kal − a∗l ak), (6.23)

(passive) G(2)kl = a∗kal + a∗l ak, (6.24)

non compact generators : G(3)kl = i(a∗ka

∗l − akal), (6.25)

(active) G(4)kl = a∗ka

∗l + akal. (6.26)

G(1) and G(2) generate the subgroup K(n) = Sp(2n,R) ∩ SO(2n), which is themaximal compact subgroup of Sp(2n,R) and has therefore a finite dimensionalunitary representation, which we will specify later on. The important point isthat G(1) and G(2) commute with all number operators a∗kak such that they are“passive” in the sense that they conserve the total photon number. In fact, everyHamiltonian, which is a real linear combination of the compact generators can inquantum optical settings be implemented using beam-splitters, phase shifters andmirrors only [RZ94]. Conversely, every array of passive optical elements is describedby such a Hamiltonian.

The non compact generators correspond to active elements such as scaling resp.“squeezing” transformations as known e.g. from down converters. On the levelof 2n × 2n matrices, G(3) and G(4) correspond to the subset Π(n) of symmetricand positive definite symplectic matrices [ADMS95]. Active transformations ofparticular importance are single-mode squeezing operations described by a unitary

Ur = er2(a2−a∗2) = ei

r2Q,P+ , (6.27)

92


which depend on the squeezing parameter r > 0. The behavior of creation andannihilation resp. canonical operators under this transformation is given by10

UraU∗r = a cosh r + a∗ sinh r, (6.28)

UrQU∗r = erQ, UrPU

∗r = e−rP. (6.29)

Decompositions of symplectic matrices: The possibility of dividing thegenerators into the two groups in Eqs.(6.23-6.26) leads us to another importantaspect of symplectic transformations — their decompositions (cf. [ADMS95]). The“polar decomposition” S = KP , suggested by the above grouping of the generators,with K ∈ K(n), P ∈ Π(n),

K(n) = Sp(2n,R) ∩ SO(2n), (6.30)

Π(n) = S ∈ Sp(2n,R)∣∣S = ST , S > 0, (6.31)

is unique for every S ∈ Sp(2n,R) and tells us that every symplectic transformationhas a decomposition into an active scaling/squeezing transformation and a passivepart.

Another important decomposition of similar type is the “Bloch-Messiah” or“Euler decomposition”, which implies that the “active part” of a symplectic trans-formation can be factorized into scaling/squeezing transformations, each of themacting on a single mode, i.e., a single degree of freedom:

Proposition 20 (Euler decomposition) Every real symplectic transformation S ∈Sp(2n,R) has a decomposition

S = K

(D 00 D−1

)K ′, (6.32)

where K,K ′ ∈ K(n) are orthogonal and symplectic 2n× 2n matrices and D > 0 isa diagonal matrix, which is uniquely determined up to permutations of the entries.

Within this decomposition, the “active part” of the transformation is thus givenby a tensor product of single-mode squeezing operations of the form in Eq.(6.27).Prop.20 is proven by applying the singular value decomposition to the blocks ap-pearing in the complex representation ΩSΩ∗ of the transformation S [Bra99, BM62].

The complex representation is in particular useful when dealing with passive,i.e., orthogonal symplectic transformations. It is straight forward to verify thatevery K ∈ K(n) is of the form

K = Ω∗(U 00 U

)Ω =

(X Y−Y X

), (6.33)

where X and Y are real and imaginary parts of the unitary U = X + iY . SinceK in the form of Eq.(6.33) satisfies the symplectic condition KσKT = σ for anyU ∈ U(n), we have that the orthogonal symplectic group K(n) is isomorphic to thegroup of unitaries U(n) ' K(n). The block structure in the complex representationimplies that the transformation does not mix annihilation with creation operators.

Let us now consider the behavior of the first and second moments of a stateunder symplectic transformations. From the definition of the displacement vectord and the covariance matrix γ in Eq.(6.18) we obtain that R 7→ SR implies

d 7→ Sd, γ 7→ SγST . (6.34)

10The Eqs.(6.28,6.29) follow from Eq.(6.27) by utilizing that eAeBe−A = B + [A,B] +12![A, [A,B]] + 1

3![A, [A, [A,B]]] + . . . .

93


Note that the latter is not a similarity transformation unless S ∈ K(n). However, itenables us to bring γ into a diagonal normal form corresponding to a decompositioninto uncorrelated normal modes [Wil36]:

Proposition 21 (Williamson) Every real and positive definite 2n× 2n matrix Ccan be diagonalized such that

SCST = diag(s1, . . . , sn, s1, . . . , sn), (6.35)

where S ∈ Sp(2n,R) and si > 0. The “symplectic eigenvalues” si appear aseigenvalues of the matrix (iσ−1C), such that spec(iσ−1C) = ±si.

Squeezing criteria: If the variances with respect to position and momentum ofa single mode are balanced, then the respective diagonal entries in the covariancematrix have to be larger than or equal to one. This is a direct consequence ofHeisenberg’s uncertainty relation.

If the covariance matrix γ has any diagonal entry smaller than one, the corre-sponding state is said to be “squeezed”. To depict it graphically, the ellipsoid inphase space, which is characterized by γ, is “squeezed” in the direction correspond-ing to the small diagonal entry, such that it can not contain any sphere consistentwith Heisenberg’s uncertainty relation. This fact will not change if we apply apassive symplectic transformation to the state, since passive transformations arejust norm-preserving rotations in phase space. However, it may happen that allthe diagonal entries of the covariance matrix become larger than one during thistransformation. Hence, ∃k : γkk < 1 is a sufficient but not necessary criterion forthe state to be squeezed, since passive transformations, which can be implementedusing beam splitters and phase shifters only, should be non-squeezing operations.To put it in mathematical terms, we will call a state with covariance matrix γsqueezed iff

∃S ∈ K(n), ∃k : (SγST )kk < 1 . (6.36)

As shown in [SMD94] the isomorphism K(n) ' U(n) allows us to simplify thiscondition:

Proposition 22 (Squeezing criterion) Consider a state of n modes with covari-ance matrix γ. This state is squeezed in the sense of Eq.(6.36) iff γ has at least oneeigenvalue smaller than one.

Proof: If there exists a symplectic transformation S ∈ K(n) such that Eq.(6.36)is fulfilled, then γ has clearly an eigenvalue smaller than one, since S ∈ K(n) isorthogonal and therefore preserves the spectrum of γ.

In order to show that a small eigenvalue is also sufficient for Eq.(6.36), we makeuse of the complex representation. Remember that S ∈ K(n) is of the form

S = Ω∗(U 00 U

)Ω , (6.37)

where U ∈ U(n) is any unitary. The fact that the orbit of a unit vector underU(n) is the whole complex unit sphere implies that the orbit of a real normalizedvector under K(n) is in turn the entire real unit sphere. In particular, if γ has aneigenvalue λ smaller than one, then there exists an element of K(n) which rotatesthe respective eigenvector onto the vector (1, 0, . . . , 0), such that (SγST )11 = λ.

The above proof also shows that the minimum eigenvalue of the covariance ma-trix quantifies the strength of squeezing with respect to the most squeezed direction.

94

Bound entangled Gaussian states

6.4 Bound entangled Gaussian states

For finite dimensional bipartite quantum systems the positivity of the partial trans-pose is sufficient for separability only for the lowest dimensional cases 2 ⊗ 2 and2⊗ 3. For all higher dimensions there exists bound entanglement.

For Gaussian systems it was shown by Simon [Sim00] and independently byDuan et al. [DGCZ00], that the PPT criterion is again sufficient for the smallestbipartite case of 1 × 1 modes. However, it remained unclear, whether this is stilltrue if we increase the number of modes. Moreover, although it was known thatthe covariance matrix carries all the information about entanglement properties inprinciple, there was no method of expressing separability in terms of this matrix.The following letter tackles these two problems:

• The definition of separability is translated into the covariance matrix represen-tation for Gaussian states and expressed as a matrix inequality for covariancematrices. This leads to a necessary separability criterion for non-Gaussiancontinuous variable states as well.

• For Gaussian states with 1 × n modes positivity of the partial transpose isnecessary and sufficient for separability.

• For bipartite Gaussian states with at least two modes per site, there existentangled PPT states. A five parameter family of such states for 2× 2 modesis explicitly constructed.

95

VOLUME 86, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 16 APRIL 2001

Bound Entangled Gaussian States

R. F. Werner* and M. M. Wolf†

Institut für Mathematische Physik, TU Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig, Germany(Received 4 October 2000)

We discuss the entanglement properties of bipartite states with Gaussian Wigner functions. For theseparability, and the positivity of the partial transpose, we establish explicit necessary and sufficientcriteria in terms of the covariance matrix of the state. It is shown that, for systems composed of asingle oscillator for Alice and an arbitrary number for Bob, positivity of the partial transpose impliesseparability. However, this implication fails with two oscillators on each side, as we show by constructinga five parameter family of bound entangled Gaussian states.

DOI: 10.1103/PhysRevLett.86.3658 PACS numbers: 03.65.Ud, 03.67.–a

I. Introduction.—Many experiments in the young fieldof quantum information physics are not carried out onfinite-dimensional quantum systems, for which most of thebasic theory has been developed, but in the quantum opticalsetting. In that setting the basic variables are quadraturesof field modes, which satisfy canonical commutation rela-tions, and hence have no finite-dimensional realizations. Itwould seem that the theory therefore becomes burdenedwith all the technical difficulties of infinite-dimensionalspaces, while theoreticians are, on the other hand, stillstruggling to answer some simple questions about qubitsystems. However, the states relevant in quantum optics areoften of a special type, and for this class the typical ques-tions of quantum information theory are luckily of the samecomplexity as for the usual finite-dimensional systems.

This simple class of states of “continuous variable sys-tems” is the class of Gaussian states, i.e., those stateswhose Wigner function is a Gaussian on phase space. Sucha state is therefore completely specified by its mean and itscovariance matrix, where the mean is irrelevant for entan-glement questions, because it can be shifted to zero by alocal unitary (phase space) translation. It turns out that thebasic entanglement properties of a Gaussian density matrix(as a state on two infinite-dimensional Hilbert spaces) canbe translated very nicely into properties of its covariancematrix (see Section II), so that problems involving Gauss-ian states are reduced to problems of finite-dimensionallinear algebra rather reminiscent of the problems involv-ing finite-dimensional density matrices.

For the latter it is well known [1,2] that the positivityof the partial transpose (“ppt”) is necessary for separabil-ity, but sufficient only for the smallest nontrivial systems,namely, systems in dimensions 2 ≠ 2 and 2 ≠ 3. In allhigher dimensions we can find “bound entangled states,”which are not separable, but nevertheless have a positivepartial transpose, and are hence not distillable [3]. In thecase of continuous variable systems the first nontrivial ex-amples of this kind were obtained in [4]. In the Gaussiansetting it was shown by Simon [5] that for bipartite systemswith one canonical degree of freedom on each side, i.e.,once again for the simplest possible systems, the equiva-lence of ppt and separability also holds. For this system

another equivalent entanglement criterion was provided in[6], and it was also shown that non-ppt states are indeeddistillable [7]. In this paper we settle the relationship be-tween separability and ppt for all higher dimensions, show-ing that the equivalence holds also for systems of 1 3 N

oscillators, but fails for all higher dimensions. We showthis by giving explicit examples for 2 3 2 oscillators.

The key idea for constructing bound entangled Gaussianstates is the notion of “minimal ppt” covariance matrices.These are defined as the covariance matrices of ppt Gauss-ian states, which are not larger (in matrix ordering) thanthe covariance matrix of any other ppt Gaussian state. It iseasy to see that a minimal ppt covariance matrix belongs toa separable state if, and only if, that state is a product state.Hence bound entangled Gaussians arise from all minimalppt covariance matrices, which are not block diagonal.Numerically, minimal ppt covariance matrices can be ob-tained very efficiently by successively subtracting rank oneoperators from a given covariance matrix. This algorithmis reminiscent of techniques for density matrices in the con-text of “best separable approximation” [8]. Running thisprocedure for 2 3 2 or larger systems generically givesbound entangled Gaussian states.

Our paper is organized as follows: In Section II we willset up the basic notation and the translation of separabilityand ppt conditions into properties of covariance matrices(for separability this appears to be new). We also describethe minimal ppt covariance matrices. In Section III weprove the equivalence for the 1 3 N case, and in Sec-tion IV we present a five parameter family of 2 3 2 boundentangled states.

II. Gaussian states and entanglement.—A system of f

canonical degrees of freedom is described classically in aphase space, which is a 2f-dimensional real vector spaceX. The canonical structure is given by a 2f 3 2f matrixs, known as the symplectic matrix, which is antisymmetricand nonsingular. With a suitable choice of coordinates(“canonical coordinates”), it can be brought into a standardform: The 2f variables are then grouped into f canonicalpairs (e.g., position and momentum), for each of whichthe symplectic matrix takes the form s 0

1210 , and all

other matrix elements vanish.

3658 0031-90070186(16)3658(4)$15.00 © 2001 The American Physical Society


The symplectic matrix also governs the canonical com-mutation relations for the corresponding quantum system:if Ra , a 1, . . . , 2f are canonical operators (for canoni-cal coordinates these are naturally grouped into f standardposition operators and f standard momentum operators),the commutation relations read

iRa , Rb sab1 . (1)

These relations may be exponentiated to the Weyl rela-tions involving unitaries Wj expij ? s ? R, wherej [ X, and j ? s ? R

Pab jasabRb . These Weyl

operators implement the phase space translations. We willassume that they act irreducibly on the given Hilbert space,i.e., that there are no further degrees of freedom. Then byvon Neumann’s Uniqueness Theorem [9] the Ra are uni-tarily equivalent to the usual position and momentum op-erators in the L 2 space over position space.

For a general density operator r we define the mean asthe vector ma trrRa, and the covariance matrix g,

gab 1 isab 2 trrRa 2 ma1 Rb 2 mb1 , (2)

which is well defined whenever all of the unbounded posi-tive operators R2

a have finite expectations in r. Becauseof the canonical commutation relations the antisymmetricpart of the right-hand side is indeed the symplectic matrix,independently of the state r. The state-dependent covari-ance matrix g is therefore real and symmetric. Moreover,g 1 is is obviously positive definite.

A Gaussian state is best defined in terms of its character-

istic function, which for a general state is j trrWj.This should be seen as the quantum Fourier transform [10]of r, and is indeed the Fourier transform of the Wignerfunction of r. Hence we call r Gaussian, if its character-istic function is of the form

trrW j expimTj 21

4jT gj . (3)

Here the coefficients were chosen such that g and m areindeed the covariance and mean of r, as is readily verifiedby differentiation. The necessary condition g 1 is $ 0,which is equivalent to g 2 is $ 0 by complex conjuga-tion, is also sufficient for Eq. (3) to define a positive opera-tor r. We note for later use that a Gaussian state is pureif, and only if, s21g2

21 [11], which is equivalentto g 1 is having the maximal number of null eigenvec-tors, i.e., the null space N F j g 1 isF 0 hasdimension dimX2. Note that this null space must al-ways be considered as a complex linear subspace of 2f ,the complexification of X. For such a complex subspacewe denote by ReN the subspace of X consisting of allreal parts of vectors in N . Then a Gaussian state is pureif, and only if, ReN X.

Let us now consider bipartite systems. The phase spaceis then split into two phase spaces X XA © XB, whereA stands for Alice and B for Bob. This is a “symplectic

direct sum,” which means that s sA © sB is block di-agonal with respect to this decomposition. In other words,Alice’s canonical operators Ra commute with all of Bob’s.The Weyl operators are naturally identified with tensorproducts: WjA © jB WjA ≠ WjB. We call thisan fA 3 fB system, if dimXA 2fA, and dimXB 2fB.

It is clear from (2) and (3) that the covariance matrix of aproduct state is block diagonal, and, conversely, a Gaussianstate with block diagonal g is a product state. Separabilityis characterized as follows:

Proposition 1: Let g be the covariance matrix of a

separable state with finite second moments. Then there

are covariance matrices gA and gB such that

g $

√gA 0

0 gB

!. (4)

Conversely, if this condition is satisfied, the Gaussian state

with covariance g is separable.

In order to show the first statement, suppose that thegiven state is decomposed into product states with co-variance gk and mean mk with convex weight lk . Thenma

Pk lkmk

a and, similarly, for the second momentswe have

gab 1 2mamb

Xk

lkgkab 1 2mk

amkb . (5)

Hence the difference between g and the block diagonalPk lkgk is the matrix

Dab 2

√Xk

lkmkamk

b 2Xk

lklmkam

b

!, (6)

which is positive definite, becauseP

jajbDab Pk lklsk 2 s2 $ 0, where sk

Pa jama .

In order to show the converse, let s be the Gaussianproduct state with covariance gA © gB, and let g0

g 2gA © gB $ 0. Then g0 is the covariance of a classicalGaussian probability distribution P, and the characteristicfunction of the given state r is the product of the character-istic function of s and the Fourier transform of P. Hencer is the convolution of s, and P in the sense of [10], whichis the average of the phase space translates WjsW j

over j with weight P. Since all these states will be productstates, r is separable.

There are different ways of characterizing the partialtranspose. One simple way is to say that with respect tosome set of canonical coordinates the momenta in Alice’ssystem are reversed, while her position coordinates and allof Bob’s canonical variables are left unchanged. In addi-tion, the order of factors in the partial transpose of RaRb

is reversed when both factors belong to Alice. When wereplace r in (2) by its partial transpose, we therefore findthe antisymmetric part of the equation unchanged, whereasgab picks up a factor 21 whenever just one of the in-dices corresponds to one of Alice’s momenta. Let us call

3659


the resulting covariance matrix by eg. Clearly, if the par-tial transpose of r is again a density operator, we musthave eg 1 is $ 0. But this is equivalent to g 1 i es $ 0,where in es the corresponding components are reversed, sothat es 2sA © sB. This form of the condition is valideven if we do not insist on canonical variables. Combiningit with the positivity condition for Gaussian states we getthe following characterization:

Proposition 2: Let g be the covariance matrix of a

state, with finite second moments, which has positive par-

tial transpose. Then

g 1 i es $ 0, where es

√2sA 0

0 sB

!. (7)

Conversely, if this condition is satisfied, the Gaussian state

with covariance g has positive partial transpose.

When r is separable, Proposition 2 shows the existenceof a block diagonal g0

gA © gB with g $ g0. SincegA and gB are covariance matrices in their own right, wehave gA 6 isA $ 0, and similarly for Bob’s side. Butthis means that g $ g0 $ 2is and g $ g0 $ 2i es, andg has positive partial transpose, as a separable densityoperator should. We have made this explicit, because itshows that it may be interesting to see how much “space”there is between g and 2is and 2i es. This leads to thecentral definition of this paper:

Definition: We say that a real symmetric matrix g is

a ppt covariance, if g 1 is $ 0 and g 1 i es $ 0, and

that it is minimal ppt, if it is a ppt covariance, and any ppt

covariance g0 with g $ g0 must be equal to g.

Note that a minimal ppt matrix g is separable if, andonly if, it is a direct sum, i.e., if the corresponding statefactorizes (and is thus a product of pure states). There is arather effective criterion for deciding whether a given pptcovariance is even minimal ppt: First, if there were anyg0 # g with g0 fi g, we can also choose g 2 g0

D tobe a rank one operator, i.e., a matrix of the form Dab

jajb . Second, we have g 1 is $ eD for sufficientlysmall positive e if, and only if, j is in the support of thepositive operator g 1 is. The same reasoning applies toes, so that g is minimal ppt if, and only if, there is noreal vector j, which is in the support of both g 1 is andg 1 i es. Rephrasing this in terms of the orthogonal com-plements of the supports, we get the following character-ization which we will use later:

Proposition 3: Let g be a ppt covariance, and let Nand fN denote the null spaces of g 1 is and g 1 i es,

respectively. Then g is minimal ppt if, and only if, ReNand Re fN span X together.

This gives an effective procedure to find a minimal pptg0 below a given g: in each step one subtracts the largestadmissible multiple of a rank one operator with vector

j orthogonal to the span of ReN and Re fN , which isthen in the supports of g 1 is and g 1 i es. This will

either increase N or fN , so that a minimal ppt covariancematrix is reached after a finite number of steps.

III. The 1 3 N case.—This section is devoted to theproof that, for Gaussian states of 1 3 N systems, whereN is arbitrarily large, ppt implies separability. It is clearfrom the previous section that this is equivalent to sayingthat every minimal ppt covariance matrix is block diagonal,i.e., belongs to a product state. So throughout this sectionwe assume that g is a minimal ppt covariance matrix.

As a first step we get rid of irrelevant pure state factorsin the following sense: Suppose that the two null spaceshave a nontrivial intersection, i.e., there is a F fi 0 with

F [ N > fN . Then s 2 esF ig 2 igF 0,so F has nonzero components only in Bob’s part of thesystem. So let XC denote the subspace of XB spanned bythe real and imaginary parts of F. Then the restriction ofthe state to the subsystem C satisfies the pure state con-dition (its covariance matrix gC 1 isC has a null vectorby construction). It follows that the density matrix factor-izes: rA,BnC,C rA,BnC ≠ rC , where rC is a pure state.(This conclusion can also be obtained purely on the levelof covariance matrices, by introducing in XB a basis ofcanonical variables containing a canonical basis of XC .)Clearly, the separability of such a state is equivalent to theseparability of r, and the covariance matrix restricted toXA © XBnC is again minimal ppt. Hence we have reducedthe problem to the analogous one for the smaller spaceXA © XBnC .

We may therefore assume that the null spaces N andfN have trivial intersection. This means that we proceed

by contradiction, since we want to prove ultimately thatthe state is a product of “irrelevant pure state factors.”

Now let 0 fi F [ N and 0 fi eF [ fN . Then, be-

cause g is Hermitian, we have eF, gF g eF, F. Us-ing the null space conditions and the skew hermiticity ofs, we can rewrite this as

eF, s 2 esF 0 . (8)

Now the vector s 2 esF must be nonzero, since other-

wise we would have F [ N > fN . This is a conditionon the XA components FA of F, since s and es differonly on that two-dimensional subspace. By the same to-

ken the XA component eFA of eF must be nonzero. Henceall vectors s 2 esF lie in the one-dimensional subspace

of XA orthogonal to eFA fi 0. The proportionality con-stant is thus a linear functional on N vanishing only forF 0, which means that N must be one dimensional.

By symmetry, dim fN 1. By Proposition 3 the spaces

ReN and Re fN together span X, and, since they are twodimensional, it follows that dimX # 4, i.e., we can haveat most a 1 3 1 system. For such systems our claim hasbeen shown by Simon [5], and is hence proved.

IV. 2 3 2 bound entangled states.— It was alreadymentioned in the introduction that numerical examples of

3660


minimal ppt covariances, which do not split into gA © gB

are easily generated by the subtraction method. In con-trast, the subtraction method for 1 3 N systems alwaysends up at a block diagonal g. This is rather striking, butnot really conclusive, because the numerical determina-tion of the null space of a matrix which may have smalleigenvalues may depend critically on rounding errors. Wehave therefore prepared the following all-integer 2 3 2

example g:

g

0BBBBBBBBBBBB@

2 0 0 0 1 0 0 0

0 1 0 0 0 0 0 21

0 0 2 0 0 0 21 0

0 0 0 1 0 21 0 0

1 0 0 0 2 0 0 0

0 0 0 21 0 4 0 0

0 0 21 0 0 0 2 0

0 21 0 0 0 0 0 4

1CCCCCCCCCCCCA

. (9)

The key to getting simple examples is symmetry, whichin turn simplifies the verification of the basic properties.The most important symmetry in the example is themultiplication operator S with diagonal matrix elements1, 1, 21, 21, 1, 21, 21, 1. It satisfies Ss 1 esS 0,and Sg gS. Consequently, g 1 is and g 2 i es

Sg 1 isS are unitarily equivalent, so it suffices tocheck the positivity and to compute the null space ofg 1 is. We note in passing that this unitary equivalenceis not necessary for bound entangled Gaussians, sincegenerically the spectra of g 1 is and g 1 i es aredifferent.

Further unitaries commuting with the covariance matrix(9) are the multiplication operator C with diagonal matrixelements 1, 21, 1, 21, 1, 21, 1, 21, the skew symmetricoperator R with R13 R24 R75 R86 1, and zeroremaining entries. All these operators have square 61,and commute with each other and the symplectic formsup to signs. Therefore, if we start with a generic vectorV1 [ N , the application of R, C, S, and products of theseoperators yields eight vectors Vi , which form a basis of 8.

Since these vectors lie in either N , fN or their complexconjugates, we know how the covariance matrix acts onthem and it is thus determined by g LV21, where Land V denote the matrices consisting of column vectorsLk gVk (expressed in terms of s, es). The above g isgenerated in this manner from

V1 21, i, 2, 23i, 1, 2i, 1, 0 . (10)

Then the condition of Proposition 1 is satisfied by con-struction, and we have only to verify that g 1 is $ 0,which is again simplified by this operator commuting withR. Explicitly, we get the eigenvalues 0, 3 2

p3, 3, 3 1p

3, each with multiplicity 2.By generalizing this example, we can construct a five

parameter family of bound entangled Gaussian states com-muting with R, S, and C in the same manner as above. Westart with a generic vector

V1 2a, ib, c, 2id, e, 2if, 1, 0, a, b, . . . , f . 0 .

(11)

Then g being real and symmetric requires d bc 1fa, and from the characteristic function of Vk , Ll 1isVl we obtain that g 1 is $ 0 if, and only if, a # ce,where equality is ruled out since this would be equivalentto detV 0.

States obtained from (11) are all of a nonblock diagonalform similar to (9), and are hence bound entangled.

Funding by the European Union Project EQUIP (Con-tract No. IST-1999-11053) and financial support from theDFG (Bonn) are gratefully acknowledged.

*Electronic address: [email protected]†Electronic address: [email protected]

[1] A. Peres, Phys. Rev. Lett. 77, 1413 (1996).[2] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.

Lett. A 223, 1 (1996).[3] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.

Lett. 80, 5239 (1998).[4] P. Horodecki and M. Lewenstein, Phys. Rev. Lett. 85, 2657

(2000).[5] R. Simon, Phys. Rev. Lett. 84, 2726 (2000).[6] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys.

Rev. Lett. 84, 2722 (2000).[7] G. Giedke, L.-M. Duan, J. I. Cirac, and P. Zoller, quant-

ph/0007061.[8] M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261

(1998).[9] J. von Neumann, Math. Ann. 104, 570 (1931).

[10] R. F. Werner, J. Math. Phys. 25, 1404 (1984).[11] A. S. Holevo, Probabilistic and Statistical Aspects of

Quantum Theory (North-Holland, Amsterdam, 1982),Chap. 5.

3661

The entangling power of passive optical elements

6.5 The entangling power of passive optical ele-

ments

It is well known that entanglement between two modes can be generated by send-ing two light field modes, which are in a pure state and squeezed in orthogonaldirections, through a beam-splitter. Hence, we can in this way convert one resource(squeezing) into another resource (entanglement) by utilizing a rather cheap passiveoptical element (a beam-splitter). The following letter investigates such processes ina very general form: The input is a general mixed Gaussian state of n modes. Theoperations are arbitrary arrays of passive optical elements (beam-splitter, phaseshifters and mirrors), and the output should be a non-PPT (and thus distillable11)state with respect to a given bipartite partition of the n modes. The main resultsare:

• We provide a simple necessary and sufficient condition for the possibility ofcreating distillable (non-PPT) entanglement between any bipartite split of ageneral n mode Gaussian state by means of passive optical elements.

• We calculate the maximal amount of entanglement which can in such a way beobtained for an arbitrary two-mode subsystem. The entanglement is measuredin terms of the “logarithmic negativity”.

• We provide the optimal entangling operation for the two-mode case.

11It was proven in [GDCZ01] that a bipartite Gaussian state is distillable iff it has non-positivepartial transpose. However, distillation is not possible with “Gaussian operations” [CG02, ESP02].

101


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Michael M. Wolf, Jens Eisert

, and Martin B. Plenio

1 Institute for Mathematical Physics, TU Braunschweig, 38106 Braunschweig, Germany

2 QOLS, Blackett Laboratory, Imperial College of Science, Technology and Medicine, London, SW7 2BW, UK(October 11, 2002)

We investigate the entangling capability of passive optical elements, both qualitatively and quantitatively.We present a general necessary and sufficient condition for the possibility of creating distillable entanglement inan arbitrary multi-mode Gaussian state with the help of passive optical elements, thereby establishing a generalconnection between squeezing and the entanglement that is attainable by non-squeezing operations. Specialattention is devoted to general two-mode Gaussian states, for which we provide the optimal entangling proce-dure, present an explicit formula for the attainable degree of entanglement measured in terms of the logarithmicnegativity, and discuss several practically important special cases.

Entangled states of light field modes may be generated bytransmitting two squeezed states through a beam splitter [1].This is one of the experimentally accessible procedures forgenerating continuous-variable entanglement in optical sys-tems [2]. Moreover, it is a particular example of a situationwhere passive optical elements exhibit their entangling powerwhen applied to Gaussian input states. It is well known thatthe presence of squeezing is necessary for obtaining entan-glement in this manner [1]. However, the degree of the at-tained entanglement is by no means the same for all inputstates: it depends to a large extent on the degree and direc-tion of squeezing of the incoming modes and on the specificproperties of the beam splitter. This raises the question underwhat circumstances such an entangling procedure is optimalin the sense of generating states which have the maximal at-tainable amount of entanglement. And in general, by meansof arbitrary passive optical elements, what are the require-ments such that entanglement can be generated between anybi-partite split of a system in a multi-mode Gaussian state?

In this letter we address the question of the entanglingpower of passive optical elements acting on any number ofmodes in an arbitrary Gaussian state, qualitatively as well asquantitatively. Passive optical operations can be implementedby using beam splitters and phase shifters [3]. These are cheapoperations and easy to implement in contrast to squeezing op-erations. Therefore we will consider squeezing as a potentialresource for entanglement and ask for the requirements andthe optimal way of entangling a squeezed state by means ofpassive operations, which we assume to be available in arbi-trary quantities. The main result and starting point is a nec-essary and sufficient condition for the possibility of creatingdistillable entanglement on general Gaussian initial states –pure or mixed – between any bi-partite split of an -modesystem with the help of passive optical elements. We thenintroduce a lower bound for the attainable degree of entangle-ment measured in terms of the logarithmic negativity [4] for -mode systems. Moreover, we derive a general formula forthe largest degree of entanglement of an arbitrary subsystemconsisting of two modes. The operations that can be imple-

mented with passive optical elements can be identified withthe non-squeezing operations. In this sense we establish aquantitative connection between the degree of squeezing ofa Gaussian state and the degree of entanglement that is attain-able with the application of non-squeezing operations. Of par-ticular interest is the case where only two modes are present.We will discuss this situation in more detail by explicitly con-structing the optimal entangling procedure and discussing sev-eral meaningful special cases.

Bn

An

nPassive optical elements

ϕ

FIG. 1. field modes in a Gaussian state can be(NPPT)-entangled with respect to a partition into modes by means of passive optical elements if and only if ,where and are the two smallest eigenvalues of the covariancematrix associated with .

We start by introducing the formalism that we will use ex-tensively. Gaussian states are completely characterized bytheir first and second moments, where only the latter, givenin terms of a covariance matrix , carry information about en-tanglement and squeezing. For this reason we will set the firstmoments to zero, which can always be achieved by unitary op-erations on individual modes. The covariance matrix is thengiven by ! #"%$&$')(+*-,/.0 , 1243576208882 , where thevector $9;:)< 280882<>=2@? 28882@?+=A consists of the canon-ical coordinates for modes and the symplectic matrix

.B CEDGF =H F = D!I (1)

governs the canonical commutation relations (CCR)J $'#2$' KLM,N. . A matrix represents an admissible covari-ance matrix if it satisfies the Heisenberg uncertainty relationsB*O,/.-P D . Symplectic transformations -QHRTS5U S pre-serve the CCR and therefore satisfy S5U . S V. [5]. Allsymplectic transformations correspond to unitary Gaussian

1

operations [6] on the level of states, in the sense that theGaussian character of arbitrary input states is preserved undersuch unitary operations. They can be decomposed [5,7] intoactive/nonlinear and passive/linear operations [8]. The lattercan be implemented by using passive optical elements such asbeam splitters and phase plates only [3], and are of the formQHWRYXZU X ,

X 9[\ C] DD ] I [^ C`_baH ac_I 8 (2)

Here] _ *^, a (

_ 2 a real) is any unitary matrix and

[- 6d CeF =E, F =F = H , F = I (3)

relates real and complex representations by mapping cre-ation/annihilation operators to position/momentum operatorsvia [>:/$ 288082$ =#A U f:/g 280882g=2@g \ 28882@g \= A U . Trans-formations of the type hQHRiXBU X with X as above willfrom now on be denoted as passive transformations. Any suchX is both symplectic and orthogonal, i.e., XUjX F , and theset of all symplectic transformations that can be implementedwith passive optical elements form a group, the maximal com-pact subgroup X :/A of the group of symplectic transforma-tions Sk :/ l2mnA [5].

A Gaussian state is said to be squeezed if there exists a ba-sis in phase space such that at least one diagonal element ofthe covariance matrix is smaller than one. From now on weorder the eigenvalues of in non-increasing order, so that thisimplies that the smallest eigenvalue o of is smaller thanone [5]. Since every passive transformation X is orthogonal,it does not affect the squeezing of a state.

Let us now turn to entanglement properties. A Gaussianstate of a bi-partite system consisting of parts p and q withre*sht modes is separable, i.e. unentangled betweenp and q , iff there exist covariance matrices +r&2s for rresp. s modes such that P9+rvu-s [9,10]. A necessaryand for 6wZx modes also sufficient condition for separability[9,11] is that the partial transpose of the state is positive semi-definite. This, in turn is equivalent to 9Py,z. , with the par-tially transposed symplectic matrix z.Z : F =>u|A4.l: F =>u|Aand ! F =#nuO: H F =#~5A being the partial transposition oper-ator that reverses all momenta on one side. It has been shownthat a Gaussian state is distillable, i.e., that its entanglementcan be revealed using local operations and classical commu-nication, iff its partial transpose is non-positive [12].

Obviously, every entangled Gaussian state is squeezedsince o P6 would mean that P F u F which in turnimplies separability. Hence, a state can only be entangled bymeans of passive operations if it is squeezed initially. The fol-lowing proposition gives a necessary and sufficient conditionfor the possibility of transforming a general Gaussian stateinto a distillable one by means of passive transformations (seeFig. 1):

Proposition 1 Let be a covariance matrix correspondingto a Gaussian state of modes. A passive transformation

^QHR XBU X leading to an entangled state having anon-positive partial transpose with respect to a partition intor*^sh modes exists iff

o o ' 62 (4)

where o 2o are the two smallest eigenvalues of .

Proof: A Gaussian state of an -mode system with covariancematrix has a positive partial transpose iff all symplecticeigenvalues of the respective partially transposed covariancematrix z Y: F =Zu9|A4 : F =Zu!A are larger than or equalto one. The symplectic eigenvalues of zW are in turn equalto the square roots of the ordinary eigenvalues of H :)54z.A .The square of the smallest symplectic eigenvalue additionallyminimized over all passive transformations is thus given by h " W @@ U 4 (2 (5)

where X z. XZU and 8 denotes the standard vector

norm. Hence, we have to show that inequality (4) is equivalentto 6 . Since

is an antisymmetric orthogonal matrix, it

maps any real unit vector onto the :/ H A -dimensional unitsphere of its orthogonal complement. The vector

: X 2%A4( 4 (l")W (@ @ (6)

therefore satisfies ")W 4 N: X 2%A4( D . Inserting Eq. (6) inEq. (5) we get h # " W (@") : X 2A : X 2%A4( (7)

P # @ ")W (") (2 (8)

where the infimum in Eq. (8) is taken over all real unit vec-tors satisfying " W 4 ( D

. This relaxes the require-ment that has to be of the form in Eq. (6) and there-fore leads to the lower bound. The minimum in Eq. (8) isnow attained for vectors lying in the two-dimensional spacecorresponding to the two smallest eigenvalues o 2o of .Hence, (n j o (*4 ' j o ( for some and )(n¡d o 4 ' j o ( H d o + j o ( . However, every leads tothe same value and we have Pho o 2 (9)

showing that o o > 6 is indeed necessary for 6 .In order to prove sufficiency, we have to show that there

always exists a passive transformation X such that is ofthe form (6) and the inequalities (8,9) thus become equali-ties. Note that this is in turn equivalent to the statement thatfor every pair of orthogonal real unit vectors £¢`¤ there isa passive transformation X such that "%¤ X z. XBU ('f6 . Toprove this equivalent statement, let us first restate the prob-lem using complex variables. Decompose 2¤ into positionand momentum components and define ¥ ¦ §5¨j*!, ¦ ©5¨and ª t¤¦ §5¨5*,N¤¦ ©5¨ such that [| ('y[| ¦ §5¨u¦ ©5¨(>

2

« ¥u ¥( and analogous for ¤ and ª . Then ¥¬ 7 ª| 0 W 7 ¤W 6 and

"%¤ X z. X U ( ImJ "%ª| ] ] \ ¥L(NK%2 (10)"N¤ ( ReJ "%ª| ¥>(NKW® D 8 (11)

We now decompose]

into two unitaries] °¯!± and

choose ¯ such that the vectors ¯ \ ª( and ¯ \ ¥>( have onlytwo non-vanishing components in rows where has oppositesigns. ± can then be chosen to be a block matrix consist-ing of identity blocks and a Zw² unitary matrix ± ¦ ¨ whichis such that it solves the reduced problem for `³ . Inorder to see that the latter is indeed always possible we fixw.l.o.g. ¥ ¦ ¨ ³:462 D A U , which can always be achieved byapplying an additional unitary in the two dimensional sub-space. Then, every two-dimensional unit vector ª ¦ ¨ forwhich Re

J "%ª ¦ ¨ ¥ ¦ ¨ (NKW D is of the form

ª ¦ ¨ H ,4::/ 0´WA2µ 4¶ · 4 5:/ Á4A U 8 (12)

Choosing

± ¦ ¨ C µ ¶)· ´ H µ ¶)· ´µ ¶)· 4 ´ µ ¶)· ´ I 2 (13)

we obtain with ¦ ¨ diag :462 H 6A6¸ Im

J "%ª ¦ ¨ ± ¦ ¨ ¦ ¨ ± \¦ ¨ ¥ ¦ ¨ (NKW;"N¤ X z. X U (2which completes the proof.

Whereas every entangled state is squeezed, Proposition 1implies that conversely a squeezed state can be entangled byusing passive optical elements supplemented by a single addi-tional vacuum mode (empty port of a beam splitter), becausethe joint covariance matrix ¹u F [10] then satisfies inequal-ity (4). Moreover, the proof of proposition 1 leads to a lowerbound for the attainable entanglement measured in terms ofthe logarithmic negativity [4]. The latter is so far the only cal-culable entanglement measure for mixed Gaussian states. Foran -mode Gaussian state º it is given by

¸»t He¼ ¶ min : D 2½ ¾ :/¿ ¶ A4A8 (14)

where the ¿ ¶ , ,y6208882 are the symplectic eigenvalues ofthe partially transposed covariance matrix. Since Oo o isthe square of the smallest symplectic eigenvalue, we obtain

» P^ÀÁ0Â: D 2 H ½ ¾ :/o o A4Ã A (15)

for the attainable entanglement, with equality if there is onlyone ¿ ¶ smaller than one. A particularly transparent situationis now the case where we consider only the entanglementpresent in an arbitrary two-mode subsystem obtained whentracing out the other modes at the end.

Proposition 2 Let !QHR ' XZU X be a passive trans-formation acting on a Gaussian state of !P modes with

covariance matrix . The maximum attainable amount of en-tanglement obtained for an arbitrary two-mode subsystem of is then given by»thÀBÁÂ J D 2 H ½ ¾ :/o o A4Ã K%2 (16)

where o 2o are the two smallest eigenvalues of .

Proof: First note that for the case of a two-mode state onlyone of the two symplectic eigenvalues ¿ 2¿ of the partiallytransposed covariance matrix zW ¦ ¨ can be smaller than one,

since :)¿ ¿ A MÄÅÆ z ¦ ¨ fÄÅÆ ¦ ¨ PY6 [13]. Moreover,the two smallest eigenvalues of any principal submatrix of 5are always larger than or equal to o 2o . Hence, it remains tobe shown that there is always a passive transformation X suchthat the two smallest eigenvalues of the covariance matrix + ¦ ¨corresponding to an arbitrary two-mode subsystem are equalto o 2o . To this end let o ( and o ( be the eigenvectors of corresponding to o and o , with complex versions

¥ ¶ (lt o ¦ §5¨¶ (*-, o ¦ ©5¨¶ (2T,+;62 8 (17)

Then there is always a unitary]

such that the vectors] ¥ ¶ (

have only two non-zero components in two arbitrary rows, say and . However, via the relation (2) this unitary corre-sponds to a passive transformation which is, by construction,such that the two-mode submatrix of + (for mode and )has eigenvalues o and o .

A special instance of proposition 2 is the case where the in-put already is a two-mode system, i.e. ` . For this casewe will now explicitly construct the optimal entangling pro-cedure. We will show that it is always sufficient to performa single phase rotation in one of the two modes, for examplein p , succeeded by a beam splitter operation on both modes.Again, it is most convenient to employ the complex version ofthe problem. In their complex forms, a beam splitter qÇ: Á anda phase shift È:/ÉlA in system p are represented by the matrices

qn: ´WA C ´ H 4 ´4 ´Ê´ I 2BÈ&:)ÉlAjC µ ¶)· DD 6 I 2 (18)

where ´ÌË J D 2 ÍA determines the transmission coefficient ofthe beam splitter, and É²Ë J D 2ÍA is the phase difference of theincoming and outgoing fields. W.l.o.g., the beam splitter itselfis assumed to induce no phase difference.

Proposition 3 Let º be a Gaussian 6&we6 -mode state with co-variance matrix . Let o ( and o ( be the eigenvectors ofthe two smallest eigenvalues of , with complex versions ¥ (and ¥ ( as in Eq. (17). The optimal entangling operation us-ing only passive optical elements is given by a phase rotationÈ:/ÉlA on mode A, followed by a beam-splitter qn: ´Ã A , suchthat ´ and É are the solutions of0: Á Im

J "%¥ .Î ¥ (NK42 (19) 5:/ÉlA4 5: Á ImJ "%¥ .Ï ¥ (NK42 (20):/ÉlA4 5: Á ImJ "%¥ .ÐW ¥ (NK2 (21)

where .Ð , .Ï , and .Î are the Pauli spin matrices.

3

Proof: In order to find the optimal entangling procedure onehas to identify a unitary ± such that

Im Ñ/"%¥ ±> ¦ ¨ ± \ ¥ (%Ò 62 (22)

and decompose it into a beam splitter and a phase shift. Themost general form for ±> ¦ ¨ ± \ Ó is given by

Ó : ´2ÉlA C 0: ÁÔµ ¶)· 5: ´WAµ ¶)· 4 5: Á H : ´WA I 2 (23)

which corresponds to ±ÕÕÈ:/ÉlAqÇ: ´Ã A . Inserting thedecomposition Ó : ´52@ÉAhT0: Á4.ÎB* :/ÉA# 5: ´WA.Ðv*4 :/ÉA# 5: ´WA.Ï into Eq. (23) one verifies that values É2´ thatsatisfy Eqs. (19-21) provide a solution of Eq. (23). Moreover,the set of equations (19-21) always has a solution, since thevector of the imaginary parts in Eqs. (19-21) can be shown tobe a unit vector if Re

J "%¥ ¥ (NKW D .We will in the following apply this result to some special

cases. The covariance matrix of the initial state of the 6lwÇ6 -mode system will be written in the block form

e C p×ÖÖ U q I 2 (24)

where p and q are the reduced covariance matrices corre-sponding to mode p and q respectively. Depending on theform of the nw² - matrices p , q , and Ö several optimal en-tangling protocols can be identified:

(i) A product of arbitrary single mode Gaussian states: IfÖ D , then a Ø D Ø D beam-splitter is required in the opti-mal entangling procedure. The phase transformation that isneeded will in general depend on the actual form of p and q .In particular:

(ia) A product of two identical single mode states: In thiscase p99q and Ö9 D , and one finds that Év´ZOÍÃ . Theoptimal entangling operation is thus a Ø D Ø D beam-splitter,which follows a É£ ÍÃ phase transformation, as expected.This is the optimal procedure for uncorrelated identical Gaus-sian input states used in several experiments [2].

(ib) A product of a Gaussian single mode state and a co-herent or thermal state: In this case where qÙ F , ÙnPM6 ,and Öy D the optimal entangling operation is again the ap-plication of a Ø D Ø D beam-splitter. No phase transformationis required.

(ii) States with covariance matrix in Simon normal form[11]: If pO9g F 2@q!!Ù F 2Ö9 diag :/Ú2Û#A , then one eigenvec-tor ¥ ¶ is real and the other is imaginary. Hence É!Ë J D 2@ÍK ,whereas the optimal beam splitter is in general not balanced.

(iia) Symmetric states: These are states with identical ther-mal reductions, meaning that pÜÕqEÜg F , gÝPE6 .These states are already optimally entangled, since »: º#AÀÁÂ J D 2 H ½ ¾:/o o A4Ã K , and the optimal entangling proce-dure is thus the identity operation.

(iib) Special cases of symmetric states are two-modesqueezed pure Gaussian states with covariance matrix in Si-mon normal form, where in addition, Ö takes the form Ö`diag :)Ú2 H ÚA with Ú :46 H g A 4@ .

In this letter we have investigated the entangling capabili-ties of passive optical elements in a general setting. We havepresented a necessary and sufficient criterion for the possibil-ity of creating distillable entanglement in a multi-mode sys-tem that has been prepared in a Gaussian state. The findingsreveal in fact a surprisingly simple close relationship betweensqueezing and attainable entanglement. We have moreoverquantified the maximal degree of entanglement that can beachieved in a two-mode subsystem, and we have identified theoptimal entangling procedure for the case of two input modes.In view of recently proposed applications of quantum infor-mation science, we hope that the presented results as well asthe employed techniques may prove useful tools in the studyof feasible sources of continuous-variable entanglement.

We would like to thank S. Scheel and K. Audenaert forfruitful discussions. This work has been supported by the ESF,the A.v.-Humboldt Foundation, the DFG, and the EuropeanUnion (EQUIP).

[1] M.G.A. Paris, Phys. Rev. A 59, 1615 (1999); S. Scheel, L.Knoll, T. Opatrny, and D.-G. Welsch, Phys. Rev. A 62, 043803(2000); M.S. Kim, W. Son, V. Buzek, and P.L. Knight, Phys.Rev. A 65, 032323 (2002); X.B. Wang, quant-ph/0204082,quant-ph/0204039.

[2] C. Silberhorn, P.K. Lam, O. Weiss, F. Konig, N. Korolkova, G.Leuchs, Phys. Rev. Lett. 86, 4267 (2001); N. Korolkova, C. Sil-berhorn, O. Glockl, S. Lorenz, C. Marquardt, and G. Leuchs,Eur. Phys. D 18, 229 (2002).

[3] M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani, Phys.Rev. Lett. 73, 58 (1994).

[4] G. Vidal and R.F. Werner, Phys. Rev. A 65, 032314 (2002).[5] R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567

(1994).[6] B. Demoen, P. Vanheuverzwijn, and A. Verbeure, Lett. Math.

Phys. 2, 161 (1977); J. Eisert and M.B. Plenio, quant-ph/0109126; J. Eisert, S. Scheel, and M.B. Plenio, quant-ph/0204052; J. Fiurasek, quant-ph/0204069; G. Giedke and J.I.Cirac, quant-ph/0204085.

[7] C. Bloch and A. Messiah, Nuclear Physics 39, 95 (1962).[8] Here, linearity refers to the Hamiltonian, which is linear/non-

linear w.r.t. creation and annihilation operators and passivemeans photon-number conserving.

[9] R.F. Werner and M.M. Wolf, Phys. Rev. Lett. 86, 3658 (2001).[10] Note that the direct sum corresponds to a split between modes

rather than to a position/momentum split.[11] R. Simon, Phys. Rev. Lett. 84, 2726 (2000). L.-M. Duan, G.

Giedke, J.I. Cirac, and P. Zoller, ibid. 84, 2722 (2000).[12] G. Giedke, L.-M. Duan, J.I. Cirac, and P. Zoller, Quant. Inf.

Comp. 1, 79 (2001).[13] Here we have used that for any symplectic matrix Þßà@á âã9 ,

and admissible covariance matrices satisfy Þßà@á äãå .

4

http://lanl.arXiv.org/abs/quant-ph/0204082






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Danksagung

An dieser Stelle mochte ich das Terrain der Quanteninformationstheorie verlassenund mich bei allen bedanken, die mich bei der Durchfuhrung dieser Arbeit un-terstutzt haben.

Zuallererst mochte ich mich bei Reinhard Werner bedanken fur die angenehme undkompetente Betreuung dieser Arbeit und die Chance in die Theorie zu wechseln undein wenig von seinem tiefen Verstandnis der Mathematischen Physik profitieren zudurfen.

Außerdem mochte ich mich bedanken bei . . .

. . . Cornelia Schmidt, fur das Korrekturlesen der Arbeit, den Euro und die Ord-nung im Chaos — Karl nobreakspace Gerd Vollbrecht, fur die vielen gemeinsamenArbeiten, die noch viel zahlreicher gewesen waren, wenn das nicht Eric Rains schonalles gemacht hatte — Tilo Eggeling fur die steten Bemuhungen das Niveau in un-serer Butze ein wenig anzuheben — Ole Kruger fur das genaue Durchsehen dieserArbeit und die Diskussionen uber Carl Friedrichs Zustande — Dirk Schlingemann,der zwischen einer Verschworungstheorie und einem Graphen immer etwas Zeit ge-funden hat, eine dumme Frage zu beantworten — Michael Keyl fur die Erkenntnis,daß es Mathematik jenseits der Linearen Algebra gibt — Eberhard Gerbracht furden Versuch diese Arbeit durch virtuoses Anwenden der Sn lesbarer zu machen —Antonio Acin, Jens Eisert, Andrew Doherty, Martin Plenio, Valerio Scarani, Bar-bara Terhal und Frank Verstraete fur die schonen gemeinsamen Arbeiten und allenAngehorigen des Instituts fur die angenehme Arbeitsatmosphare.

. . . Eicke fur die zahlreichen Ausfluge in die Spieltheorie.

. . . allen Tangueras y Tangueros dieser Welt, allen voran Jasmin, und Lydie Auvrayfur einen Traum von Musik.

. . . meinen Eltern, die mir all dies ermoglicht haben.

partial transposition in quantum information theory · 2017. 8. 14. · partial transposition in...

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