Physics Letters A 306 (2002) 14–20

www.elsevier.com/locate/pla

The generalized partial transposition criterion for separability ofmultipartite quantum states

Kai Chen∗, Ling-An Wu

Laboratory of Optical Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, PR China

Received 9 August 2002; received in revised form 31 October 2002; accepted 4 November 2002

Communicated by P.R. Holland

Abstract

We present a generalized partial transposition separability criterion for the density matrix of a multipartite quantumsystem. This criterion comprises as special cases the famousPeres–Horodeckicriterion and the recentrealignmentcriterionin [O. Rudolph, quant-ph/0202121] and [K. Chen, L.A. Wu, quant-ph/0205017]. It involves only straightforward matrixmanipulations and is easy to apply. A quantitative measure of entanglement based on this criterion is also obtained. 2002 Elsevier Science B.V. All rights reserved.

PACS:03.67.-a; 03.65.Ud; 03.65.Ta

Keywords:Separability; Density matrix; Multipartite quantum system; Measure of entanglement

1. Introduction

Since the well-known papers of Einstein et al. [3],Schrödinger [4] and Bell [5], quantum entangled stateshave greatly enriched quantum mechanics and have re-cently found wide applications in the rapidly expand-ing field of quantum information processing. Quantumteleportation, quantum cryptography, quantum densecoding and parallel computation [6–8] have spurred aflurry of activity in the effort to fully exploit the poten-tial of quantum entanglement. Although their applica-tions have already been demonstrated experimentally,the physical character and mathematical structure ofentangled states are only partially known, and a full

* Corresponding author.E-mail address:kchen@aphy.iphy.ac.cn (K. Chen).

comprehensive understanding is still a challenge forthe theorists.

The most familiar entangled state is the singlet purestate of a pair of spin-1/2 particles given by Bohm [9]

(1)ψs = 1√2

(|↑〉|↓〉 − |↓〉|↑〉),which cannot be reduced or factorized to a direct prod-uct of the states for the two particles. Due to uncon-trolled interactions with the environment, a practicalcomposite system generally evolves to a mixed state.How do we know if a quantum state is entangled, andhow entangled is it still after the intervention of noise?

To answer these questions we must have a physicalacceptable definition of entangled states. From aphysically meaningful and practical point of view,the state of a composite quantum system is called

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)01538-4

K. Chen, L.-A. Wu / Physics Letters A 306 (2002) 14–20 15

disentangledor separableif it can be prepared in a“ local” or “ classical” way. A separable multipartitesystem can be expressed as an ensemble realizationof pure product states|ψi〉A|φi〉B · · · |ϕi〉Z occurringwith a certain probabilitypi :

(2)ρAB···Z =∑i

piρAi ⊗ ρBi ⊗ · · · ⊗ ρZi ,

where ρAi = |ψi〉A〈ψi |, ρBi = |φi〉B〈φi |, . . . , ρZi =|ϕi〉Z〈ϕi |, ∑

i pi = 1, and|ψi〉A, |φi〉B, . . . , |ϕi〉Z arenormalized pure states of subsystemsA,B, . . . , andZ,respectively, [10]. If no convex linear combination ex-ists for a givenρAB···Z , then the state is called “entan-gled” and includes quantum correlation.

For a pure stateρAB···Z , it is trivial and straightfor-ward to judge its separability:

A pure stateρAB···Z is separable if and only if

(3)ρAB···Z = ρA ⊗ ρB ⊗ · · · ⊗ ρZ,

where ρA,B,...,Z are the reduced density matricesdefined as

ρA = TrB,C,...,Z(ρAB···Z),ρB = TrA,C,...,Z(ρAB···Z), . . . ,ρZ = TrA,B,...,Y (ρAB···Z).

However, for a generic mixed stateρAB···Z , findinga decomposition as in Eq. (2) or proving that it doesnot exist is a non-trivial task (we refer to recentreviews [11–13] and references therein). There hasbeen considerable effort in recent years to analyzethe separability and quantitative character of quantumentanglement. The Bell inequalities satisfied by aseparable system give the first necessary condition forseparability [5]. In 1996, Peres made an important stepforward and showed that partial transpositions withrespect to one and more subsystems of the densitymatrix for a separable state are positive,

(4)ρTX � 0,

whereX ⊂ {A,B, . . . ,Z}. Thus theρTX should havenon-negative eigenvalues (this is known as thePPTcriterion orPeres–Horodeckicriterion) [14]. This wasimmediately shown by Horodecki et al. [15] to besufficient for bipartite systems of 2× 2 and 2× 3.Meanwhile, they also found a necessary and suffi-cient condition for separability by establishing a close

connection between positive map theory and separa-bility [15]. In view of the quantitative character forentanglement, Wootters succeeded in computing the“entanglement of formation” [16] and thus obtaineda separability criterion for 2× 2 mixtures [17]. The“ reduction criterion” proposed independently in [18]and [19] gives another necessary criterion which isequivalent to thePPT criterion for 2× n compos-ite systems but is generally weaker. Pittenger et al.gave also a sufficient criterion for separability con-nected with the Fourier representations of density ma-trices [20]. Later, Nielsen et al. [21] presented anothernecessary criterion called themajorization criterion:the decreasingly ordered vector of the eigenvalues forρAB is majorized by that ofρA or ρB alone for a sep-arable state. A new method of constructingentangle-ment witnessesfor detecting entanglement was givenin [15] and [22,23]. There are also some necessary andsufficient criteria of separability for low rank cases ofthe density matrix, as shown in [24,25]. In addition, itwas shown in [26] and [27] that a necessary and suffi-cient separability criterion is also equivalent to certainsets of equations.

However, despite these advances, a practical andconvenient computable criterion for generic bipar-tite systems is mainly limited to thePPT, reductionand majorization criteria, as well as a recent exten-sion of thePPT criterion based on semidefinite pro-grams [28]. Very recently Rudolph [1] and the au-thors [2] proposed a new operational criterion for sepa-rability: therealignmentcriterion (named thus follow-ing the suggestion of [29] which is also called thecom-putational cross normcriterion given in Ref. [1]). Thecriterion is very simple to apply and shows dramaticability to detect most of the bound entangled states [2]and even genuinely tripartite entanglement [29]. Thisis a surprising new result, for the bound entangledstates [30] are “weakly” inseparable and in the pastit was very hard to establish with certainty their in-separability [19]. Soon after, Horodecki et al. showedthat thePPT criterion and realignment criterion canbe equivalent to any permutation of the indices of thedensity matrix [29].

In this Letter we generalize our realignment crite-rion to multipartite quantum systems in arbitrary di-mensions; then, from a different aspect, we give a sys-tematic construction for thegeneralized partial trans-positioncriterion. The constructions are given in Sec-

16 K. Chen, L.-A. Wu / Physics Letters A 306 (2002) 14–20

tion 2 where the strongPPT and realignment criteriaare shown as two special cases of the criterion. Onlyinvolving straightforward matrix manipulations, it isalso very easy to apply. A quantitative measure of en-tanglement based on the criterion for detecting entan-glement is also obtained in Section 3. A brief summaryand some discussions are given in the last section.

2. The criteria for separability

In this section we will give two criteria for sepa-rability of the density matrix. We first introduce themultipartite generalization of the realignment crite-rion, then present the generalized partial transpositioncriterion as a further generalization of thePPTand re-alignment criteria.

2.1. Some notation

The various matrix operations we shall use can befound in [31,32], with the following notation:

Definition. For eachm×n matrixA= [aij ], whereaijis the matrix entry ofA, we define the vector vec(A)as

vec(A)= [a11, . . . , am1, a12, . . . , am2, . . . ,

a1n, . . . , amn]T .LetZ be anm×m block matrix with block sizen×n.We define a “realignment” operationR to changeZ toa realigned matrix̃Z of sizem2 × n2 that contains thesame elements asZ but in different positions as

(5)R(Z)≡ Z̃ ≡

vec(Z1,1)T

...

vec(Zm,1)T

...

vec(Z1,m)T

...

vec(Zm,m)T

.

For example, a 2× 2 bipartite density matrixρ canbe transformed as:

ρ =

ρ11 ρ12 ρ13 ρ14ρ21 ρ22 ρ23 ρ24

ρ31 ρ32 ρ33 ρ34ρ41 ρ42 ρ43 ρ44

(6)−→ R(ρ)=

ρ11 ρ21 ρ12 ρ22

ρ31 ρ41 ρ32 ρ42

ρ13 ρ23 ρ14 ρ24

ρ33 ρ43 ρ34 ρ44

.

2.2. The realignment criterion

Motivated by the Kronecker product approximationtechnique for a matrix [33,34], we developed a verysimple method to obtain the realignment criterionin [2].

The realignment criterion.If an m × n bipar-tite density matrixρAB is separable, then for them2 × n2matrix R(ρAB) the trace norm‖R(ρAB)‖ ≡∑q

i=1σi(R(ρAB)), which is the sum of all the singu-lar values ofR(ρAB), should be� 1, or equivalentlylog‖R(ρAB)‖ � 0, whereq = min(m2, n2).

This criterion is strong enough to detect most ofthe bound entangled states in the literature, as shownin [2].

For multipartite systems, we have a natural gener-alization that was also partially introduced in [29]:

The multipartite realignment criterion.If ann-partite density matrixρ is separable, then

(7)∥∥(R(k) ⊗ I(n−k))ρ

∥∥ � 1, k = 2,3, . . . , n,

where the subscript indices mean that we act byR(k)

on k chosen subsystems, while leaving untouched theremainingn− k subsystems.

HereR(k) can be a realigned matrix according toall the possible bipartite cuts for thek subsystems, inaddition toall their combinations. It is surprisinglystrong enough to detect the genuinely tripartite boundentangled state [29] which is bi-separable with respectto any bipartite cuts for the 3 subsystems. The opera-tion (R(2) ⊗ I(n−2)) is also shown to be equivalent tocertain permutations of the indices of the density ma-trix [29].

2.3. The generalized partial transposition criterion

We will now derive the main result of this Letter:a generalized partial transposition criterion for sepa-rability of multipartite quantum systems in arbitrarydimensions.

K. Chen, L.-A. Wu / Physics Letters A 306 (2002) 14–20 17

2.3.1. The main theoremSince the density matrixρp for a d-dimensional

pure separable state is a self-adjoint Hermitiand × d

matrix with only one eigenvalue 1, it is evident thatρpis of rank 1 and we have naturally

(8)ρp = u⊗ u† = u† ⊗ u,

whereu is ad × 1 columnvector satisfyingu†u≡ 1.This is possible sinceu can be chosen to be thenormalized eigenvector ofρp . For the convenience oflater use we define two new operatorsTr , Tc and theirmultiplication:

Tr :A−→ row transposition ofA

(9)⇐⇒A−→ (vec(A)

)t,

Tc :A−→ column transposition ofA

(10)⇐⇒A−→ vec(A),

(11)TcTr or TrTc :A−→At,

where t represents standard transposition operation.For example, a 2× 2 matrixA can be transformed to

A=(a11 a12a21 a22

)−→ Tr (A)= (a11 a21 |a12 a22),

−→ Tc(A)=

a11a21

a12a22

.

As for a generic density matrix we have

ρ =∑i,j

ρij |i〉〈j | =∑i,j

ρij |i〉 ⊗ 〈j | =∑i,j

ρij 〈j | ⊗ |i〉

where|i〉, |j 〉 are suitably selected normalized orthog-onal bases and〈i|, 〈j | are the corresponding transpo-sitions, respectively (here, for simplicity of notation,〈i|, 〈j | are not regarded as the corresponding conju-gate transpositions). Thus the operations ofTr andTccan be realized conveniently:

(12)

ρTr−→

∑i,j

ρij 〈j | ⊗ 〈i| Tc−→∑i,j

ρij |j 〉 ⊗ 〈i| = ρt ,

(13)

ρTc−→

∑i,j

ρij |j 〉 ⊗ |i〉 Tr−→∑i,j

ρij |j 〉 ⊗ 〈i| = ρt .

Now we arrive at the following separability criterionfor a multipartite system:

Theorem 1. If an n-partited1 × d2 × · · · × dn densitymatrix ρ is separable, then the generalized partialtranspositions ofρ satisfy

(14)

∥∥ρTY∥∥ � 1, ∀Y ⊂ {rA, cA, rB, cB, . . . , rZ, cZ︸ ︷︷ ︸2n

},

whereTrk or Tck (k = A,B, . . . ,Z) means transpo-sition with respect to the row or column for thekthsubsystem. The superscript indicesTY represent par-tial transpositions of every element of setY on cho-sen subsystems, while leaving untouched the remain-ing subsystems.

Proof. Applying Eq. (2) for separable states and theproperty Eq. (8) for pure states, we have

(15)

ρ =∑i

piuAi ⊗ u

A†i ⊗ uBi ⊗ u

B†i ⊗ · · · ⊗ uZi ⊗ u

Z†i .

The transformation ofρTY is to make partial transpo-sitions of the rows or columns corresponding to somesubsystems. Without loss of generality we supposethat we only make a row transposition to theA sub-system:

(16)

ρT{rA} =∑i

pi(uAi

)† ⊗ (uAi

)t ⊗ uBi

⊗ uB†i ⊗ · · · ⊗ uZi ⊗ u

Z†i .

We shall need the following property (see Chapters 3,4 of [32]) of the Kronecker product of them × m

matrixA and then× n matrixB (Property 1), as wellas that of the trace norm of two matricesA andB ofthe same size (Property 2):

Property 1. The non-zero singular values ofA ⊗ B

are the positive numbers{σi(A)σj (B)} whereσi(A)and σj (B) are non-zero singular values ofA andBarranged in decreasing order, respectively,

Property 2. ‖A+B‖ � ‖A‖ + ‖B‖,

so that

(17)

∥∥ρT{rA }∥∥ �∑i

pi∥∥(uAi

)† ⊗ (uAi

)t ⊗ uBi

⊗ uB†i ⊗ · · · ⊗ uZi ⊗ u

Z†i

∥∥.

18 K. Chen, L.-A. Wu / Physics Letters A 306 (2002) 14–20

It is straightforward thatuAi , (uAi )

†, (uAi )t and (uAi )

∗have only one singular value 1, respectively, due to the

normalization conditionuAi†uAi = 1. The same holds

for the case ofA replaced withB,C, . . . , or Z. Thuswe obtain∥∥(uAi

)† ⊗ (uAi

)t ⊗ uBi ⊗ uB†i ⊗ · · · ⊗ uZi ⊗ u

Z†i

∥∥ = 1

by applying Property 1. Therefore it is evident that

(18)∥∥ρT{rA}∥∥ �

∑i

pi = 1.

When applying column transposition, the standardtransposition with respect to one subsystem or evena combination of transpositions to a subsetY of{rA, cA, rB, cB, . . . , rZ, cZ}, we follow a similar pro-cedure and obtain the same conclusion as the above-derived. Thus we obtain the final result of Eq. (14).✷2.3.2. Relationship with other necessary criteria

We shall now show that Theorem 1 actually encom-passes previous strong computational criteria for sep-arability.

2.3.2.1. The PPT criterion For partial transpositionwith respect to one or more subsystems of a separablestate, we have

(19)ρTX � 0,

and‖ρTX ‖ = Tr(ρTX )= 1 due to the Hermitian prop-erty of ρTX . It is obvious for anyX ⊂ {A,B, . . . ,Z}that there is aY in the generalized partial transpositioncriterion satisfying

(20)∥∥ρTY∥∥ = 1.

We only need to substituteX with its pairwisecorrespondence toY ⊂ {rA, cA, rB, cB, . . . , rZ, cZ}.For example,

X={B,C}�

Y={rB, cB, rC, cC}.

2.3.2.2. The multipartite realignment criterionFromthe property given in Ref. [34], we have a Kro-necker product decomposition for a density matrixof ρ = ∑

i αAi ⊗ βBi whereαAi ,β

Bi are, in general, not

density matrices. Thus one finds

(21)R(ρ)=∑i

vec(αAi

)vec

(βBi

)T,

it is easy to find that the above operationR(ρ)

corresponds toY = {cA, rB} from the properties ofEqs. (9) and (10). For the generic realignment oper-ation (R(k) ⊗ I(n−k)), we will have a similar corre-spondence so that Theorem 1 also includes the mul-tipartite realignment criterion of Eq. (7) as a specialcase.

2.3.2.3. The Horodecki indices permutation criterionAs for the criterion proposed in [29], Horodecki etal. show that thePPT criterion and the realignmentcriterion are equivalent to certain permutations ofdensity matrix indices up to some unitary matrixwhich keeps the trace norm invariant. Moreover, itis obvious that the realignment criterion is identicalto the operation ofR(2) ⊗ I(n−2) in the language ofgeneric realignment, so it is also a special case ofTheorem 1 according to Section 2.3.2.2.

We can see that the generalized partial transpositioncriterion is a powerful computational criterion sinceit includes as special cases the strongPPT and thegeneric realignment criterion. This gives us a wholeframework to detect entanglement with great conve-nience of manipulation. In addition, it also containssome new criteria for recognizing entanglement. Forinstance, making only the row transpositions or col-umn transpositions of various subsystems is differentfrom thePPT and realignment criteria. However, de-spite these virtues, it is still not sufficient for detect-ing all the entangled states. For example, it still failsto recognize the 2× 4 Horodecki bound entangledstate [30] by direct computation.

3. Entanglement measure induced by theseparability criterion

With the power of our new criterion, we expectthat it should be able to give some information onthe degree of entanglement. Among the quantitativemeasures, the entanglement of formationEF (ρ) [16]and distillable entanglementED(ρ) [16] are two ofthe most meaningful ones. The former quantifies theasymptotic pure-state entanglement required to create

K. Chen, L.-A. Wu / Physics Letters A 306 (2002) 14–20 19

ρ while the latter quantifies which can be extractedfrom ρ, by means of local operations and classicalcommunication (LOCC). But, in practice it is verydifficult to compute the two measures for a genericquantum state. There is only one exception for thesuccess of computation ofEF (ρ) for two-qubits [17].

Recently, Vidal and Werner showed that the nega-tivity is an entanglement monotone [35] and thereforea good entanglement measure [36]. The negativity isdefined as

(22)N (ρ)≡ ‖ρTA‖ − 1

2.

Following this idea we introduce a computationalmeasure based on our criterion as

E(ρ)≡ sup‖ρTY ‖ − 1

2,

(23)Y ⊂ {rA, cA, rB, cB, . . . , rZ, cZ}.For separable states,‖ρTY ‖ � 1 due to Theorem 1.WheneverY corresponds to a standard partial trans-position of one or more of the subsystems, we have‖ρTY ‖ = 1 due to the positivity ofρTY . Thus we haveE(ρ) = 0 for all the separable states. On the otherhand,E(ρ) is convex due to the convexity of the tracenorm. Furthermore, we have the following result:

Theorem 2. Applying a local unitary transformationleavesE(ρ) invariant, i.e.,

(24)E(ρ′) =E(ρ),

whereρ′ = (UA ⊗UB ⊗· · ·⊗UZ)ρ(U†A ⊗U

†B ⊗· · ·⊗

U†Z) andUA,UB, . . . ,UZ are unitary operators acting

onA,B, . . . ,Z subsystems, respectively.

Proof. If ‖ρTY ‖ = ‖(ρ′)TY ‖ for ∀Y , we certainlyhave Eq. (24). Consider one term ofρ′:

ρ′i = (UA ⊗UB ⊗ · · · ⊗UZ)

(αAi ⊗ βBi ⊗ · · · ⊗ ηZi

)× (

U†A ⊗U

†B ⊗ · · · ⊗U

†Z

)(25)=UAα

Ai U

†A ⊗UBβ

Bi U

†B ⊗ · · · ⊗UZη

Zi U

†Z,

where we supposeρ = ∑i α

Ai ⊗ βBi ⊗ · · · ⊗ ηZi

which can be obtained according to [37] or can bedone by applying repeatedly the Kronecker productdecomposition for a bipartite-cut of a matrix [33,34]. It should be remarked thatαAi ,β

Bi , . . . , η

Zi are,

in general, not density matrices. Without loss ofgenerality, we can just perform a row transposition ontheA subsystem, thus(ρ′i

)T{rA } = (vec

(UAα

Ai U

†A

))t ⊗UBβBi U

†B

(26)⊗ · · · ⊗UZηZi U

†Z.

Applying the property

(27)vec(XYZ)= (ZT ⊗X

)vec(Y ) (see [32])

we arrive at(ρ′i

)T{rA } =((U∗A ⊗UA

)vec

(αAi

))t ⊗UBβBi U

†B

⊗ · · · ⊗UZηZi U

†Z

= (Id ⊗UB ⊗ · · · ⊗UZ)

×((

vec(αAi

))t ⊗ βBi ⊗ · · · ⊗ ηZi

)(28)× (

U†A ⊗Ut

A ⊗U†B ⊗ · · · ⊗U

†Z

).

Since a local unitary transformation acts as a wholefactor onρ, we have∥∥(ρ′)T{rA}∥∥=

∥∥∥∥(Id ⊗UB ⊗ · · · ⊗UZ)

×∑i

((vec

(αAi

))t ⊗ βBi ⊗ · · · ⊗ ηZi

)

× (U

†A ⊗Ut

A ⊗U†B ⊗ · · · ⊗U

†Z

)∥∥∥∥=

∥∥∥∥∑i

((vec

(αAi

))t ⊗ βBi ⊗ · · · ⊗ ηZi

)∥∥∥∥(29)= ∥∥(ρ)T{rA }∥∥.

Here we have used the unitarily invariant property ofthe trace norm.

The same procedure can be used to perform col-umn transposition, partial transposition of some sub-systems, and any combinations of these transpositions.The proof is thus completed.✷

Since the negativity gives an upper bound forthe distillable entanglement, we also obtain an upperboundE(ρ) for the distillable entanglementED(ρ),becauseE(ρ) takes the maximum for all the possibleY transpositions which naturally contain the partialtransposition operations.

20 K. Chen, L.-A. Wu / Physics Letters A 306 (2002) 14–20

4. Conclusion

Summarizing, we have presented a computationalnecessary criterion for separability of multipartitequantum systems in arbitrary dimensions which wecall a “generalized partial transposition criterion”.This criterion unifies the previous knownPPT crite-rion, the recently found realignment criterion and thepermutation criterion in a single concise framework. Itprovides a very powerful necessary condition for sepa-rability and is quite easy to apply. A quantitative mea-sure of entanglement based on the criterion is also ob-tained. Moreover, the measure gives an upper boundfor distillable entanglement.

Comparing with previous works, we have signif-icantly expanded our ability to distinguish directlythe entanglement and separability of any multipartitequantum state in arbitrary dimensions. We expect thatour method can shed some light on the final solutionof the separability problem and provide a more suit-able better measure for the degree of entanglement.

Acknowledgements

We would like to thank Dr. H. Fan and Dr. L. Yangfor valuable discussions. K.C. is grateful to encour-agement from Prof. Guozhen Yang. This work wassupported by the National “973” Program for Funda-mental Research, the Knowledge Innovation Programof the Chinese Academy of Sciences and the NationalNatural Science Foundation of China.

References

[1] O. Rudolph, quant-ph/0202121.[2] K. Chen, L.A. Wu, quant-ph/0205017.[3] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47 (1935) 777.[4] E. Schrödinger, Natürwissenschaften 23 (1935) 807.[5] J.S. Bell, Physics (N.Y.) 1 (1964) 195.[6] J. Preskill, The Theory of Quantum Information and Quantum

Computation, California Inst. of Technology, Pasadena, CA,2000,http://www.theory.caltech.edu/people/preskill/ph229/.

[7] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quan-tum Information, Cambridge Univ. Press, Cambridge, 2000.

[8] D. Bouwmeester, A. Ekert, A. Zeilinger (Eds.), The Physicsof Quantum Information: Quantum Cryptography, QuantumTeleportation and Quantum Computation, Springer, New York,2000.

[9] D. Bohm, Phys. Rev. 85 (1952) 166.[10] R.F. Werner, Phys. Rev. A 40 (1989) 4277.[11] M. Horodecki, P. Horodecki, R. Horodecki, in: Springer Tracts

in Modern Physics, Vol. 173, Springer, Berlin, 2001, p. 151.[12] M. Lewenstein, D. Bruss, J.I. Cirac, B. Kraus, M. Kus,

J. Samsonowicz, A. Sanpera, R. Tarrach, J. Mod. Opt. 47(2000) 2841.

[13] B.M. Terhal, Theor. Comput. Sci. 287 (2002) 313.[14] A. Peres, Phys. Rev. Lett. 77 (1996) 1413.[15] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223

(1996) 1.[16] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters,

Phys. Rev. A 54 (1996) 3824.[17] W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245.[18] M. Horodecki, P. Horodecki, Phys. Rev. A 59 (1999) 4206.[19] N.J. Cerf, C. Adami, R.M. Gingrich, Phys. Rev. A 60 (1999)

898.[20] A.O. Pittenger, M.H. Rubin, Phys. Rev. A 62 (2000) 032313.[21] M.A. Nielsen, J. Kempe, Phys. Rev. Lett. 86 (2001) 5184.[22] B. Terhal, Phys. Lett. A 271 (2000) 319.[23] M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev.

A 62 (2000) 052310.[24] P. Horodecki, M. Lewenstein, G. Vidal, I. Cirac, Phys. Rev.

A 62 (2000) 032310.[25] S. Albeverio, S.M. Fei, D. Goswami, Phys. Lett. A 286 (2001)

91.[26] S.J. Wu, X.M. Chen, Y.D. Zhang, Phys. Lett. A 275 (2000)

244.[27] P.X. Chen, L.M. Liang, C.Z. Li, M.Q. Huang, Phys. Rev. A 63

(2001) 052306.[28] A.C. Doherty, P.A. Parrilo, F.M. Spedalieri, Phys. Rev. Lett. 88

(2002) 187904.[29] M. Horodecki, P. Horodecki, R. Horodecki, quant-ph/0206008.[30] P. Horodecki, Phys. Lett. A 232 (1997) 333.[31] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge Univ.

Press, New York, 1985.[32] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cam-

bridge Univ. Press, New York, 1991.[33] C.F. Van Loan, N.P. Pitsianis, in: M.S. Moonen, G.H. Golub

(Eds.), Linear Algebra for Large Scale and Real Time Appli-cations, Kluwer, 1993, pp. 293–314.

[34] N.P. Pitsianis, Ph.D. Thesis, The Kronecker Product in Ap-proximation and Fast Transform Generation, Cornell Univer-sity, New York, 1997.

[35] G. Vidal, J. Mod. Opt. 47 (2000) 355.[36] G. Vidal, R.F. Werner, Phys. Rev. A 65 (2002) 032314.[37] L. De Lathauwer, B. De Moor, J. Vandewalle, SIAM J. Matrix

Anal. Appl. 21 (2000) 1253.