SEPARABILITY CRITERION FOR MULTIPARTITE QUANTUM STATES BASED ON THE BLOCH REPRESENTATION OF DENSITY MATRICES ALI SAIF M. HASSANa and PRAMOD S. JOAGb Department of Physics, University of Pune, Pune, India-411007.

arXiv:0704.3942v5 [quant-ph] 13 Apr 2008

We give a new separability criterion, a necessary condition for separability of N partite quantum states. The criterion is based on the Bloch representation of a N -partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N -partite quantum states in H = Hd1 ? Hd2 ? · · ·?HdN . The criterion can test whether a N -partite state is entangled and can be applied to di?erent partitions of the N -partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a su?ciency condition for separability of a 3-partite state, straightforwardly generalizable to the case N > 3, under certain condition. We also give a necessary and su?cient condition for separability of a class of N -qubit states which includes N -qubit PPT states. Keywords : Bloch representation of quantum state, separability criteria, matrization of tensors, PPT entangled states.

1

Introduction

The question of quantifying entanglement of multipartite quantum states is fundamental to the whole ?eld of quantum information and in general to the physics of multicomponent quantum systems. Whereas entanglement of pure bipartite states is well understood, the classi?cation of mixed states according to the degree and character of their entanglement is still a matter of intense research [1,2]. A N -partite state acting on H = Hd1 ? Hd2 ? · · · ? HdN is separable [3] ( or fully separable) if it can be written as a convex sum of tensor products of subsystem states

N

ρ=

w

a b

pw ρ(1) w

?

ρ(2) w

···?

(N ) ρw

=

w

pw

j =1

(j ) ρw , pw > 0; w

pw = 1 .

(1)

Electronic address: alisaif@physics.unipune.ernet.in Electronic address: pramod@physics.unipune.ernet.in

1

A state is called k separable if we can write ρ=

w (a1 ) (a2 ) (ak ) pw ρw ? ρw · · · ? ρw

(2)

where ai ; i = 1, 2, . . . , k are the disjoint subsets of {1, 2, . . . , N } and ρ(ai ) acts on the tensor product space made up by the factors of H labeled by the members of ai . The understanding of multipartite entanglement has progressed by dealing with some special classes of states like the density operators supported on the symmetric subspace of H [4]. A lower bound on concurrence on the multipartite mixed states is obtained [5]. K. Chen and L. Wu have given a generalized partial transposition and realignment criterion to detect entanglement of a multipartite quantum state [6]. There are two de?nitions commonly used for the entanglement of multipartite quantum states, the one from Ref. [7] (ABLS) and the one introduced in [8] (DCT). In DCT, all possible partitions of N parties are considered and it is tested for each partition if the state is fully separable there or not. A state is called N partite entangled if it is not separable for any partition. If a state is separable for a bipartite partition, it is called biseparable. In ABLS, a state is called biseparable if it is a convex combination of biseparable states, possibly concerning di?erent partitions. A N -partite entangled state is one which is not biseparable. In this paper we derive a necessary condition for the separability of multipartite quantum states for arbitrary ?nite dimensions of the subsystem Hilbert spaces and without any further restriction on them. The criterion is based on the Bloch representation of a multipartite quantum state, which has been used in previous works to characterize the separability of bipartite density matrix, in particular, our work is a generalization of de Vicente’s work on bipartite systems [9]. We make use of the algebra of higher order tensors, in particular the matrization of a tensor [10,11,12,13,14,15,16,17]. The paper is organized as follows. In section II we present the Bloch representation of a N -partite quantum state. In section III we obtain the main results on separability of a N -partite quantum state. In section IV we give a su?cient condition for the separability of a 3-partite quantum state generalizable to the case N > 3. In section V we investigate our separability criterion for mixed states, in particular, bound entangled states. Finally we summarize in section VI.

2

Bloch Representation of a N -Partite Quantum State

Bloch representation [18,19,20,21,22] of a density operator acting on the Hilbert space of a d-level quantum system Cd is given by [9] 1 ρ = (Id + si λ i ) (3) d i 2

Eq.(3) is the expansion of ρ in the Hilbert-Schmidt basis {Id , λi ; i = 1, 2, . . . , d2 ? 1} where λi are the traceless hermitian generators of SU (d) satisfying T r (λiλj ) = 2δij and are characterized by the structure constants of the corresponding Lie algebra, fijk , gijk which are, respectively, completely antisymmetric and completely symmetric. 2 λi λj = δij Id + ifijk λk + gijk λk d

2

(4)

s = (s1 , s2 , . . . , sd2 ?1 ) in Eq.(3) are the vectors in Rd ?1 , constrained by the positive semide?niteness of ρ, called Bloch vectors [21]. The set of all Bloch vectors that consti2 tute a density operator is known as the Bloch vector space B (Rd ?1 ). The problem of 2 determining B (Rd ?1 ) where d ≥ 3 is still open [19,20]. However, for pure states (ρ = ρ2 ) the following relations hold. ||s||2 = d(d ? 1) ; 2

2

si sj gijk = (d ? 2)sk

(5)

where ||.||2 is the Euclidean norm in Rd ?1 . 2 2 It is known [23,24] that B (Rd ?1 ) is a subset of the ball DR (Rd ?1 ) of radius R =

d(d?1) 2

, which is the minimum ball containing it, and that the ball Dr (Rd is included in B (R

d2 ?1

2 ?1

) of radius

r=

d 2(d?1)

). That is, ) ? B (R d

2 ?1

Dr ( R d

2 ?1

) ? DR ( R d

2 ?1

)

(6)

In order to give the Bloch representation of a density operator acting on the Hilbert space Cd1 ?Cd2 ?· · ·?CdN of a N -partite quantum system, we introduce following notation. We use k , ki (i = 1, 2, · · · ) to denote a subsystem chosen from N subsystems, so that k , ki (i = 1, 2, · · · ) take values in the set N = {1, 2, · · · , N }. The variables αk or αki for a given k or ki span the set of generators of SU (dk ) or SU (dki ) group (Eqs.(3) and (4)) for } for the ki th subsystem. the k th or ki th subsystem, namely the set {λ1ki , λ2ki , · · · , λd2 ki ? 1 For two subsystems k1 and k2 we de?ne

(k1 ) λα = (Id1 ? Id2 ? · · · ? λαk1 ? Idk1 +1 ? · · · ? IdN ) k

1

(k2 ) λα = (Id1 ? Id2 ? · · · ? λαk2 ? Idk2 +1 ? · · · ? IdN ) k

2

(k1 ) (k2 ) λα λαk = (Id1 ? Id2 ? · · · ? λαk1 ? Idk1 +1 ? · · · ? λαk2 ? Idk2 +1 ? IdN ) k

1 2

(7)

where λαk1 and λαk2 occur at the k1 th and k2 th places (corresponding to k1 th and k2 th subsystems respectively) in the tensor product and are the αk1 th and αk2 th generators of 2 SU (dk1 ), SU (dk2 ), αk1 = 1, 2, . . . , d2 k1 ? 1 and αk2 = 1, 2, . . . , dk2 ? 1 respectively. Then we can write 3

ρ=

1 ΠN k dk

{?N k Idk +

k ∈N αk

k) sαk λ( αk + {k1 ,k2 } αk1 αk2

1 2 M

(k1 ) (k2 ) tαk1 αk2 λα λαk + · · · + k

1 2

{k1 ,k2 ,··· ,kM } αk1 αk2 ···αkM (k )

(k1 ) (k2 ) kM ) λαk · · · λ( tαk1 αk2 ···αkM λα αk +· · ·+ k

α1 α2 ···αN

(2) (N ) tα1 α2 ···αN λ(1) α1 λα2 · · · λαN }.

where s is a Bloch vector corresponding to k th subsystem, s a tensor of order one de?ned by sαk = dk dk k) T r [ρλ( T r [ρk λαk ], αk ] = 2 2

(k )

=

d2 ?1 [sαk ]αk k =1

(8) which is

(9a)

where ρk is the reduced density matrix for the k th subsystem. Here {k1 , k2 , · · · , kM }, 2 ≤ N N ways, contributing M terms in the M ≤ N, is a subset of N and can be chosen in M sum {k1 ,k2 ,··· ,kM } in Eq.(8), each containing a tensor of order M . The total number of terms in the Bloch representation of ρ is 2N . We denote the tensors occurring in the sum {k1 ,k2 ,··· ,kM } = [tαk1 αk2 ···αkM ] which are de?ned by {k1 ,k2 ,··· ,kM } , (2 ≤ M ≤ N ) by T tαk1 αk2 ...αkM = = dk1 dk2 . . . dkM (k1 ) (k2 ) kM ) T r [ρλα λαk · · · λ( αkM ] k1 2 2M

dk1 dk2 . . . dkM (9b) T r [ρk1 k2 ...kM (λαk1 ? λαk2 ? · · · ? λαkM )] 2M where ρk1 k2 ...kM is the reduced density matrix for the subsystem {k1 k2 . . . kM }. We call The tensor in last term in Eq. (8) T (N ) .

3

Separability Conditions

Before we obtain the main results we need following de?nition. Throughout the paper, we use the bold letter for vector and normal letter for components of a vector, matrix and tensor elements. A rank-1 tensor is a tensor that consists of the outer product of a number of vectors. For M th order tensor T (M ) and M vectors u(1) , u(2) , . . . , u(M ) this means that ti1 i2 ...iM = (1) (2) (M ) ui1 ui2 . . . uiM for all values of the indices. This is concisely written as T (M ) = u(1) ? u(2) ? · · · ? u(M ) [17,11]. Also, given two tensors T (M ) and S (N ) of order M and N respectively, with dimensions I1 × I2 × · · · × IM and J1 × J2 × · · · × JN respectively, their outer product is de?ned as [16,10] (T (M ) ? S (N ) )i1 i2 ...iM j1 j2 ...jN = ti1 i2 ...iM sj1 j2 ...jN 4 (10)

Proposition 1 : A pure N -partite quantum state with Bloch representation (8) is fully separable (product state) if and only if T {k1 ,k2 ,··· ,kM } = s(k1 ) ? s(k2 ) ? · · · ? s(kM ) (11)

for 2 ≤ M ≤ N. In particular T (N ) = s(1) ? s(2) ? · · · ? s(N ) holds. Here {k1 , k2 , . . . , kM } ? {1, 2, . . . , N }, and s(k) is the Bloch vector of k th subsystem reduced density matrix. Proof : Notice that Eq.(8) can be rewritten as ρ = ρ(1) ? ρ(2) ? · · · ? ρ(N ) + 1 { d1 d2 · · · dN

(k1 ) (k2 ) [tαk1 αk2 ? sαk1 sαk2 ]λα λαk + · · · + k

1 2

{k1 ,k2 } αk1 αk2

+···+

α1 α2 ···αN

(2) (N ) [tα1 α2 ···αN ? sα1 sα2 . . . sαN ]λ(1) α1 λα2 · · · λαN }.

(12)

For full separability, the sum of all the terms apart from the ?rst term must vanish. Note that for every subsystem k = 1, 2, . . . , N the set {Id , λi ; i = 1, 2, . . . , d2 k ? 1} forms (k ) (k1 ) (k2 ) an orthonormal Hilbert-Schmidt basis for the k th subsystem. Hence λαk ; λαk1 λαk2 . . . ; (1) (2) (N ) (k ) (k ) (k ) ; . . . ; λα1 λα2 · · · λαN are the vectors belonging to the orthonormal prodλαk11 λαk22 · · · λαkM M uct basis of the Hilbert-Schmidt space of the whole N -partite system. By orthonormality of the tensor product of λ’s occurring in di?erent terms, the required sum will vanish if and only if coe?cients of each term vanish separately, that is if and only if tαk1 αk2 ···αkM = sαk1 sαk2 . . . sαkM ; 2 ≤ M ≤ N, that is, T {k1 ,k2 ,··· ,kM } = s(k1 ) ? s(k2 ) ? · · · ? s(kM ) ; 2 ≤ M ≤ N. In fact, the condition (11) for all N parts is enough to decide the separability of pure N -partite quantum states, as the following proposition shows. Proposition 1a : A pure N -partite quantum state with Bloch representation (8) is fully separable (product state) if and only if T (N ) = s(1) ? s(2) ? · · · ? s(N ) , where s(k) is the Bloch vector of k th subsystem reduced density matrix. Proof : Suppose ρ is a product state ρ = ρ1 ? ρ2 ? · · · ? ρN . Then, tα1 α2 ...αN = d1 d2 . . . dN T r [(ρ1 ? ρ2 ? · · · ? ρN )(λα1 ? λα2 ? · · · ? λαN )] 2N 5

=

d1 d2 . . . dN T r [(ρ1 λα1 ) ? (ρ2 λα2 ) ? · · · ? (ρN λαN )] 2N d1 d2 . . . dN [T r (ρ1 λα1 )T r (ρ2 λα2 ) · · · T r (ρN λαN )] 2N = [sα1 sα2 · · · sαN ].

=

Suppose the condition holds, that is, [s(1) ? s(2) ? · · · ? s(N ) ]α1 α2 ...αN = tα1 α2 ...αN . Then, d1 d2 . . . dN [T r (ρ1 λα1 )T r (ρ2 λα2 ) · · · T r (ρN λαN )] 2N

[s(1) ? s(2) ? · · · ? s(N ) ]α1 α2 ...αN = = =

d1 d2 . . . dN T r [(ρ1 λα1 ) ? (ρ2 λα2 ) ? · · · ? (ρN λαN )] 2N

d1 d2 . . . dN T r [(ρ1 ? ρ2 ? · · · ? ρN )(λα1 ? λα2 ? · · · ? λαN )] 2N = tα1 α2 ...αN = d1 d2 . . . dN T r [ρ(λα1 ? λα2 ? · · · ? λαN )]. 2N

The equality T r [(ρ1 ? ρ2 ? · · · ? ρN )(λα1 ? λα2 ? · · · ? λαN )] = T r [ρ(λα1 ? λα2 ? · · · ? λαN )]

2 is satis?ed for all elements in the orthonormal basis {?N k =1 λαk }, 0 ≤ αk ≤ dk ? 1, (αk = 0 for Idk ) where {λαk } are the d2 k ? 1 generators of SU (dk ). This means that the joint probabilities obtained from the ensemble of measurements of (λα1 · · · λαN ) for states ρ and ρ = ρ1 ? ρ2 ? · · · ? ρN are equal. This implies

ρ = ρ1 ? ρ2 ? · · · ? ρN .

Note that this criterion is easily amenable with experiments. In order to check it for an element of T (N ) we have to measure the corresponding generators on each subsystem and then check whether the product of the averages equals the average of the products. Thus in order to check whether a given pure state is a product state we have to check whether T (N ) = s(1) ? s(2) ? · · ·? s(N ) , where the Bloch vectors s(1) , s(2) , . . . , s(N ) can be constructed from the reduced density matrices ρ1 , ρ2 , · · · , ρN for subsystems 1, 2, · · · , N (sαk = dk T r (ρk λαk ), k ∈ {1, 2, · · · , N }, see Eq.(9a)). 2 In the case of mixed states we can characterize separability from the Bloch representation point of view as follows. 6

A N -partite quantum state with Bloch representation (8) is fully separable if and only 2 (k ) if there exist vectors uw ∈ Rdk ?1 satisfying Eq.(5), and weights pw satisfying 0 ≤ pw ≤ 1 and w pw = 1 such that

R

T and

(N )

=

w

pw

N k =1

k) u( w ,

s(k) =

w

k) pw u ( w

(13a)

R

T

{k1 ,k2 ,··· ,kM }

=

w

pw

M i=1

(ki ) uw

(13b)

for 2 ≤ M ≤ N ; for all subsets {k1 , k2 , . . . , kM } ? {1, 2, . . . , N }, where s(k) is the Bloch vector of the mixed state density matrix for k th subsystem and (k ) uw represent the Bloch vector of the pure state of the k th subsystem contributing to the w th term in Eq. (1). This follows from proposition 1 and Eq. (1). However, in view of proposition (1a), the necessary and su?cient condition is given by Eq.(13a), so that Eq.(13b) can be dropped. The above result can not be used directly, as it amounts to rewriting Werner’s de?nition of separability in a di?erent way. However, it allows us to derive a necessary condition for separability for N -partite quantum states. We need some concepts in multilinear algebra. Consider a tensor T (N ) ∈ RI1 ×I2 ×···×IN (N ) , where Ik = d2 (n = 1, 2, · · · , N ) [10] is a matrix k ? 1. The nth matrix unfolding of T (N ) (N ) In ×(In+1 In+2 ...IN I1 I2 ...In?1 ) T(n) ∈ R . T(n) contains the element ti1 i2 ...iN at the position with row index in (in = 1, 2, · · · , In ) and column index (in+1 ? 1)In+2 In+3 . . . IN I1 I2 . . . In?1 + (in+2 ? 1)In+3 In+4 . . . IN I1 I2 . . . In?1 + · · · + (iN ? 1)I1 I2 . . . In?1 + (i1 ? 1)I2 I3 . . . In?1 + (i2 ? 1)I3 I4 . . . In?1 + · · · + in?1 . For n = 1, we take the last term in?1 = i0 = iN . This ordering is called backward cyclic [16]. To facilitate understanding, put N points on a circle and label them successively (N ) by i1 , i2 , · · · , iN . The consecutive terms in the expression for the column index in T(n) corresponding to ti1 ,i2 ,··· ,iN become quite apparent using this circle. (3) For T (3) ∈ RI1 ×I2 ×I3 the matrix unfolding T(1) contains the elements ti1 i2 i3 (ik = 1, 2, · · · , Ik ; k = 1, 2, 3) at the position with row number i1 and column number equal to (3) (i2 ? 1)I3 + i3 , T(2) contains ti1 i2 i3 at the position with row number i2 and column number equal to (i3 ? 1)I1 + i1 and T(3) contains ti1 i2 i3 at the position with row number i3 and column number equal to (i1 ? 1)I2 + i2 . 7

(3)

As an example [25], de?ne a tensor T (3) ∈ R3×2×3 , by t111 = t112 = t211 = ?t212 = 1, t213 = t311 = t313 = t121 = t122 = t221 = ?t222 = 2, t223 = t321 = t323 = 4, t113 = t312 = (3) t123 = t322 = 0. The matrix unfolding T(1) is given by ? ? 1 1 0 2 2 0 (3) T(1) = ? 1 ?1 2 2 ?2 4 ? . 2 0 2 4 0 4 Note that there are N possible matrix unfoldings of T (N ) . The matrix unfolding is called the matrization of the tensor [10,17]. We can now de?ne the Ky Fan norm of the tensor T (N ) (of order N ) over N matrix unfoldings of a tensor, as ||T (N ) ||KF = max{||T(n) ||KF }, n = 1, . . . , N ;

(N ) (N ) (N ) (N )

(14)

where ||T(n) ||KF is the Ky Fan norm of matrix T(n) de?ned as the sum of singular values of T(n) [26]. It is straightforward to check that ||T (N ) ||KF de?ned in (14) satis?es all the conditions of a norm and is also unitarily invariant [9,26]. The tensors in Eq.(13a) are called Kruskal tensors with the restriction 0 ≤ pw ≤ 1, w pw = 1 [14,16]. We are interested in ?nding the matrix unfoldings and Ky Fan norms of T (N ) occurring in Eq.(13a). The k th matrix unfolding for Kruskal tensor is [17] T(k) = U (k) Σ(U (N ) ⊙ U (N ?1) ⊙ · · · ⊙ U (k+1) ⊙ U (k?1) ⊙ · · · ⊙ U (1) )T .

(k ) (k ) (k ) (N )

(15)

Here U (k) = [u1 u2 . . . uR ] ∈ RIk ×R ; k = 1, 2, . . . N and R is the rank of Kruskal (k ) tensor [14,12,17], i.e. the number of terms in Eq.(13a). ui is a vector in RIk and is the ith column vector in the matrix U (k) . Σ is the R × R diagonal matrix, Σ =diag[p1 . . . pR ]. The symbol ⊙ denotes the Khatri-Rao product of matrices [17] U ∈ RI ×R and V ∈ RJ ×R de?ned as U ⊙ V = [u1 ? v1 u2 ? v2 . . . uR ? vR ] ∈ RIJ ×R where ui and vi , i = 1, 2, . . . R are column vectors of matrices U and V respectively. Eq.(15) can be rewritten as T(k) = U (k) Σ[v1 v2 . . . vR ]T = U (k) ΣV (k)

?) (k where vi ; i = 1, 2, . . . , R are the column vectors of the matrix ?) ? T (k (N ) (N ?1) ? V (k) ∈ RIN IN ?1 IN ?2 ...Ik+1 Ik?1 ...I1 ×R and vi = ui ? ui (k ?1) (1) (N ) ui ? · · · ? ui . Using Eq.(16) we can write T(k) as R (N ) ?) (k ?) (k ?) (k ?

T

(16) ? · · · ? ui

(k +1)

ui

(N ?2)

?

T(k) =

w =1

(N )

k ) (k ) pw u( w vw ; k = 1, 2, . . . , N.

?

T

(17)

Theorem 1 : If a N -partite quantum state of d1 d2 . . . dN dimension with Bloch representation (8) is fully separable, then ||T (N ) ||KF ≤ 1 N Π dk (dk ? 1). 2N k=1 8 (18)

Proof : If the state ρ is separable then T (N ) has to admit a decomposition of the form Eqs.(13) with ||uw ||2 = tensors, Eq.(14),

(k ) dk (dk ?1) ,k 2

= 1, 2, . . . , N. From de?nition of KF norm of

||T (N ) ||KF = max{||T(k) ||KF } ; k = 1, . . . , N. From Eq.(17), ||T (N ) ||KF = max{||

w k ) (k ) pw u( w vw ||KF } ; k = 1, . . . , N ?

T

(N )

≤ max{

w

k ) (k ) pw ||u( w vw ||KF } = max{ w

2

?

T

pw

1 N ? T k ) (k ?( ?w ) ||KF } Πk dk (dk ? 1)||u w v N 2

2 2 2

where d2 k ? R 1 ?1 ? · · · ? R Using

w

?) (k ) (k u ?w , v ?w are d2 1 ?1

unit vectors in Rdk ?1 and RdN ?1 ? RdN ?1 ?1 ? · · · ? Rdk+1 ?1 ? ?)T (k ) (k respectively, so that ||u ?w v ?w ||KF = 1 for all k = 1, 2, . . . , N .

1 ΠN d (d 2N k =1 k k

pw = 1 we get ||T (N ) ||KF ≤

? 1).

For a subsystem we get, Corollary 1 : If the reduced density matrix of a subsystem consisting of M out of N parts is separable then ||T {k1 ,k2 ,··· ,kM } ||KF ≤

1 2M

ΠM k =1 dk (dk ? 1).

N The negation of the above condition, that is, ||T (N ) ||KF > 21 N Πk =1 dk (dk ? 1), is a su?cient condition of entanglement of N -partite quantum state. This leads to a hierarchy of inseparability conditions which test entanglement in all the subsystems. For N = 2

the condition ||T (N ) ||KF ≤ 212 d1 (d1 ? 1)d2 (d2 ? 1) has been shown in Ref. [9], to be a su?cient condition for entanglement associated with any bipartite density matrix. Note that for N -qubits, di = 2, i = 1, 2, . . . , N , the above criterion becomes, for a separable state, ||T (N ) ||KF ≤ 1. d Consider a N qudit system Hs = ?N k =1 Hk in a state ρ, supported in the symmetric subspace of Hs . It is straightforward to see that all the tensors in the Bloch representation of ρ are supersymmetric, that is (see Eqs.(8) and (9)), tαk1 αk2 ···αkM = tP (αk1 )P (αk2 )···P (αkM ) , 2 ≤ M ≤ N, where P is any permutation over indices {αk1 , αk2 , · · · , αkM }. We have, neglecting the constant multipliers, tαk1 αk2 ···αkM = T r [ρk1 k2 ···kM λαk1 ? λαk2 ? · · · ? λαkM ]

= T r [(P T ρk1 k2 ···kM P )(P T λαk1 ? λαk2 ? · · · ? λαkM P )] = T r [ρk1 k2 ···kM (λP (αk1 ) ? λP (αk2 ) ? · · · ? λP (αkM ) )] = tP (αk1 )P (αk2 )···P (αkM ) 9

= T r [ρk1 k2 ···kM P P T λαk1 ? λαk2 ? · · · ? λαkM P P T ]

where P is the appropriate permutation matrix permuting the λ matrices within the tensor product [26], P T being the transpose of P satisfying P T = P ?1 . In particular T (N ) is supersymmetric. All matrix unfoldings of a supersymmetric tensor have the same set of singular values [10] and hence the same KF norm. Thus, for a N -qudit system in a state supported in the symmetric subspace, it is enough to calculate the KF norm for any (N ) one of the N matrix unfoldings to get max{||T(k) ||KF }.

4

A Su?cient Condition for Separability of a 3Partite Quantum State

Consider the Bloch representation of a tripartite state ρ acting on Hd1 ? Hd2 ? Hd3 , d1 ≤ d2 ≤ d3 . 1 (?3 Id + d1 d2 d3 k=1 k +

α1 α3

ρ=

rα1 λ(1) α1 +

α1 α2

sα2 λ(2) α2 +

α3

qα3 λ(3) α3 +

α1 α2

(2) tα1 α2 λ(1) α1 λα2

(3) tα1 α3 λ(1) α1 λα3 + α2 α3

(3) tα2 α3 λ(2) α2 λα3 + α1 α2 α3

(2) (3) tα1 α2 α3 λ(1) α1 λα2 λα3 ,

(19a)

where r, s and q are the Bloch vectors of three subsystems respectively , T {?,ν } = [tα? αν ] the correlation matrix between the subsystems ?, ν ; {?, ν } ? {1, 2, 3} and T (3) = [tα1 α2 α3 ] the correlation tensor among three subsystems. Before stating proposition 2, we need the following de?nition and result. Kruskal decomposition of a tensor T (N )

R

T

(N )

=

j =1

ξj uj ? uj ? · · · ? uj

(i) (i)

(1)

(2)

(N )

is called completely orthogonal if uk , ul = δkl , i = 1, 2, · · · , N ; k, l = 1, 2, · · · , R [13], where , denotes the scalar product of two vectors. If T (N ) has completely orthogonal Kruskal decomposition, then it is straightforward to show that

R

||T

(N )

||KF =

j =1

ξj ,

(20)

where R is the rank of T (N ) and ξj , j = 1, 2, · · · , R are the coe?cients occurring in the completely orthogonal Kruskal decomposition of T (N ) . In the proof of proposition 2, we assume that completely orthogonal Kruskal decomposition of T (k) , k > 2 is available. A completely orthogonal Kruskal decomposition may not be available for an arbitrary tensor 10

[13]. The general conditions under which the completely orthogonal Kruskal decomposition is possible is an open problem. We conjecture that completely orthogonal kruskal decomposition is available for all tensors in the Bloch representation of a quantum state, but we do not have a proof. As it stands, this issue has to be settled case by case. Proposition 2 : If a tripartite state ρ acting on Hd1 ? Hd2 ? Hd3 , d1 ≤ d2 ≤ d3 , with Bloch representation (19a), where T (3) has the completely orthogonal Kruskal decomposition, satis?es

2(d1 ? 1) ||r||2+ d1

2(d2 ? 1) ||s||2 + d2

2(d3 ? 1) ||q||2+ d3

{?,ν }

4(d? ? 1)(dν ? 1) {?,ν } ||T ||KF d? dν

+ then ρ is separable.

8(d1 ? 1)(d2 ? 1)(d3 ? 1) ||T ||KF ≤ 1, d1 d2 d3

(21)

Proof : The idea of the proof is as follows. (i) We ?rst decompose all the tensors in the Bloch representation of ρ as the completely orthogonal Kruskal decomposition in terms of the outer products of the vectors in the Bloch spaces of the subsystems (coherence vectors). (ii)We prove that we can decompose ρ using the Kruskal decompositions described in (i) above, as the linear combination of separable density matrices, which is a convex combination if the coe?cient of identity is positive. This condition is the same as the condition stated in the proposition. Let T {?,ν } ; {?, ν } ? {1, 2, 3} in Eq.(19a) have singular value decomposition T {?,ν } = (?) (ν ) T (?) (ν ) ; with ||ai ||2 = ||ai ||2 = 1 , for {?, ν } ? {1, 2, 3} and let T in i σi ai (ai ) Eq. (19a) have the completely orthogonal Kruskal decomposition T = j ξj uj ? vj ? wj [17,14,27] with ||uj ||2 = ||vj ||2 = ||wj ||2 = 1. We de?ne ? ai

(?)

=

d? 2(d? ?1)

ai

(?)

, ? ∈ {1, 2, 3}

so that we can rewrite T {?,ν } = . Similarly, we de?ne 4(d? ? 1)(dν ? 1) d? dν

T σi ? a? aν i) i (? i

(22a)

11

u ?j =

d1 2(d1 ?1)

uj ; v ?j =

d2 2(d2 ?1)

vj ; w ?j =

d3 2(d3 ?1)

wj , so that we can write

T =

8(d1 ? 1)(d2 ? 1)(d3 ? 1) d1 d2 d3

ξj u ?j ? v ?j ? w ?j

j

(22b)

If we substitute Eqs.(22a) and (22b) in ρ Eq.(19a), we get ρ= 1 (?3 Id + d1 d2 d3 k=1 k rα1 λ(1) α1 +

α1 α2 (?) (ν )

sα2 λ(2) α2 +

α3

qα3 λ(3) α3

+

{?,ν } α? αν

4(d? ? 1)(dν ? 1) d? dν

(?) (ν ) λαν ai )αν λα σi (? ai )α? (? ? i

+

8(d1 ? 1)(d2 ? 1)(d3 ? 1) d1 d2 d3 α

(?)

(2) (3) ξ j (u ?j )α1 (v ?j )α2 (w ? j )α3 λ(1) α1 λα2 λα3 j

2

(19b)

1 α2 α3

The coherence vectors ? ai occur in Dr (Rd? ?1 ), ? ai occur in Dr (Rdν ?1 ), u ?j occur in 2 2 d2 ? 1 d1 ?1 d2 ?1 ) and w ? j occur in Dr (R 3 ) (see Eq.(6)), so that they Dr ( R ), v ?j occur in Dr (R correspond to Bloch vectors. We can decompose ρ Eq.(19b) as the following convex combination of the density matrices 1 ′′′ ′ ′ ′ ρj , ρ′j , ρ′′ j , ρj ; ?i , ?i , τi , τi , πi , πi ; ρr , ρs , ρq and d1 d2 d3 Id1 d2 d3 ; 8(d1 ? 1)(d2 ? 1)(d3 ? 1) ξj ′′′ (ρj +ρ′j +ρ′′ j +ρj )+ d1 d2 d3 4

′ 4(d1 ? 1)(d3 ? 1) σi (τi + τi′ ) + d1 d3 2

(ν )

2

ρ=

j

i

4(d1 ? 1)(d2 ? 1) σi (?i +?′i ) d1 d2 2

+

i

i

′′ 4(d2 ? 1)(d3 ? 1) σi ′ (πi + πi ) d2 d3 2

+

2(d1 ? 1) ||r||2ρr + d1 ?

2(d2 ? 1) ||s||2ρs + d2 2(d3 ? 1) ||q||2 ? d3

2(d3 ? 1) ||q||2ρq + (1 ? d3

2(d1 ? 1) ||r||2 d1

2(d2 ? 1) ||s||2 ? d2 ?

{?,ν }

4(d? ? 1)(dν ? 1) {?,ν } ||T ||KF d? dν (23)

8(d1 ? 1)(d2 ? 1)(d3 ? 1) Id d d ||T ||KF ) 1 2 3 . d1 d2 d3 d1 d2 d3

where ρj in Bloch representation is 12

ρj = +

α1 α2

1 ?3 k =1 Idk + d1 d2 d3

(u ?j )α1 λ(1) α1 +

α1 α2

(v ?j )α2 λ(2) α2 +

α3

(w ? j )α3 λ(3) α3

(2) (u ?j )α1 (v ?j )α2 λ(1) α1 λα2 + α1 α3

(3) (u ?j )α1 (w ? j )α3 λ(1) α1 λα3 + α2 α3

(3) (v ?j )α2 (w ? j )α3 λ(2) α2 λα3

+

α1 α2 α3

(2) (3) (u ?j )α1 (v ?j )α2 (w ? j )α3 λ(1) α1 λα2 λα3

=

1 (Id + d1 d2 d3 1

(u ?j )α1 λ(1) α1 ) ? (Id2 +

α1 α2

(v ?j )α2 λ(2) α2 ) ? (Id3 +

α3

(w ? j )α3 λ(3) α3 ).

(24)

Note that ||T ||KF in Eq.(23) is de?ned via Eq.(20), which is based on completely orthogonal Kruskal decomposition of T . The Bloch vectors, correlation matrices and correlation tensors of the density matrices ′′′ ′ ′ ′ ρj , ρ′j , ρ′′ j , ρj ; ?i , ?i , τi , τi , πi , πi ; ρr , ρs , ρq are For ρj , rj = u ?j , sj = v ?j , qj = w ? j , Tj Tj = u ?j ? v ?j ? w ? j. For ρ′j , r′ j = u ?j , s′ j = ?v ?j , q′ j = ?w ? j , Tj Tj

′{2,3} T =v ?j w ?j , T ′j = u ?j ? v ?j ? w ?j. ′{1,2} T = ?u ?j v ?j , Tj ′{1,3} T = ?u ?j w ?j {1,2} T =u ?j v ?j , Tj {1,3} T =u ?j w ?j , Tj {2,3} T =v ?j w ?j

For ρ′′ j, ?j , s′′ j = v ?j , q′′ j = ?w ? j , Tj r′′ j = ?u Tj

′′{2,3} ′′{1,2} T = ?u ?j v ?j , Tj ′′{1,3} T =u ?j w ?j

T ?j ? v ?j ? w ? j. = ?v ?j w ?j , T ′′ j = u

For ρ′′′ j , ?j , q′′′ j = w ? j , Tj ?j , s′′′ j = ?v r′′′ j = ?u Tj

′′′{2,3} T , T ′′′ j = u ?j ? v ?j ? w ?j. = ?v ?j w ?j ′′′{1,2} T =u ?j v ?j , Tj ′′′{1,3} T = ?u ?j w ?j

For ?i , r? ai , s? ai , q? i =? i = ? i = 0 , Ti

(1) (2) ?{1,2}

=? ai ? ai 13

(1) (2)T

, Ti

?{1,3}

=0

Ti

?{2,3}

= 0 , Ti? = 0.

For ?′i , ai , q? ai , s? r? i = 0 , Ti i = ?? i = ?? ′ ′ ′ ? {1,3} ? {2,3} Ti = 0 , Ti = 0 , Ti? = 0. For τi ,

τ rτ ai , sτ ai , Ti i =? i = 0 , qi = ? τ {2,3} τ Ti = 0 , Ti = 0. (1) (3) τ { 1,2}

′

(1)

′

(2)

′

?′ {1,2}

=? ai ? ai

(1) (2)T

= 0 , Ti

τ {1,3}

=? ai ? ai

(1) (3)T

For τi′

τ rτ ai , sτ ai , Ti i = ?? i = 0 , qi = ?? ′ τ {2,3} τ′ Ti = 0 , Ti = 0.

′

(1)

′

′

(3)

τ ′ {1,2}

= 0 , Ti

τ ′ {1,3}

=? ai ? ai

(1) (3)T

For π ,

π rπ ai , qπ ai , T π {1, 2}i = 0 , Ti i = 0 , si = ? i = ? π {2,3} (2) (3)T Ti =? ai ? ai , Tiπ = 0. (2) (3) π {1,3}

=0

For π ′ ,

π ai , qπ ai , Ti rπ i = ?? i = 0 , si = ?? ′ π {2,3} (2) (3)T π′ Ti =? ai ? ai , Ti = 0.

′ ′

(2)

′

(3)

π ′ {1,2}

= 0 , Ti

π ′ {1,3}

=0

For ρr , rr = For ρs , rs = 0 , ss = For ρq , rq = 0 , sq = 0 , qq =

q d3 2(d3 ?1) ||q||2 d2 s 2(d2 ?1) ||s||2 r d1 2(d1 ?1) ||r||2

, sr = 0 , qr = 0 , Tr

{?,ν }

= 0 ; ?{?, ν } ? {1, 2, 3} , Tr = 0.

, qs = 0 , Ts

{?,ν }

= 0 ; ?{?, ν } ? {1, 2, 3} , Ts = 0.

, Tq

{?,ν }

= 0 ; ?{?, ν } ? {1, 2, 3} , Tq = 0.

′′′ ′ ′ ′ If we write all matrices ρ′j , ρ′′ j , ρj ; ?i , ?i , τi , τi , πi , πi ; ρr , ρs , ρq (as we have done for ρj in Eq.(24)) in the Bloch representation and substitute them in Eq.(23) we get

14

ρ as in Eq.(19b). To understand this let us see how the ?rst term in Eq.(23) adds up to give the ′′′ last term in Eq.(19b). The de?nition of ρj , ρ′j , ρ′′ j , ρj (denoting the Bloch vectors by s1 , s2 , s3 , s4 , ....) can be summarized in the tabular form

Table 1 Correspondence between the ?rst term in Eq.(23) and the last term in Eq. (19b). s1 u ?j u ?j ?u ?j ?u ?j s2 v ?j ?v ?j v ?j ?v ?j s3 w ?j ?w ?j ?w ?j w ?j s1 s2 u ?j v ?j ?u ?j v ?j ?u ?j v ?j u ?j v ?j s1 s3 u ?j w ?j ?u ?j w ?j u ?j w ?j ?u ?j w ?j s2 s3 v ?j w ?j v ?j w ?j ?v ?j w ?j ?v ?j w ?j s1 s2 s3 u ?j v ?j w ?j u ?j v ?j w ?j u ?j v ?j w ?j u ?j v ?j w ?j

ρj ρ′j ρ′′ j ρ′′′ j

′′′ The contribution of each column to ρj + ρ′j + ρ′′ j + ρj is zero except the last column which reproduces the last term in Eq.(19b). We can get the contributions of each term ′′′ in ρj , ρ′j , ρ′′ j , ρj to their sum by just keeping track of their signs. Thus we only need the following table (dropping j )

Table 2 Contributions of various terms in ρ, ρ′ , ρ′′ , ρ′′′ to their sum. s1 + + ? ? s2 + ? + ? s3 + ? ? + s1 s2 + ? ? + s1 s3 + ? + ? s2 s3 + + ? ? s1 s2 s3 + + + +

ρ ρ′ ρ′′ ρ′′′

In the same way, the contributions of the terms involving ?, τ, π are obtained by using the table corresponding to table 2 for the bipartite case [9]. ?, τ, π which contain tensors of order two correspond to three 2-partite subsystems 12,13 and 23 . The corresponding tables are

Table 3 Contributions to ? + ?′ 15

? ?′

s1 + ?

s2 + ?

s3 0 0

s1 s2 + +

s1 s3 0 0

s2 s3 0 0

s1 s2 s3 0 0

Table 4 Contributions to τ + τ ′ τ τ′ s1 + ? s2 0 0 s3 + ? s1 s2 0 0 s1 s3 + + s2 s3 0 0 s1 s2 s3 0 0

Table 5 Contributions to π + π ′ π π′ s1 0 0 s2 + ? s3 + ? s1 s2 0 0 s1 s3 0 0 s2 s3 + + s1 s2 s3 0 0

Tables 2, 3, 4, 5 encode the procedure to construct the possible separable state given in Eq.(23). We now note the following points (i) If the condition (21) holds, then the coe?cient of the matrix Id1 d2 d3 in Eq.(23) is positive which ensures that the decomposition (23) of ρ is positive semide?nite. ′′′ (ii) By virtue of Eq.(6), all the coherence vectors occurring in ρ′j , ρ′′ j , ρj ; ′ ?i , ?′i , τi , τi′ , πi , πi ; ρr , ρs , ρq belong to the corresponding Bloch spaces.

′′′ ′ ′ ′ By (i) and (ii) we conclude that ρ′j , ρ′′ j , ρj ; ?i , ?i , τi , τi , πi , πi ; ρr , ρs , ρq constitute density matrices. Further, all these matrices satisfy condition (11) so that, via proposition 1, all these matrices correspond to pure separable states, equal to the tensor products of their reductions. Therefore, they constitute density matrices and they are separable and so must be ρ.

We can generalize proposition 2 to the N -partite case by constructing the tables successively for N = 4, 5, 6, · · · . First note that the number of ρ s in the ?rst term of Eq.(23) lifted to the N -partite case is 2N ?1 . For N = 4 we have eight. The corresponding table is

16

Table 6 Generalization of Table 1 to N = 4. ρ(1) ρ(2) ρ(3) ρ(4) ρ(5) ρ(6) ρ(7) ρ(8) s1 + + + + ? ? ? ? s2 + + ? ? + + ? ? s3 + ? + ? + ? + ? s4 + ? ? + ? + + ? s1 s2 + + ? ? ? ? + + s1 s3 + ? + ? ? + ? + s1 s4 + ? ? + + ? ? + s2 s3 + ? ? + + ? ? + s2 s4 + ? + ? ? + ? + s3 s4 + + ? ? ? ? + + s1 s2 s3 + ? ? + ? + + ? s1 s2 s4 + ? + ? + ? + ?

(Table 6. Continued) s1 s3 s4 + + ? ? + + ? ? s2 s3 s4 + + + + ? ? ? ? s1 s2 s3 s4 + + + + + + + +

We see that the contribution of each column to the sum i ρ(i) is zero except the last one corresponding to the Kruskal decomposition of T (N ) occurring in the Bloch representation of the given state ρ. For general case of N -partite state we construct the table for ρ(i) , i = 1, 2, · · · , 2N ?1 as follows. First column consists of 2N ?2 plus signs followed by 2N ?2 minus signs. Second column comprises alternating 2N ?3 plus and minus signs. Continuing in this way upto 2N ?N = 1 we get alternating plus and minus signs in the (N ? 1)th column. We set the N th column to ensure that there are zero or even number of minus signs in each row. Rest of the columns can be constructed by appropriate multiplications. This procedure can be checked on table 6. We denote the sequence of such tables for N = 2, 3, 4, · · · as Ti , i = 2, 3, 4, · · · . The tables corresponding to (N ? 1), (N ? 2), ..., 2 partite subsystems giving rise to the remaining terms in the equation (23), lifted to N -partite case, are obtained from TN ?1 , TN ?2 , ..., T3 , T2 , exactly as described in the proof of proposition 2. In this way we N ?1 N 2N ?1?i +1. can lift eq.(23) to the N -partite case, with the total numbers of terms i =0 i 17

Once this is done, the rest of the proof for N -partite case follows as in proposition 2. Thus we have Proposition (2a): If a N -partite state ρ acting on H = Hd1 ? Hd2 ? · · · ? HdN , d1 ≤ d2 ≤ · · · ≤ dN with Bloch representation (8), where all T (k) , k > 2 have the completely orthogonal Kruskal decomposition, satisfy 2(dk ? 1) ||sk ||2 + dk 4(d? ? 1)(dν ? 1) {?,ν } ||T ||KF d? dν

k

{?,ν }

+

{?,ν,κ}

8(d? ? 1)(dν ? 1)(dκ ? 1) {?,ν,κ} ||T ||KF + · · · + d? dν dκ 2N ΠN i (di ? 1) ||T (N ) ||KF ≤ 1, N Πi di (21)

{k1 ,k2 ,··· ,kM }

2M Πki (dki ? 1) {k1 ,k2 ,··· ,kM } ||T ||KF + · · · + Πki dki

then ρ is separable. For a N -qubit system Theorem 1 and proposition 2a together imply Corollary 2 : Let a N -qubit state have a Bloch representation ρ= 1 (k ) (?N k =1 I2 + N 2 α

(2) (N ) tα1 ···αN λ(1) α1 λα2 · · · λαN ),

1 ···αN

and let the tensor in the second term have the completely orthogonal Kruskal decomposition. Then ρ is separable if and only if ||T (N ) ||KF ≤ 1.

5

Examples

We now investigate our separability criterion (18) for mixed states. We consider N -qubit state 1?p (N ) I + p| ψ ψ | , 0 ≤ p ≤ 1 (25) ρnoisy = 2N where |ψ is a N -qubit W state or GHZ state. We test for N = 3, 4, 5 and 6 qubits. We get,

18

Table 7 The values of p above which the states are entangled. |GHZ |W p> p> 0.35355 0.3068 0.2 0.3018 0.17675 0.30225 0.1112 0.3045 N 3 4 5 6

Entanglement in various partitions of W noisy state Eq.(25) is obtained by using (N ? n) qubit reduced W noisy state ρnoisy (W ) =

(N ?n)

n N ?n 1?p IN ?n + p|0N ?n 0N ?n | + p|WN ?n WN ?n | N ? n 2 N N

(26)

For N = 6 and n = 2 we found that the state is entangled for 0.491 < p ≤ 1. For N qutrits (d = 3) we test for ρnoisy = where |ψ =

1 √ d d k =1 (N )

1?p I + p| ψ ψ | 3N

(27)

|kkk . . . is the maximally entangled state for N qutrits.

(N )

For N = 3 and N = 4 (qutrits) the state ρnoisy in Eq. (27) is entangled for 0.2285 < p ≤ 1, 0.2162 < p ≤ 1, The state ρnoisy = N =3 N =4 (28)

1?p I + p| ψ ψ | (29) 24 (|112 + |123 + |214 + |234 ) in the space C2 ? C3 ? C4 is found to be where |ψ = 1 2 entangled for 0.24152 < p ≤ 1. All of the above examples involve NPT states. Now we apply our criterion to PPT entangled states for which PPT criterion is not available. We apply our criterion to the three qutrit bound entangled state considered by L. Clarisse and P. Wocjan [27], given by ρc ? |ψ ψ | where ρc is the chess-board state given in [27] and |ψ is an uncorrelated ancilla. Our criterion detects the entanglement of this 19

state as ||T (12) || = 3.75 > 3. Further, the four qutrit state ρ = (1 ? β )ρc ? ρc + βI/81 considered by the same authors yields entanglement for 0 ≤ β ≤ 0.2, after tracing out either subsystems 1 and 2 or subsystems 3 and 4. Now we consider the important example of the Smolin state [28,29], which is a four qubit bound entangled state given by ρunlock ABCD = 1 4

4 i i i i |ψAB ψAB | ? |ψCD ψCD | i=1

(30)

i i where |ψAB and |ψCD are the Bell states. ρunlock ABCD has the Bloch representation 3 ? 4 1 unlock ?4 ρABCD = 16 (I + i=1 σi ) so that Corollary 2 applies (note that the requirement of completely orthogonal Kruskal decomposition is trivially satis?ed). We ?nd for this state ||T (4) ||KF = 3 > 1 con?rming its entanglement.

Our last example is the four qubit bound entangled state due to W. D¨ ur [30,31] ρBE 4 1 1 = (| ψ ψ | + 5 2

4

(Pi + P i ))

i=1

where |ψ is a 4-party (GHZ) state , Pi is the projector onto the state |φi , which is a product state equal to |1 for party i and |0 for the rest , and P i is obtained from Pi by replacing all zeros by ones and vice versa. We get ||T (4) ||KF = 1.4 > 1 con?rming the entanglement of this state.

6

Summary

In conclusion we have presented a new criterion for separability of N partite quantum states based on the Bloch representation of states. This criterion is quite general, as it applies to all N -partite quantum states living in H = Hd1 ? Hd2 ? · · · ? HdN , where, in general, the Hilbert space dimensions of various parts are not equal. Most of the previous such criteria had restricted domain of applicability like the states supported on symmetric subspace [4] or, are, in general, restricted to bipartite case. In proposition 2 we have given a su?cient condition for the separability of a tripartite state under the condition that the tensors occurring in the Bloch representation of the state have completely orthogonal Kruskal decomposition. This result can be generalized to the N -partite case. Via corollary 2 we give a necessary an su?cient condition to test the separability of a class of N -qubit states which includes N -qubit PPT states. Smolin state (30) is an important example in this class. The key idea in our work is the matrization of multidimensional tensors, in particular, Kruskal matrization. We have de?ned a new tensor norm as the maximum of the KF norms of all the matrix unfoldings of a tensor, which is easily computed. We have also shown that this norm can be calculated even more e?ciently for 20

a N -qudit state supported in the symmetric subspace. It will be interesting to seek a relation of this tensor norm with other entanglement measures. Again, the entanglement measures like concurrence known so far are successfully applied to pure states, bipartite or multipartite, while our tensor norm can be easily computed for arbitrary N -partite quantum state. Finally, our result on full separability (proposition 1) of N -partite pure states can be easily moulded for the k -separability of an N -partite pure state. In fact it is straightforward to construct an algorithm giving the complete factorization of the N partite pure state (see the paragraph following the proof of proposition 1). It is also easy to see that theorem 1 can be applied to any partition of a N -partite system via the Bloch representation in terms of the generators of the appropriate SU groups. Most important is the observation that all the tensors in the Bloch representation can be computed using the measured values of the basis operators {λαk } so that our detectiblity criterion is experimentally implementable. Acknowledgments We thank Guruprasad Kar and R.Simon for encouragement. We thank Julio I. de Vicente for a very useful communication. We thank Guruprasad Kar and Sibasish Ghosh for suggesting the last two examples. ASMH thanks Sana’a University for ?nancial support. References [1] M. B. Plenio and S. Virmani (2007), An introduction to entanglement measures, Quantum Inf. Comput., Vol. 7, pp. 001-051. ˙ [2] K. Zyczkowski and I. Bengstsson (2006), An introduction to quantum entanglement: a geometric approach, quant-ph/0606228. [3] R.F. Werner (1989), Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model , Phys. Rev. A , 40, pp. 4277-4281. [4] A. R. Usha Devi, R. Prabhu, and A. K. Rajagopal (2007), Characterizing Multiparticle Entanglement in Symmetric N-Qubit States via Negativity of Covariance Matrices , Phys. Rev. Lett., 98, pp. 060501. [5] Florian Mintert, Marek Ku? s, and Andreas Buchleitner (2005), Concurrence of mixed multipartite quantum states, Phys. Rev. Lett., 95, pp. 260502. [6] K. Chen and L. Wu (2002), The generalized partial transposition criterion for separability of multipartite quantum states, Phys. Lett. A, 306, pp. 14-20. [7] A. Ac? ?n, D. Bruss, M. Lewenstein and A. Sanpera (2001), Classi?cation of mixed three-qubit states , Phys. Rev. Lett., 87, pp. 040401 . 21

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