Quantum Entanglement: Separability, Measure, Fidelity of Teleportation, and Distillation

Abstract

Quantum entanglement plays crucial roles in quantum information processing. Quantum entangled states have become the key ingredient in the rapidly expanding field of quantum information science. Although the nonclassical nature of entanglement has been recognized for many years, considerable efforts have been taken to understand and characterize its properties recently. In this review, we introduce some recent results in the theory of quantum entanglement. In particular separability criteria based on the Bloch representation, covariance matrix, normal form and entanglement witness, lower bounds, subadditivity property of concurrence and tangle, fully entangled fraction related to the optimal fidelity of quantum teleportation, and entanglement distillation will be discussed in detail.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Ming Li 1 and Shao-Ming Fei 2, 3 and Xianqing Li-Jost 3

Recommended by NaiHuan Jing

1, College of Mathematics and Computational Science, China University of Petroleum, 257061 Dongying, China
2, Department of Mathematics, Capital Normal University, 100037 Beijing, China
3, Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

Received 29 August 2009; Accepted 2 December 2009

1. Introduction

Entanglement is the characteristic trait of quantum mechanics, and it reflects the property that a quantum system can simultaneously appear in two or more different states [1]. This feature implies the existence of global states of composite system which cannot be written as a product of the states of individual subsystems. This phenomenon [2], now known as "quantum entanglement," plays crucial roles in quantum information processing [3]. Quantum entangled states have become the key ingredient in the rapidly expanding field of quantum information science, with remarkable prospective applications such as quantum computation [3, 4], quantum teleportation [5-9], dense coding [10], quantum cryptographic schemes [11-13], entanglement swapping [14-18],, and remote states preparation (RSP) [19-24]. All such effects are based on entanglement and have been demonstrated in pioneering experiments.

It has become clear that entanglement is not only the subject of philosophical debates, but also a new quantum resource for tasks which cannot be performed by means of classical resources. Although considerable efforts have been taken to understand and characterize the properties of quantum entanglement recently, the physical character and mathematical structure of entangled states have not been satisfactorily understood yet [25, 26]. In this review we mainly introduce some recent results related to our researches on several basic questions in this subject.

(1) Separability of Quantum States

We first discuss the separability of a quantum states; namely, for a given quantum state, how we can know whether or not it is entangled.

For pure quantum states, there are many ways to verify the separability. For instance, for a bipartite pure quantum state the separability is easily determined in terms of its Schmidt numbers. For multipartite pure states, the generalized concurrence given in [27] can be used to judge if the state is separable or not. In addition separable states must satisfy all possible Bell inequalities [28].

For mixed states we still have no general criterion. The well-known PPT (partial positive transposition) criterion was proposed by Peres in 1996 [29]. It says that for any bipartite separable quantum state the density matrix must be positive under partial transposition. By using the method of positive maps Horodecki et al. [30] showed that the Peres' criterion is also sufficient for 2×2 and 2×3 bipartite systems. And for higher dimensional states, the PPT criterion is only necessary. Horodecki [31] has constructed some classes entangled states with positive partial transposes for 3×3 and 2×4 systems. States of this kind are said to be bound entangled (BE). Another powerful operational criterion is the realignment criterion [32, 33]. It demonstrates a remarkable ability to detect many bound entangled states and even genuinely tripartite entanglement [34]. Considerable efforts have been made in finding stronger variants and multipartite generalizations for this criterion [35-39]. It was shown that PPT criterion and realignment criterion are equivalent to the permutations of the density matrix's indices [34]. Another important criterion for separability is the reduction criterion [40, 41]. This criterion is equivalent to the PPT criterion for 2×N composite systems. Although it is generally weaker than the PPT, the reduction criteria have tight relation to the distillation of quantum states.

There are also some other necessary criteria for separability. Nielsen and Kempe [42] presented a necessary criterion called majorization: the decreasing ordered vector of the eigenvalues for ρ is majorized by that of ^ρ_A1 or ^ρ_A2 alone for a separable state. That is, if a state ρ is separable, then ^{λρ[arrow down]} [precedes]^{λρ_A1 [arrow down]} , ^{λρ[arrow down]} [precedes]^{λρ_A2 [arrow down]} . Here ^{λρ[arrow down]} denotes the decreasing ordered vector of the eigenvalues of ρ . A d -dimensional vector ^{x[arrow down]} is majorized by ^{y[arrow down]} , ^{x[arrow down]} [precedes]^{y[arrow down]} , if ^{∑j=1kxj[arrow down]} ≤^{∑j=1kyj[arrow down]} for k=1,...,d-1 and the equality holds for k=d . Zeros are appended to the vectors ^{λρ_A1 ,_A2 [arrow down] such that their dimensions are equal to the one of λρ[arrow down] .}

In [31], another necessary criterion called range criterion was given. If a bipartite state ρ acting on the space _{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} is separable, then there exists a family of product vectors _ψi [ecedil]7; _[varphi]i such that (i) they span the range of ρ ; (ii) the vector {_ψi [ecedil]7;^{[varphi]i*}i=1k} spans the range of ^ρ_TB , where * denotes complex conjugation in the basis in which partial transposition was performed and ^ρ_TB is the partially transposed matrix of ρ with respect to the subspace B . In particular, any of the vectors _ψi [ecedil]7;^[varphi]i* belongs to the range of ρ .

Recently, some elegant results for the separability problem have been derived. In [43-45], a separability criteria based on the local uncertainty relations (LURs) was obtained. The authors show that, for any separable state ρ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} , [figure omitted; refer to PDF] where ^GkA or ^GkB are arbitrary local orthogonal and normalized operators (LOOs) in _{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} . This criterion is strictly stronger than the realignment criterion. Thus more bound entangled quantum states can be recognized by the LUR criterion. The criterion is optimized in [46] by choosing the optimal LOOs. In [47] a criterion based on the correlation matrix of a state has been presented. The correlation matrix criterion is shown to be independent of PPT and realignment criterion [48], that is, there exist quantum states that can be recognized by correlation criterion while the PPT and realignment criterion fail. The covariance matrix of a quantum state is also used to study separability in [49]. It has been shown that the LUR criterion, including the optimized one, can be derived from the covariance matrix criterion [50].

(2) Measure of Quantum Entanglement

One of the most difficult and fundamental problems in entanglement theory is to quantify entanglement. The initial idea to quantify entanglement was connected with its usefulness in terms of communication [51]. A good entanglement measure has to fulfill some conditions [52]. For bipartite quantum systems, we have several good entanglement measures such as Entanglement of Formation (EOF), Concurrence, and Tangle ctc. For two-qubit systems it has been proved that EOF is a monotonically increasing function of the concurrence and an elegant formula for the concurrence was derived analytically by Wootters [53]. However with the increasing dimensions of the subsystems the computation of EOF and concurrence become formidably difficult. A few explicit analytic formulae for EOF and concurrence have been found only for some special symmetric states [54-58].

The first analytic lower bound of concurrence for arbitrary dimensional bipartite quantum states was derived by Mintert et al. in [59]. By using the positive partial transposition (PPT) and realignment separability criterion, analytic lower bounds on EOF and concurrence for any dimensional mixed bipartite quantum states have been derived in [60, 61]. These bounds are exact for some special classes of states and can be used to detect many bound entangled states. In [62] another lower bound on EOF for bipartite states has been presented from a new separability criterion [63]. A lower bound of concurrence based on local uncertainty relations (LURs) criterion is derived in [64]. This bound is further optimized in [46]. The lower bound of concurrence for tripartite systems has been studied in [65]. In [66, 67] the authors presented lower bounds of concurrence for bipartite systems by considering the "two-qubit" entanglement of bipartite quantum states with arbitrary dimensions. It has been shown that this lower bound has a tight relationship with the distillability of bipartite quantum states. Tangle is also a good entanglement measure that has a close relation with concurrence, as it is defined by the square of the concurrence for a pure state. It is also meaningful to derive tight lower and upper bounds for tangle [68].

In [69] Mintert et al. proposed an experimental method to measure the concurrence directly by using joint measurements on two copies of a pure state. Then Walborn et al. presented an experimental determination of concurrence for two-qubit states [70, 71], where only one-setting measurement is needed, but two copies of the state have to be prepared in every measurement. In [72] another way of experimental determination of concurrence for two-qubit and multiqubit states has been presented, in which only one copy of the state is needed in every measurement. To determine the concurrence of the two-qubit state used in [70, 71], also one-setting measurement is needed, which avoids the preparation of the twin states or the imperfect copy of the unknown state, and the experimental difficulty is dramatically reduced.

(3) Fidelity of Quantum Teleportation and Distillation

Quantum teleportation, or entanglement-assisted teleportation, is a technique used to transfer information on a quantum level, usually from one particle (or series of particles) to another particle (or series of particles) in another location via quantum entanglement. It does not transport energy or matter, nor does it allow communication of information at super luminal (faster than light) speed.

In [5-7], Bennett et al. first presented a protocol to teleport an unknown qubit state by using a pair of maximally entangled pure qubit state. The protocol is generalized to transmit high-dimensional quantum states [8, 9]. The optimal fidelity of teleportation is shown to be determined by the fully entangled fraction of the entangled resource which is generally a mixed state. Nevertheless similar to the estimation of concurrence, the computation of the fully entangled fraction for a given mixed state is also very difficult.

The distillation protocol has been presented to get maximally entangled pure states from many entangled mixed states by means of local quantum operations and classical communication (LQCC) between the parties sharing the pairs of particles in this mixed state [73-76]. Bennett et al. first derived a protocol to distill one maximally entangled pure Bell state from many copies of not maximally entangled quantum mixed states in [73] in 1996. The protocol is then generalized to distill any bipartite quantum state with higher dimension by M. Horodecki and P. Horodecki in 1999 [77]. It is proven that a quantum state can be always distilled if it violates the reduced matrix separability criterion [77].

This review mainly contains three parts. In Section 2 we investigate the separability of quantum states. We first introduce several important separability criteria. Then we discuss the criteria by using the Bloch representation of the density matrix of a quantum state. We also study the covariance matrix of a quantum density matrix and derive separability criterion for multipartite systems. We investigate the normal forms for multipartite quantum states at the end of this section and show that the normal form can be used to improve the power of these criteria. In Section 3 we mainly consider the entanglement measure concurrence. We investigate the lower and upper bounds of concurrence for both bipartite and multipartite systems. We also show that the concurrence and tangle of two entangled quantum states will be always larger than that of one, even if both of the two states are bound entangled (not distillable). In Section 4 we study the fully entangled fraction of an arbitrary bipartite quantum state. We derive precise formula of fully entangled fraction for two-qubit system. For bipartite system with higher dimension we obtain tight upper bounds which can not only be used to estimate the optimal teleportation fidelity but also help to improve the distillation protocol. We further investigate the evolution of the fully entangled fraction when one of the bipartite system undergoes a noisy channel. We give a summary and conclusion in the last section.

2. Separability Criteria and Normal Form

A multipartite pure quantum state _ρ12...N ∈_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} is said to be fully separable if it can be written as

[figure omitted; refer to PDF] where _ρ1 and _ρ2 ,...,_ρN are reduced density matrices defined as _ρ1 =_{Tr 23...N} [_ρ12...N ] , _ρ2 =_{Tr 13...N} [_ρ12...N ],...,_ρN =_{Tr 12...N-1} [_ρ12...N ] . This is equivalent to the condition [figure omitted; refer to PDF] where |_ψ1 ...∈_{[Hamiltonian (script capital H)]1} , |_[varphi]2 ...∈_{[Hamiltonian (script capital H)]2} ,...,|_μN ...∈_{[Hamiltonian (script capital H)]N} .

A multipartite quantum mixed state _ρ12...N ∈_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} is said to be fully separable if it can be written as

[figure omitted; refer to PDF] where ^{_ρi 1} ,^{_ρi 2} ,...,^{_ρi N} are the reduced density matrices with respect to the systems 1,2,...,N , respectively, _qi >0 , and _∑iqi =1 . This is equivalent to the condition [figure omitted; refer to PDF] where |^ψi1 ...,|^[varphi]i2 ...,...,|^μiN ... are normalized pure states of systems 1,2,...,N , respectively, _pi >0 , and _∑ipi =1 .

For pure states, the definition (2.1) itself is an operational separability criterion. In particular, for bipartite case, there are Schmidt decompositions.

Theorem 2.1 (see Schmidt decomposition in [78]).

Suppose that |ψ...∈[Hamiltonian (script capital H)]A [ecedil]7;[Hamiltonian (script capital H)]B is a pure state of a composite system, AB , then there exist orthonormal states |iA ... for system A and orthonormal states |iB ... for system B such that [figure omitted; refer to PDF] where λi are nonnegative real numbers satisfying ∑iλi 2 =1 , known as Schmidt coefficients.

|_iA ... and |_iB ... are called Schmidt bases with respect to _{[Hamiltonian (script capital H)]A} and _{[Hamiltonian (script capital H)]B} . The number of nonzero values _λi is called Schmidt number, also known as Schmidt rank, which is invariant under unitary transformations on system A or system B . For a bipartite pure state |ψ... , |ψ... is separable if and only if the Schmidt number of |ψ... is one.

For multipartite pure states, one has no such Schmidt decomposition. In [79] it has been verified that any pure three-qubit state |Ψ... can be uniquely written as

[figure omitted; refer to PDF] with normalization condition _λi ≥0, 0≤ψ≤π , where _∑iμi =1 , _μi ≡^λi2 . Equation (2.6) is called generalized Schmidt decomposition.

For mixed states it is generally very hard to verify whether a decomposition like (2.3) exists. For a given generic separable density matrix, it is also not easy to find the decomposition (2.3) in detail.

2.1. Separability Criteria for Mixed States

In this section we introduce several separability criteria and the relations among themselves. These criteria have also tight relations with lower bounds of entanglement measures and distillation that will be discussed in the next section.

2.1.1. Partial Positive Transpose Criterion

The positive partial transpose (PPT) criterion provided by Peres [29] says that if a bipartite state _ρAB ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} is separable, then the new matrix ^ρAB_TB with matrix elements defined in some fixed product basis as [figure omitted; refer to PDF] is also a density matrix (i.e., it has nonnegative spectrum). The operation _TB , called partial transpose, just corresponds to the transposition of the indices with respect to the second subsystem B . It has an interpretation as a partial time reversal [80].

Afterwards Horodecki et al. showed that Peres' criterion is also sufficient for 2×2 and 2×3 bipartite systems [30]. This criterion is now called PPT or Peres-Horodecki (P-H) criterion. For high-dimensional states, the P-H criterion is only necessary. Horodecki has constructed some classes of families of entangled states with positive partial transposes for 3×3 and 2×4 systems [31]. States of this kind are said to be bound entangled (BE).

2.1.2. Reduced Density Matrix Criterion

Cerf et al. [81] and M. Horodecki and P. Horodecki [82], independently, introduced a map Γ:ρ[arrow right]_{Tr B} [_ρAB ][ecedil]7;I-_ρAB (I[ecedil]7;_{Tr A} [_ρAB ]-_ρAB ), which gives rise to a simple necessary condition for separability in arbitrary dimensions, called the reduction criterion. If _ρAB is separable, then [figure omitted; refer to PDF] where _ρA =_{Tr B} [_ρAB ] , _ρB =_{Tr A} [_ρAB ] . This criterion is simply equivalent to the P-H criterion for 2×n composite systems. It is also sufficient for 2×2 and 2×3 systems. In higher dimensions the reduction criterion is weaker than the P-H criterion.

2.1.3. Realignment Criterion

There is yet another class of criteria based on linear contractions on product states. They stem from the new criterion discovered in [33, 83] called computable cross-norm (CCN) criterion or matrix realignment criterion which is operational and independent on PPT test [29]. If a state _ρAB is separable, then the realigned matrix ...(ρ) with elements ...(ρ_)ij,kl =_ρik,jl has trace norm not greater than one: [figure omitted; refer to PDF] Quite remarkably, the realignment criterion can detect some PPT entangled (bound entangled) states [33, 83] and can be used for construction of some nondecomposable maps. It also provides nice lower bound for concurrence [61].

2.1.4. Criteria Based on Bloch Representations

Any Hermitian operator on an N -dimensional Hilbert space [Hamiltonian (script capital H)] can be expressed according to the generators of the special unitary group SU(N) [84]. The generators of SU(N) can be introduced according to the transition-projection operators _Pjk =|j......k| , where |i... , i=1,...,N , are the orthonormal eigenstates of a linear Hermitian operator on [Hamiltonian (script capital H)] . Set [figure omitted; refer to PDF] where 1≤l≤N-1 and 1≤j<k≤N . We get a set of ^N2 -1 operators [figure omitted; refer to PDF] which satisfies the relations [figure omitted; refer to PDF] and thus generate the SU(N) [85].

Any Hermitian operator ρ in [Hamiltonian (script capital H)] can be represented in terms of these generators of SU(N) as

[figure omitted; refer to PDF] where _IN is a unit matrix and r=(_r1 ,_r2 ,...,_{r^N2 -1} )∈^{...N2 -1} and r is called Bloch vector. The set of all the Bloch vectors that constitute a density operator is known as the Bloch vector space B(^{...N2 -1} ) .

A matrix of the form (2.13) is of unit trace and Hermitian, but it might not be positive. To guarantee the positivity restrictions must be imposed on the Bloch vector. It is shown that B(^{...N2 -1} ) is a subset of the ball _DR (^{...N2 -1} ) of radius R=2(1-1/N) , which is the minimum ball containing it, and that the ball _Dr (^{...N2 -1} ) of radius r=2/N(N-1) is included in B(^{...N2 -1} ) [86], that is,

[figure omitted; refer to PDF]

Let the dimensions of systems A, B , and C be _dA =_N1 , _dB =_N2 , and _dC =_N3 , respectively. Any tripartite quantum states _ρABC ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} [ecedil]7;_{[Hamiltonian (script capital H)]C} can be written as

[figure omitted; refer to PDF] where _λi (1) , _λj (2) are the generators of SU(_N1 ) and SU(_N2 ) ; _Mi , _M...j , and _Mij are operators of _{[Hamiltonian (script capital H)]C} .

Theorem 2.2.

Let r∈^{...N12 -1} , s∈^{...N22 -1} and |r|≤2/_N1 (_N1 -1) , |s|≤2/_N2 (_N2 -1) . For a tripartite quantum state ρ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} [ecedil]7;_{[Hamiltonian (script capital H)]C} with representation (2.15), one has [87] [figure omitted; refer to PDF]

Proof.

Since r∈^{...N12 -1} , s∈^{...N22 -1} and |r|≤2/_N1 (_N1 -1) , |s|≤2/_N2 (_N2 -1) , we have that _A1 ≡(1/2)((2/_N1 )I-^{∑i=1N12 -1}_riλi (1)) and _A2 ≡(1/2)((2/_N2 )I-^{∑j=1N22 -1}_sjλj (2)) are positive Hermitian operators. Let A=_A1 [ecedil]7;_A2 [ecedil]7;_IN3 . Then AρA≥0 and (AρA^)[dagger] =AρA . The partial trace of AρA over _{[Hamiltonian (script capital H)]A} (and _{[Hamiltonian (script capital H)]B} ) should be also positive. Hence [figure omitted; refer to PDF]

Formula (2.16) is valid for any tripartite state. By setting s=0 in (2.16), one can get a result for bipartite systems.

Corollary 2.3.

Let _ρAB ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} , which can be generally written as _ρAB =_IN1 [ecedil]7;_M0 +^{∑j=1N12 -1}_λj [ecedil]7;_Mj , then, for any r∈^{...N12 -1} with |r|≤2/_N1 (_N1 -1) , _M0 -^{∑j=1N12 -1}_rjMj ≥0 .

A separable tripartite state _ρABC can be written as [figure omitted; refer to PDF] From (2.13) it can also be represented as

[figure omitted; refer to PDF] where (^ai(1) ,^ai(2) ,...,^{ai(N12 -1)} ) and (^bi(1) ,^bi(2) ,...,^{bi(N22 -1)} ) are real vectors on the Bloch sphere satisfying |_ai ...^|2 =^{∑j=1N12 -1} (^ai(j))2 =2(1-1/_N1 ) and |_bi ...^|2 =^{∑j=1N22 -1} (^bi(j))2 =2(1-1/_N2 ) .

Comparing (2.15) with (2.19), we have

[figure omitted; refer to PDF]

For any (^N12 -1)×(^N12 -1) real matrix R(1) and (^N22 -1)×(^N22 -1) real matrix R(2) satisfying (1/(_N1 -1⁾² )I-R(1^)T R(1)≥0 and (1/(_N2 -1⁾² )I-R(2^)T R(2)≥0 , we define a new matrix

[figure omitted; refer to PDF] where T is a transformation acting on an (^N12 -1)×(^N22 -1) matrix M by

[figure omitted; refer to PDF]

Using ... , we define a new operator _γ... :

[figure omitted; refer to PDF] where ^{M0[variant prime]} =_M0 , ^{Mk[variant prime]} =^{∑m=1N12 -1}_Rkm (1)_Mm , ^{M...l[variant prime]} =^{∑n=1N22 -1}_Rln (2)_M...n , and ^{Mij[variant prime]} =(T(M)_)ij =(R(1)M^RT (2)_)ij .

Theorem 2.4.

If _ρABC is separable, then [87] _γ... (_ρABC )≥0 .

Proof.

From (2.20) and (2.23) we get [figure omitted; refer to PDF] A straightforward calculation gives rise to [figure omitted; refer to PDF] As (1/(_N1 -1⁾² )I-R(1^)T R(1)≥0 and (1/(_N2 -1⁾² )I-R(2^)T R(2)≥0 , we get [figure omitted; refer to PDF] Therefore _γ... (_ρABC ) is still a density operator, that is, _γ... (_ρABC )≥0 .

Theorem 2.4 gives a necessary separability criterion for general tripartite systems. The result can be also applied to bipartite systems. Let _ρAB ∈_{[Hamiltonian (script capital H)]A} [ecedil]7; _{[Hamiltonian (script capital H)]B} , _ρAB =_IN1 [ecedil]7;_M0 +^{∑j=1N12 -1}_λj [ecedil]7;_Mj . For any real (^N12 -1)×(^N12 -1) matrix ... satisfying (1/(_N1 -1⁾² )I-^...T ...≥0 and any state _ρAB , we define [figure omitted; refer to PDF] where ^{Mj[variant prime]} =_∑k...jkMk .

Corollary 2.5.

For _ρAB ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} , if there exists an ... with (1/(_N1 -1⁾² )I-^...T ...≥0 such that _γ... (_ρAB )<0 , then _ρAB must be entangled.

For 2×N systems, the above corollary is reduced to the results in [88]. As an example we consider the 3×3 istropic states [figure omitted; refer to PDF] If we choose ... to be Diag{1/2,1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2} , we get that _ρI is entangled for 0.5<p≤1 .

For tripartite case, we take the following 3×3×3 mixed state as an example: [figure omitted; refer to PDF] where |ψ...=(1/3)(|000...+|111...+|222...)(...000|+...111|+...222|) . Taking R(1)=R(2)=Diag{1/2,1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2} , we have that ρ is entangled for 0.6248<p≤1 .

In fact the criterion for 2×N systems [88] is equivalent to the PPT criterion [89]. Similarly Theorem 2.4 is also equivalent to the PPT criterion for 2×2×N systems.

2.1.5. Covariance Matrix Criterion

In this subsection we study the separability problem by using the covariance matrix approach. We first give a brief review of covariance matrix criterion proposed in [49]. Let ^{[Hamiltonian (script capital H)]dA} and ^{[Hamiltonian (script capital H)]dB} be d -dimensional complex vector spaces and _ρAB a bipartite quantum state in ^{[Hamiltonian (script capital H)]dA} [ecedil]7;^{[Hamiltonian (script capital H)]dB} . Let _Ak (resp., _Bk ) be ^d2 observables on ^{[Hamiltonian (script capital H)]dA} (resp., ^{[Hamiltonian (script capital H)]dB} ) such that they form an orthonormal normalized basis of the observable space, satisfying Tr [_AkAl ]=_δk,l (resp., Tr [_BkBl ]=_δk,l ). Consider the total set {_Mk }={_Ak [ecedil]7;I,I[ecedil]7;_Bk } . It can be proven that [44]

[figure omitted; refer to PDF]

The covariance matrix γ is defined with entries

[figure omitted; refer to PDF] which has a block structure [49]

[figure omitted; refer to PDF] where A=γ(_ρA ,{_Ak }), B=γ(_ρB ,{_Bk }), _Cij =..._Ai [ecedil]7;_Bj...ρAB -..._Ai...ρA ..._Bj...ρB , _ρA =_{Tr B} [_ρAB ] , and _ρB =_{Tr A} [_ρAB ] . Such covariance matrix has a concavity property: for a mixed density matrix ρ=_∑kpkρk with _pk ≥0 and _∑kpk =1 , one has γ(ρ)≥_∑kpk γ(_ρk ) .

For a bipartite product state _ρAB =_ρA [ecedil]7;_ρB , C in (2.32) is zero. Generally if _ρAB is separable, then there exist states |_ak ......_ak | on ^{[Hamiltonian (script capital H)]dA} , |_bk ......_bk | on ^{[Hamiltonian (script capital H)]dB} and _pk such that

[figure omitted; refer to PDF] where _κA =∑_pk γ(|_ak ......_ak |,{_Ak }) , _κB =∑_pk γ(|_bk ......_bk |,{_Bk }) .

For a separable bipartite state, it has been shown that [49]

[figure omitted; refer to PDF]

Criterion (2.34) depends on the choice of the orthonormal normalized basis of the observables. In fact the term ^∑i=1d2 |_Cii | has an upper bound _||C||KF which is invariant under unitary transformation and can be attained by choosing proper local orthonormal observable basis, where _||C||KF stands for the Ky Fan norm of C , _||C||KF =Tr [C^C[dagger] ] , with [dagger] denoting the transpose and conjugation. It has been shown in [46] that if _ρAB is separable, then

[figure omitted; refer to PDF]

From the covariance matrix approach, we can also get an alternative criterion. From (2.32) and (2.33) we have that if _ρAB is separable, then [figure omitted; refer to PDF] Hence all the 2×2 minor submatrices of X must be positive. Namely, one has [figure omitted; refer to PDF] that is, (A-_κA)ii (B-_κB)jj ≥^Cij2 . Summing over all i , j and using (2.30), we get

[figure omitted; refer to PDF]

That is,

[figure omitted; refer to PDF] where _||C||HS stands for the Euclid norm of C , that is, _||C||HS =Tr [C^C[dagger] ] .

Formulae (2.35) and (2.39) are independent and could be complement. When [figure omitted; refer to PDF] (2.39) can recognize the entanglement but (2.35) cannot. When [figure omitted; refer to PDF] (2.35) can recognize the entanglement while (2.39) cannot.

The separability criteria based on covariance matrix approach can be generalized to multipartite systems. We first consider the tripartite case _ρABC ∈^{[Hamiltonian (script capital H)]dA} [ecedil]7;^{[Hamiltonian (script capital H)]dB} [ecedil]7;^{[Hamiltonian (script capital H)]dC} . Take ^d2 observables _Ak on _{[Hamiltonian (script capital H)]A} , respectively, _Bk on _{[Hamiltonian (script capital H)]B} , respectively, _Ck on _{[Hamiltonian (script capital H)]C} . Set {_Mk }={_Ak [ecedil]7;I[ecedil]7;I,I[ecedil]7;_Bk [ecedil]7;I,I[ecedil]7;I[ecedil]7;_Ck } . The covariance matrix defined by (2.31) has then the following block structure:

[figure omitted; refer to PDF] where A=γ(_ρA ,{_Ak }) , B=γ(_ρB ,{_Bk }) , C=γ(_ρC ,{_Ck }) , _Dij =..._Ai [ecedil]7;_Bj...ρAB -..._Ai...ρA ..._Bj...ρB , _Eij =..._Ai [ecedil]7;_Cj...ρAC -..._Ai...ρA ..._Cj...ρC , and _Fij =..._Bi [ecedil]7;_Cj...ρBC -..._Bi...ρB ..._Cj...ρC .

Theorem 2.6.

If _ρABC is fully separable, then [90] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Proof.

For a tripartite product state _ρABC =_ρA [ecedil]7;_ρB [ecedil]7;_ρC , D, E, and F in (2.42) are zero. If _ρABC is fully separable, then there exist states |_ak ......_ak | in ^{[Hamiltonian (script capital H)]dA} , |_bk ......_bk | in ^{[Hamiltonian (script capital H)]dB} , and |_ck ......_ck | in ^{[Hamiltonian (script capital H)]dC} , and _pk such that γ(ρ)≥_κA [ecedil]5; _κB [ecedil]5; _κC , where _κA =∑_pk γ(|_ak ......_ak |,{_Ak }) , _κB =∑_pk γ(|_bk ......_bk |,{_Bk }) , and _κC =∑_pk γ(|_ck ......_ck |,{_Ck }) , that is, [figure omitted; refer to PDF] Thus all the 2×2 minor submatrices of Y must be positive. Selecting one with two rows and columns from the first two block rows and columns of Y , we have [figure omitted; refer to PDF] that is, (A-_κA)ii (B-_κB)jj ≥|_Dij^|2 . Summing over all i , j and using (2.30), we get [figure omitted; refer to PDF] which proves (2.43). Equations (2.44) and (2.45) can be similarly proved.

From (2.50) we also have (A-_κA)ii +(B-_κB)ii ≥2|_Dii | . Therefore [figure omitted; refer to PDF] Note that ^∑i=1d2_Dii ≤^∑i=1d2 |_Dii | . By using that Tr [MU]≤_||M||KF =Tr [M^M[dagger] ] for any matrix M and any unitary U [91], we have ^∑i=1d2_Dii ≤_||D||KF .

Let D=^U[dagger] ΛV be the singular value decomposition of D . Make a transformation of the orthonormal normalized basis of the local orthonormal observable space _A...i =_∑lUilAl and _B...j =_∑m^Vjm*_Bm . In the new basis we have [figure omitted; refer to PDF] Then (2.52) becomes [figure omitted; refer to PDF] which proves (2.46). Equations (2.47) and (2.48) can be similarly treated.

We consider now the case that _ρABC is bipartite separable.

Theorem 2.7.

If _ρABC is a bipartite separable state with respect to the bipartite partition of the sub-systems A and BC (resp., AB and C ; resp., AC and B ), then (2.43), (2.44) and (2.46), (2.47) (resp., (2.44), (2.45) and (2.47), (2.48); resp., (2.43), (2.45) and (2.46), (2.48)) must hold [90].

Proof.

We prove the case that _ρABC is bipartite separable with respect to the A system and BC systems partition. The other cases can be similarly treated. In this case the matrices D and E in the covariance matrix (2.42) are zero. _ρABC takes the form _ρABC =_∑mpm^ρAm [ecedil]7;^ρBCm . Define _κA =∑_pm γ(^ρAm ,{_Ak }) , _κBC =∑_pm γ(^ρBCm ,{_Bk [ecedil]7;I,I[ecedil]7;_Ck }) . _κBC has a form [figure omitted; refer to PDF] where _κB =∑_pk γ(|_bk ......_bk |,{_Bk }) and _κC =∑_pk γ(|_ck ......_ck |,{_Ck }) , (^{F[variant prime]}_)ij =_∑mpm (..._Bi [ecedil]7; _Cj...^ρBCm -..._Bi...^ρBm ..._Cj...^ρCm ) . By using the concavity of covariance matrix we have [figure omitted; refer to PDF] Accounting to the method used in proving Theorem 2 , we get (2.43), (2.44), and (2.46), (2.47).

From Theorems 2.6 and 2.7 we have the following corollary.

Corollary 2.8.

If two of the inequalities (2.43), (2.44), and (2.45) (or (2.46), (2.47), and (2.48)) are violated, then the state must be fully entangled.

The result of Theorem 2.6 can be generalized to general multipartite case ρ∈^{[Hamiltonian (script capital H)]d(1)} [ecedil]7;^{[Hamiltonian (script capital H)]d(2)} [ecedil]7;...[ecedil]7;^{[Hamiltonian (script capital H)]d(N)} . Define ^A...αi =I[ecedil]7;I[ecedil]7;..._λα [ecedil]7;I[ecedil]7;...[ecedil]7;I , where _λ0 =I/d , _λα (α=1,2,...,^d2 -1 ) are the normalized generators of SU(d) satisfying Tr [_λαλβ ]=_δαβ and acting on the i th system ^{[Hamiltonian (script capital H)]d(i)} , i=1,2,...,N . Denote {_Mk } as the set of all ^A...αi . Then the covariance matrix of ρ can be written as

[figure omitted; refer to PDF] where _...9C;ii =γ(ρ,{^A...ki }) and (_...9C;ij)mn =...^A...mi [ecedil]7;^A...nj ...-...^A...mi ......^A...nj ... for i≠j .

For a product state _ρ12...N , _...9C;ij , i≠j , in (2.57) are zero matrices. Define [figure omitted; refer to PDF] Then for a fully separable multipartite state ρ=_∑kpk |^ψk1 ......^ψk1 |[ecedil]7;|^ψk2 ......^ψk2 |[ecedil]7;...[ecedil]7;|^ψkN ......^ψkN | , one has [figure omitted; refer to PDF] from which we have the following separability criterion for multipartite systems.

Theorem 2.9.

If a state ρ∈^{[Hamiltonian (script capital H)]d(1)} [ecedil]7;^{[Hamiltonian (script capital H)]d(2)} [ecedil]7;...[ecedil]7;^{[Hamiltonian (script capital H)]d(N)} is fully separable, then the following inequalities [figure omitted; refer to PDF] must be fulfilled for any i≠j [90].

2.2. Normal Form of Quantum States

In this subsection we show that the correlation matrix (CM) criterion can be improved from the normal form obtained under filtering transformations. Based on CM criterion entanglement witness in terms of local orthogonal observables (LOOs) [92] for both bipartite and multipartite systems can be also constructed.

For bipartite case, ρ∈[Hamiltonian (script capital H)]=_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} with dim _{[Hamiltonian (script capital H)]A} =M , dim _{[Hamiltonian (script capital H)]B} =N , and M≤N is mapped to the following form under local filtering transformations [93]:

[figure omitted; refer to PDF] where _FA/B ∈GL(M/N,...) are arbitrary invertible matrices. This transformation is also known as stochastic local operations assisted by classical communication (SLOCC). By the definition it is obvious that filtering transformation will preserve the separability of a quantum state.

It has been shown that under local filtering operations one can transform a strictly positive ρ into a normal form [94]:

[figure omitted; refer to PDF] where _ξi ≥0 and ^GiA and ^GiB are some traceless orthogonal observables. The matrices _FA and _FB can be obtained by minimizing the function [figure omitted; refer to PDF] where A=^FA[dagger]_FA and B=^FB[dagger]_FB . In fact, one can choose ^FA0 ≡|det (_ρA )^|1/2M (_ρA^)-1 , and ^FB0 ≡|det (^{ρB[variant prime]} )^|1/2N (^{ρB[variant prime])-1} , where ^{ρB[variant prime]} =_{Tr A} [I[ecedil]7;(_ρA^)-1 ρI[ecedil]7;(_ρA^)-1 ] . Then by iterations one can get the optimal A and B . In particular, there is a matlab code available in [95].

For bipartite separable states ρ , the CM separability criterion [96] says that

[figure omitted; refer to PDF] where T is an (^M2 -1)×(^N2 -1) matrix with _Tij =MN·Tr [ρ^λiA [ecedil]7;^λjB ] , _||T||KF stands for the trace norm of T , ^λkA/B s are the generators of SU(M/N) and have been chosen to be normalized, and Tr [^{λk(A/B)λl(A/B)} ]=_δkl .

As the filtering transformation does not change the separability of a state, one can study the separability of ρ... instead of ρ . Under the normal form (2.62) the criterion (2.64) becomes

[figure omitted; refer to PDF]

In [44] a separability criterion based on local uncertainty relation (LUR) has been obtained. It says that, for any separable state ρ ,

[figure omitted; refer to PDF] where ^GkA/B s are LOOs such as the normalized generators of SU(M/N) and ^GkA =0 for k=^M2 +1,...,^N2 . The criterion is shown to be strictly stronger than the realignment criterion [61]. Under the normal form (2.62) criterion (2.66) becomes [figure omitted; refer to PDF] that is,

[figure omitted; refer to PDF] As MN(M-1)(N-1)≤MN-(M+N)/2 holds for any M and N , from (2.65) and (2.68) it is obvious that the CM criterion recognizes entanglement better when the normal form is taken into account.

We now consider multipartite systems. Let ρ be a strictly positive density matrix in [Hamiltonian (script capital H)]=_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} and dim _{[Hamiltonian (script capital H)]i} =_di . ρ can be generally expressed in terms of the SU(n) generators _λαk [97] as

[figure omitted; refer to PDF] where ^{λ_αk {_μk }} =_Id1 [ecedil]7;_Id2 [ecedil]7;...[ecedil]7;_λαk [ecedil]7;_{Idμk +1} [ecedil]7;...[ecedil]7;_IdN with _λαk appears at the _μk th position and

[figure omitted; refer to PDF]

The generalized CM criterion says that if ρ in (2.69) is fully separable, then

[figure omitted; refer to PDF] for 2≤M≤N, {_μ1 ,_μ2 ,...,_μM }⊂{1,2,...,N} . The KF norm is defined by [figure omitted; refer to PDF] where _...AF;(m) is a kind of matrix unfolding of ^{...AF;{_μ1 ,_μ2 ,...,_μM }} .

The criterion (2.71) can be improved by investigating the normal form of (2.69).

Theorem 2.10.

By filtering transformations of the form [figure omitted; refer to PDF] where _Fi ∈GL(_di ,...),i=1,2,...N , followed by normalization, any strictly positive state ρ can be transformed into a normal form [98]: [figure omitted; refer to PDF]

Proof.

Let _D1 ,_D2 ,...,_DN be the sets of density matrices of the N subsystems. The cartesian product _D1 ×_D2 ×...×_DN consisting of all product density matrices _ρ1 [ecedil]7;_ρ2 [ecedil]7;...[ecedil]7;_ρN with normalization Tr [_ρi ]=1 , i=1,2,...,N , is a compact set of matrices on the full Hilbert space [Hamiltonian (script capital H)] . For the given density matrix ρ we define the following function of _ρi : [figure omitted; refer to PDF] The function is well defined on the interior of _D1 ×_D2 ×...×_DN where det _ρi >0 . As ρ is assumed to be strictly positive, we have Tr [ρ(_ρ1 [ecedil]7;_ρ2 [ecedil]7;...[ecedil]7;_ρN )]>0 . Since _D1 ×_D2 ×...×_DN is compact, we have Tr [ρ(_ρ1 [ecedil]7;_ρ2 [ecedil]7;...[ecedil]7;_ρN )]≥C>0 with a lower bound C depending on ρ .

It follows that f[arrow right]∞ on the boundary of _D1 ×_D2 ×...×_DN where at least one of the _ρi s satisfies det _ρi =0 . It follows further that f has a positive minimum on the interior of _D1 ×_D2 ×...×_DN with the minimum value attained for at least one product density matrix _τ1 [ecedil]7;_τ2 [ecedil]7;...[ecedil]7;_τN with det _τi >0 , i=1,2,...,N . Any positive density matrix _τi with det _τi >0 can be factorized in terms of Hermitian matrices _Fi as [figure omitted; refer to PDF] where _Fi ∈GL(_di ,...) . Denote that F=_F1 [ecedil]7;_F2 [ecedil]7;...[ecedil]7;_FN , so that _τ1 [ecedil]7;_τ2 [ecedil]7;...[ecedil]7;_τN =^F[dagger] F . Set ρ...=Fρ^F[dagger] and define [figure omitted; refer to PDF]

We see that, when ^Fi[dagger]_ρiFi =_τi , f... has a minimum and [figure omitted; refer to PDF]

Since f... is stationary under infinitesimal variations about the minimum, it follows that [figure omitted; refer to PDF] for all infinitesimal variations [figure omitted; refer to PDF] subjected to the constraint det (_Idi +δ_ρi )=1 , which is equivalent to Tr [δ_ρi ]=0 , i=1,2,...,N , using det (^eA )=^{eTr [A]} for a given matrix A . Thus, δ_ρi can be represented by the SU generators, δ_ρi =_∑k δ^ckiλki . It follows that Tr [ρ...^{λ_αk {_μk }} ]=0 for any _αk and _μk . Hence the terms proportional to ^{λ_αk {_μk }} in (2.69) disappear.

Corollary 2.11.

The normal form of a product state in [Hamiltonian (script capital H)] must be proportional to the identity.

Proof.

Let ρ be such a state. From (2.74), we get that [figure omitted; refer to PDF] Therefore for a product state ρ we have [figure omitted; refer to PDF]

As an example for separability of multipartite states in terms of their normal forms (2.74), we consider the PPT entangled edge state [79] [figure omitted; refer to PDF] mixed with noises [figure omitted; refer to PDF] Select a=2, b=3 , and c=0.6 . Using the criterion in [97] we get that _ρp is entangled for 0.92744<p≤1 . But after transforming _ρp to its normal form (2.74), the criterion can detect entanglement for 0.90285<p≤1 .

Here we indicate that the filtering transformation does not change the PPT property. Let ρ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} be PPT, that is, ^ρ_TA ≥0, and ^ρ_TB ≥0 . Let ρ... be the normal form of ρ . From (2.61) we have [figure omitted; refer to PDF] For any vector |ψ... , we have [figure omitted; refer to PDF] where |^{ψ[variant prime]} ...=(^FAT [ecedil]7;^FB[dagger] )|ψ.../Tr [(_FA [ecedil]7;_FB )ρ(_FA [ecedil]7;_FB^)[dagger] ] . ^ρ..._TB ≥0 can be proved similarly. This property is also valid for multipartite case. Hence a bound entangled state will be bound entangled under filtering transformations.

2.3. Entanglement Witness Based on Correlation Matrix Criterion

Entanglement witness (EW) is another way to describe separability. Based on CM criterion we can further construct entanglement witness in terms of LOOs. EW [92] is an observable of the composite system such that (i) nonnegative expectation values in all separable states and (ii) at least one negative eigenvalue (can recognize at least one entangled state). Consider bipartite systems in ^{[Hamiltonian (script capital H)]AM} [ecedil]7;^{[Hamiltonian (script capital H)]BN} with M≤N .

Theorem 2.12.

For any properly selected LOOs ^GkA and ^GkB , [figure omitted; refer to PDF] is an EW [98], where α=MN/((M-1)(N-1)+1) and [figure omitted; refer to PDF]

Proof.

Let ρ=^{∑l,m=0N2 -1}_Tlm^λlA [ecedil]7;^λmB be a separable state, where ^λkA/B are normalized generators of SU(M/N) with ^λ0A =(1/M)_IM , ^λ0B =(1/N)_IN . Any other LOOs ^GkA/B fulfilling (2.88) can be obtained from these λ s through orthogonal transformations ^...AA;A/B , ^GkA/B =^{∑l=0N2 -1...AA;klA/B}_λl , where ^...AA;A/B =(100^...A/B ) and ^...A/B are (^N2 -1)×(^N2 -1) orthogonal matrices. We have [figure omitted; refer to PDF] where we have used Tr [...T]≤_||T||KF for any unitary ... in the first inequality and the CM criterion in the second inequality.

Now let ρ=(1/MN)(_IMN +^{∑i=1M2 -1}_si^λiA [ecedil]7; _IN +^{∑j=1N2 -1}_rjIM [ecedil]7; ^λjB +^{∑i=1M2 -1∑j=1N2 -1}_Tij^λiA [ecedil]7; ^λjB ) be a state in ^{[Hamiltonian (script capital H)]AM} [ecedil]7;^{[Hamiltonian (script capital H)]BN} which violates the CM criterion. Denote _σk (T) as the singular values of T . By singular value decomposition, one has T=^U[dagger] Λ^V* , where Λ is a diagonal matrix with _Λkk =_σk (T) . Now choose LOOs to be ^GkA =_∑lUkl^λlA , ^GkB =_∑mVkm^λmB for k=1,2,...,^N2 -1 and ^G0A =(1/M)_IM , ^G0B =(1/N)_IN . We obtain [figure omitted; refer to PDF] where the CM criterion has been used in the last step.

As the CM criterion can be generalized to multipartite form [97], we can also define entanglement witness for multipartite system in ^{[Hamiltonian (script capital H)]1_d1} [ecedil]7;^{[Hamiltonian (script capital H)]2_d2} [ecedil]7;...[ecedil]7;^{[Hamiltonian (script capital H)]N_dN} . Set d(M)=max {_dμi ,i=1,2,...,M} . Choose LOOs ^{Gk{_μi }} for 0≤k≤d(M⁾² -1 with ^{G0{_μi }} =(1/_dμi )_Idμi and define

[figure omitted; refer to PDF] where ^β(M) =^∏i=1M_dμi /(1+^∏i=1M (_dμi -1)), 2≤M≤N . One can prove that (2.91) is an EW candidate for multipartite states. First we assume that _{||^...AF;(M) ||KF} =_{||...AF;(m0 ) ||KF} . Note that, for any _{...AF;(m0 )} , there must exist an elementary transformation P such that ^{∑k=1d(M)2 -1...AF;kk...k{_μ1μ2 ..._μM }} =Tr [_{...AF;(m0 )} P] . Then for an N-partite separable state we have [figure omitted; refer to PDF] for any 2≤M≤N , where we have taken into account that P is orthogonal and Tr [MU]≤_||M||KF for any unitary U at the first inequality. The second inequality is due to the generalized CM criterion.

By choosing proper LOOs, it is also easy to show that ^W(M) has negative eigenvalues. For example, for three-qubit case, taking the normalized Pauli matrices as LOOs, one finds a negative eigenvalue of ^W(M) , (1-3)/2 .

3. Concurrence and Tangle

In this section, we focus on two important measures: concurrence and tangle (see [99, 100]). An elegant formula for concurrence of two-qubit states is derived analytically by Wootters [53, 101]. This quantity has recently been shown to play an essential role in describing quantum phase transition in various interacting quantum many-body systems [102, 103] and may affect macroscopic properties of solids significantly [104, 105]. Furthermore, concurrence also provides an estimation [106, 107] for the entanglement of formation (EOF) [76], which quantifies the required minimally physical resources to prepare a quantum state.

Let _{[Hamiltonian (script capital H)]A} (resp., _{[Hamiltonian (script capital H)]B} ) be an M -(resp., N -) dimensional complex vector space with |i... , i=1,...,M (resp., |j..., j=1,...,N ), as an orthonormal basis. A general pure state on _{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} is of the form

[figure omitted; refer to PDF] where _aij ∈... satisfy the normalization ^{∑i=1M∑j=1N}_aij^aij* =1 .

The concurrence of (3.1) is defined by [27, 108, 109] [figure omitted; refer to PDF] where _ρA =_{Tr B} [|ψ......ψ|] . The definition is extended to general mixed states ρ=_∑ipi |_ψi ......_ψi | by the convex roof

[figure omitted; refer to PDF]

For two-qubits systems, the concurrence of |Ψ... is given by

[figure omitted; refer to PDF] where |Ψ......=_σy [ecedil]7;_σy |^Ψ* ... , |^Ψ* ... is the complex conjugate of |Ψ... , and _σy is the Pauli matrix, _σy =(0-ii0).

For a mixed two-qubit quantum state ρ , the entanglement of formation E(ρ) has a simple relation with the concurrence [53, 101] [figure omitted; refer to PDF] where h(x)=-x _{log 2} x-(1-x)_{log 2} (1-x) , [figure omitted; refer to PDF] where the _λi s are the eigenvalues, in decreasing order, of the Hermitian matrix ρρ...ρ and ρ...=(_σy [ecedil]7;_σy )^ρ* (_σy [ecedil]7;_σy ) .

Another entanglement measure called tangle is defined by

[figure omitted; refer to PDF] for a pure state |ψ... . For mixed state ρ=_∑ipi |_ψi ......_ψi | , the definition is given by

[figure omitted; refer to PDF]

For multipartite state |ψ...∈_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} , dim _{[Hamiltonian (script capital H)]i} =_di , i=1,...,N , the concurrence of |ψ... is defined by [110, 111]

[figure omitted; refer to PDF] where α labels all different reduced density matrices.

Up to constant factor (3.9) can be also expressed in another way. Let H denote a d -dimensional vector space with basis |i... , i=1,2,...,d . An N -partite pure state in H[ecedil]7;...[ecedil]7;H is generally of the form

[figure omitted; refer to PDF]

Let α and ^{α[variant prime]} (resp., β and ^{β[variant prime]} ) be subsets of the subindices of a , associated to the same sub Hilbert spaces but with different summing indices. α (or ^{α[variant prime]} ) and β (or ^{β[variant prime]} ) span the whole space of the given subindix of a . The generalized concurrence of |Ψ... is then given by [27]

[figure omitted; refer to PDF] where m=^2N-1 -1 and _∑p stands for the summation over all possible combinations of the indices of α and β .

For a mixed multipartite quantum state, ρ=_∑ipi |_ψi ......_ψi | in _{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} , the corresponding concurrence is given by the convex roof:

[figure omitted; refer to PDF]

3.1. Lower and Upper Bounds of Concurrence

Calculations of the concurrence for general mixed states are extremely difficult. However, one can try to find the lower and the upper bounds to estimate the exact values of the concurrence [46, 61, 64, 65].

3.1.1. Lower Bound of Concurrence from Covariance Matrix Criterion

In [61] a lower bound of C(ρ) has been obtained as

[figure omitted; refer to PDF] where _TA and R stand for partial transpose with respect to subsystem A and the realignment, respectively. This bound is further improved based on local uncertainty relations [64]

[figure omitted; refer to PDF] where ^GiA and ^GiB are any set of local orthonormal observables, ^Δρ2 (X)=Tr [^X2 ρ]-(Tr [Xρ]⁾² .

Bound (3.14) again depends on the choice of the local orthonormal observables. This bound can be optimized, in the sense that a local orthonormal observable-independent up bound of the right-hand side of (3.14) can be obtained.

Theorem 3.1.

Let ρ be a bipartite state in ^{[Hamiltonian (script capital H)]MA} [ecedil]7;^{[Hamiltonian (script capital H)]NB} . Then C(ρ) satisfies [90] [figure omitted; refer to PDF]

Proof.

The other orthonormal normalized basis of the local orthonormal observable space can be obtained from _Ai and _Bi by unitary transformations U and V : _A...i =_∑lUilAl and _B...j =_∑m^Vjm*_Bm . Select U and V so that C=^U[dagger] ΛV is the singular value decomposition of C . Then the new observables can be written as _A...i =_∑lUilAl , _B...j =-_∑m^Vjm*_Bm . We have [figure omitted; refer to PDF] Substituting above relation to (3.14), one gets (3.15).

Bound (3.15) does not depend on the choice of local orthonormal observables. It can be easily applied and realized by direct measurements in experiments. It is in accord with the result in [46] where optimization of entanglement witness based on local uncertainty relation has been taken into account. As an example, let us consider the 3×3 bound entangled state [76]

[figure omitted; refer to PDF] where _I9 is the 9×9 identity matrix, |_ξ0 ...=(1/2)|0...(|0...-|1...) , |_ξ1 ...=(1/2)(|0...-|1...)|2... , |_ξ2 ...=(1/2)|2...(|1...-|2...) , |_ξ3 ...=(1/2)(|1...-|2...)|0... , and |_ξ4 ...=(1/3)(|0...+|1...+|2...)(|0...+|1...+|2...) . We simply choose the local orthonormal observables to be the normalized generators of SU(3) . Formula (3.13) gives C(ρ)≥0.050 . Formula (3.14) gives C(ρ)≥0.052 [64], while formula (3.15) yields a better lower bound C(ρ)≥0.0555 .

If we mix the bound entangled state (3.17) with |ψ...=(1/3)^∑i=02 |ii... , that is,

[figure omitted; refer to PDF] then it is easily seen that (3.15) gives a better lower bound of concurrence than formula (3.13) (Figure 1).

Figure 1: Lower bounds from (3.15) (dashed line) and (3.13) (solid line) for mixed state (3.18).

[figure omitted; refer to PDF]

3.1.2. Lower Bound of Concurrence from "Two-Qubit" Decomposition

In [67] the authors derived an analytical lower bound of concurrence for arbitrary bipartite quantum states by decomposing the joint Hilbert space into many 2[ecedil]7;2 dimensional subspaces, which does not involve any optimization procedure and gives an effective evaluation of entanglement together with an operational sufficient condition for the distill ability of any bipartite quantum states.

(1) Lower Bound of Concurrence for Bipartite States

The lower bound _τ2 of concurrence for bipartite states has been obtained in [67]. For a bipartite quantum state ρ in H[ecedil]7;H , the concurrence C(ρ) satisfies [figure omitted; refer to PDF] where _Cmn (ρ)=max {0,^λmn(1) -^λmn(2) -^λmn(3) -^λmn(4) } with ^λmn(1) ,...,^λmn(4) being the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρ_ρ...mn with _ρ...mn =(_Lm [ecedil]7;_Ln )^ρ* (_Lm [ecedil]7;_Ln ) , while _Lm and _Ln are the generators of SO(d) .

The lower bound _τ2 in (3.19) in fact characterizes all two qubits' entanglement in a high dimensional bipartite state. One can directly verify that there are at most 4×4=16 nonzero elements in each matrix ρ_ρ...mn . These elements constitute a 4×4 matrix [varrho](_σy [ecedil]7;_σy )^[varrho]* (_σy [ecedil]7;_σy ) , where _σy is the Pauli matrix, the matrix [varrho] is a submatrix of the original ρ :

[figure omitted; refer to PDF] i≠j , and k≠l , with subindices i and j associated with the first space, k and l with the second space. The two-qubit submatrix [varrho] is not normalized but positive semidefinite. _...9E;mn are just the concurrences of these states (3.20).

The bound _τ2 provides a much clearer structure of entanglement, which not only yields an effective separability criterion and an easy evaluation of entanglement, but also helps one to classify mixed-state entanglement.

(2) Lower Bound of Concurrence for Multipartite States

We first consider tripartite case. A general pure state on H[ecedil]7;H[ecedil]7;H is of the form [figure omitted; refer to PDF] with [figure omitted; refer to PDF] or equivalently [figure omitted; refer to PDF] where _ρ1 =_{Tr 23} [ρ], _ρ2 =_{Tr 13} [ρ], and _ρ3 =_{Tr 12} [ρ] are the reduced density matrices of ρ=|Ψ......Ψ| .

Define ^Cαβ12|3 (|Ψ...)=|_aijkapqm -_aijmapqk | , ^Cαβ13|2 (|Ψ...)=|_aijkapqm -_aiqkapjm | , and ^Cαβ23|1 (|Ψ...)=|_aijkapqm -_apjkaiqm | , where α and β of ^Cαβ12|3 (resp., ^Cαβ13|2 , resp., ^Cαβ23|1 ) stand for the subindices of a associated with the subspaces 1, 2 , and 3 (resp., 1, 3 , and 2; resp., 2, 3 , and 1 ). Let ^{L_i1i2 ..._iN} denote the generators of group SO(_di1di2 ..._diN ) associated to the subsystems _i1 ,_i2 ,...,_iN . Then for a tripartite pure state (3.21), one has [figure omitted; refer to PDF] where ^Sαβ12|3 =(^Lα12 [ecedil]7;^Lβ3 ) , ^Sαβ13|2 =(^Lα13 [ecedil]7;^Lβ2 ) , and ^Sαβ23|1 =(^Lβ1 [ecedil]7;^Lα23 ) .

Theorem 3.2.

For an arbitrary mixed state ρ in H[ecedil]7;H[ecedil]7;H , the concurrence C(ρ) satisfies [112] [figure omitted; refer to PDF] where _τ3 (ρ) is a lower bound of C(ρ) [figure omitted; refer to PDF] where λ(1^)αβ12|3 , λ(2^)αβ12|3 , λ(3^)αβ12|3 , and λ(4^)αβ12|3 are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρ^{ρ...αβ12|3} with ^{ρ...αβ12|3} =^{Sαβ12|3ρ*Sαβ12|3} . ^Cαβ13|2 (ρ) and ^Cαβ23|1 (ρ) are defined in a similar way to ^Cαβ12|3 (ρ) .

Proof.

Set |_ξi ...=_pi |_ψi ... , ^xαβi =|..._ξi |^Sαβ12|3 |^ξi* ...| , ^yαβi =|..._ξi |^Sαβ13|2 |^ξi* ...| , and ^zαβi =|..._ξi |^Sαβ1|23 |^ξi* ...| . We have, from Minkowski inequality, [figure omitted; refer to PDF]

Noting that for nonnegative real variables _xα , _yα , _zα and given that X=^∑α=1N_xα , Y=^∑α=1N_Yα , and Z=^∑α=1N_zα , by using Lagrange multipliers, one obtains that the following inequality holds: [figure omitted; refer to PDF] Therefore we have [figure omitted; refer to PDF]

The values of ^Cαβ12|3 (ρ)≡min _∑i^xαβi , ^Cαβ13|2 (ρ)≡min _∑i^yαβi , and ^Cαβ23|1 (ρ)≡min _∑i^zαβi can be calculated by using the similar procedure in [53]. Here we compute the value of ^Cαβ12|3 (ρ) in detail. The values of ^Cαβ13|2 (ρ) and ^Cαβ23|1 (ρ) can be obtained analogously.

Let _λi and |_χi ... be eigenvalues and eigenvectors of ρ , respectively. Any decomposition of ρ can be obtained from a unitary ^d3 ×^d3 matrix _Vij , |_ξj ...=^∑i=1d3Vij* (_λi |_χi ...) . Therefore one has ..._ξi |^Sαβ12|3 |^ξj* ...=(V_Yαβ^VT_)ij , where the matrix _Yαβ is defined by (_Yαβ)ij =..._χi |^Sαβ12|3 |^χj* ... . Namely, ^Cαβ12|3 (ρ)=min _∑i |[V_Yαβ^VT_]ii | , which has an analytical expression [53], that ^Cαβ12|3 (ρ)=max {0,λ(1^)αβ12|3 -_∑j>1 λ(j^)αβ12|3 } , where ^λαβ12|3 (k) are the square roots of the eigenvalues of the positive Hermitian matrix _Yαβ^{Yαβ[dagger]} , or equivalently the non-Hermitian matrix ρ_ρ...αβ , in decreasing order. Here as the matrix ^Sαβ12|3 has ^d2 -4 rows and ^d2 -4 columns that are identically zero, the matrix ρ_ρ...αβ has a rank not greater than 4, that is, ^λαβ12|3 (j)=0 for j≥5 . From (3.29) we have (3.25).

Theorem 3.2 can be directly generalized to arbitrary multipartite case.

Theorem 3.3.

For an arbitrary N -partite state ρ∈H[ecedil]7;H[ecedil]7;...[ecedil]7;H , the concurrence defined in (4.1) satisfies [112], [figure omitted; refer to PDF] where _τN (ρ) is the lower bound of C(ρ) , _∑p stands for the summation over all possible combinations of the indices of α,β , ^Cαβp (ρ)=max {0,λ(1^)αβp -λ(2^)αβp -λ(3^)αβp -λ(4^)αβp } , and λ(i^)αβp , i=1,2,3,4 , are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρ^ρ...αβp where ^ρ...αβp =^{Sαβpρ*Sαβp} .

Lower Bound and Separability

An N-partite quantum state ρ is fully separable if and only if there exist _pi with _pi ≥0, _∑ipi =1 , and pure states ^ρij =|^ψij ......^ψij | such that [figure omitted; refer to PDF]

It is easily verified that, for a fully separable multipartite state ρ , _τN (ρ)=0 . Thus _τN (ρ)>0 indicates that there must be some kinds of entanglement inside the quantum state, which shows that the lower bound _τN (ρ) can be used to recognize entanglement.

As an example, we consider a tripartite quantum state [79] ρ=((1-p)/8)_I8 +p|W......W| , where _I8 is the 8×8 identity matrix, and |W...=(1/3)(|100...+|010...+|001...) is the tripartite W state. Select an entanglement witness operator to be ...B2;=(1/2)_I8 -|GHZ......GHZ| , where |GHZ...=(1/2)(|000...+|111...) is to be the tripartite GHZ-state. From the condition Tr [...B2;ρ]<0 , the entanglement of ρ is detected for (3/5)<p≤1 in [79]. In [97] the authors have obtained the generalized correlation matrix criterion which says that if an N-qubit quantum state is fully separable, then the inequality _{||^...AF;N ||KF} ≤1 must hold, where _{||^...AF;N ||KF} =max {_{||^...AF;nN ||KF} } , ^...AF;nN is a kind of matrix unfold of _{tα1α2 ...αN} defined by _{tα1α2 ...αN} =Tr [ρ^{σ_α1 (1)σ_α2 (2)} ...^{σ_αN (N)} ] , and ^{σ_αi (i)} stands for the Pauli matrix. Now using the generalized correlation, matrix criterion the entanglement of ρ is detected for 0.3068<p≤1 . From Theorem 3.2, we have the lower bound _τ3 (ρ)>0 for 0.2727<p≤1 . Therefore the bound (3.71) detects entanglement better than these two criteria in this case. If we replace W with GHZ state in ρ , then the criterion in [97] detects the entanglement of ρ for 0.35355<p≤1 , while _τ3 (ρ) detects, again better, the entanglement for 0.2<p≤1 .

Nevertheless for PPT states ρ , we have _τ3 (ρ)=0 , which can be seen in the following way. A density matrix ρ is called PPT if the partial transposition of ρ over any subsystem(s) is still positive. Let ^ρ_Ti denote the partial transposition with respect to the i th subsystem. Assume that there is a PPT state ρ with τ(ρ)>0 . Then at least one term in (3.25), say ^{C_α0β0 12|3} (ρ) , is not zero. Define _ρα0β0 =^{L_α0 12} [ecedil]7;^{L_β0 3} ρ(^{L_α0 12} [ecedil]7;^{L_β0 3)[dagger]} . By using the PPT property of ρ , we have

[figure omitted; refer to PDF] Noting that both ^{L_α0 12} and ^{L_β0 3} are projectors to two-dimensional subsystems, _ρα0β0 can be considered as a 4×4 density matrix, while a PPT 4×4 density matrix _ρα0β0 must be a separable state, which contradicts with ^{C_α0β0 12|3} (ρ)≠0 .

Relation between Lower Bounds of Bi- and Tripartite Concurrence

_τ3 is basically different from _τ2 as _τ3 characterizes also genuine tripartite entanglement that can not be described by bipartite decompositions. Nevertheless, there are interesting relations between them.

Theorem 3.4.

For any pure tripartite state (3.21), the following inequality holds [112]: [figure omitted; refer to PDF] where _τ2 is the lower bound of bipartite concurrence (3.19), _τ3 is the lower bound of tripartite concurrence (3.25), and _ρ12 =_{Tr 3} [ρ] , _ρ13 =_{Tr 2} [ρ] , _ρ23 =_{Tr 1} [ρ] , and ρ=|Ψ_...123 ...Ψ| .

Proof.

Since ^Cαβ2 ≤(_λαβ (1)⁾² ≤^∑i=14 (_λαβ (i)⁾² =Tr [ρ_ρ...αβ ] for ρ=_ρ12 , ρ=_ρ13 , and ρ=_ρ23 , we have [figure omitted; refer to PDF] where we have used the similar analysis in [67, 113] to obtain the equalities _∑α,β Tr [_ρ12 (_ρ...12)αβ ]=1-Tr [^ρ12 ]-Tr [^ρ22 ]+Tr [^ρ32 ] , _∑α,β Tr [_ρ13 (_ρ...13)αβ ]=1-Tr [^ρ12 ]+Tr [^ρ22 ]-Tr [^ρ32 ] , and _∑α,β Tr [_ρ23 (_ρ...23)αβ ]=1+Tr [^ρ12 ]-Tr [^ρ22 ]-Tr [^ρ32 ] . The last equality is due to the fact that ρ is a pure state.

In fact, the bipartite entanglement inside a tripartite state is useful for distilling maximally entangled states. Assume that there are two of the qualities {τ(_ρ12 ),τ(_ρ13 ),τ(_ρ23 )} larger than zero; say τ(_ρ12 )>0 and τ(_ρ13 )>0 . According to [67], one can distill two maximal entangled states |_ψ12 ... and |_ψ13 ... which belong to _{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} and _{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]3} , respectively. In terms of the result in [114], one can use them to produce a GHZ state.

3.1.3. Estimation of Multipartite Entanglement

For a pure N -partite quantum state |ψ...∈_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} , dim _{[Hamiltonian (script capital H)]i} =_di , i=1,...,N , the concurrence of bipartite decomposition between subsystems 12...M and M+1...N is defined by

[figure omitted; refer to PDF] where ^ρ12...M2 =_{Tr M+1...N} [|ψ......ψ|] is the reduced density matrix of ρ=|ψ......ψ| by tracing over subsystems M+1...N . On the other hand, the concurrence of |ψ... is defined by (3.9).

For a mixed multipartite quantum state ρ=_∑ipi |_ψi ......_ψi |∈_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} , the corresponding concurrences of (3.35) and (3.9) are then given by the convex roof

[figure omitted; refer to PDF] and (3.12). We now investigate the relation between these two kinds of concurrences.

Lemma 3.5.

For a bipartite density matrix ρ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} , one has [figure omitted; refer to PDF] where _ρA/B =_{Tr B/A} [ρ] are the reduced density matrices of ρ .

Proof.

Let ρ=_∑ijλij |ij......ij| be the spectral decomposition, where _λij ≥0, _∑ijλij =1 . Then _ρ1 =_∑ijλij |i......i| , _ρ2 =_∑ijλij |j......j| . Therefore [figure omitted; refer to PDF]

This lemma can be also derived in another way [46, 115].

Theorem 3.6.

For a multipartite quantum state ρ∈_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} with N≥3 , the following inequality holds [116]: [figure omitted; refer to PDF] where the maximum is taken over all kinds of bipartite concurrence.

Proof.

Without loss of generality, we suppose that the maximal bipartite concurrence is attained between subsystems 12...M and (M+1)...N .

For a pure multipartite state |ψ...∈_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;...[ecedil]7;_{[Hamiltonian (script capital H)]N} , Tr [^ρ12...M2 ]=Tr [^ρ(M+1)...N2 ] . From (3.37) we have [figure omitted; refer to PDF] that is, _CN (|ψ......ψ|)≥^2(3-N)/2_C2 (|ψ......ψ|) .

Let ρ=_∑ipi |_ψi ......_ψi | attain the minimal decomposition of the multipartite concurrence. One has [figure omitted; refer to PDF]

Corollary 3.7.

For a tripartite quantum state ρ∈_{[Hamiltonian (script capital H)]1} [ecedil]7;_{[Hamiltonian (script capital H)]2} [ecedil]7;_{[Hamiltonian (script capital H)]3} , the following inequality holds: [figure omitted; refer to PDF] where the maximum is taken over all kinds of bipartite concurrence.

In [46, 64], from the separability criteria related to local uncertainty relation, covariance matrix, and correlation matrix, the following lower bounds for bipartite concurrence are obtained:

[figure omitted; refer to PDF] [figure omitted; refer to PDF] where the entries of the matrix C , _Cij =...^λiA [ecedil]7;^λjB ...-...^λiA [ecedil]7;_IdB ......_IdA [ecedil]7;^λjB ... , _Tij =_dAdB /2...^λiA [ecedil]7;^λjB ... , ^λkA/B stand for the normalized generator of SU(_dA /_dB ) , that is, Tr [^λkA/BλlA/B ]=_δkl and ...X...=Tr [ρX] . It is shown that the lower bounds (3.43) and (3.44) are independent of (3.13).

Now we consider a multipartite quantum state ρ∈_{[Hamiltonian (script capital H)]1} [ecedil]7; _{[Hamiltonian (script capital H)]2} [ecedil]7; ... [ecedil]7; _{[Hamiltonian (script capital H)]N} as a bipartite state belonging to ^{[Hamiltonian (script capital H)]A} [ecedil]7;^{[Hamiltonian (script capital H)]B} with the dimensions of the subsystems A and B being _dA =_ds1ds2 ..._dsm and _dB =_dsm+1dsm+2 ..._dsN , respectively. By using Corollary 3.7, (3.13), (3.43), and (3.44), one has the following lower bound.

Theorem 3.8.

For any N-partite quantum state ρ [116], [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ^ρi 's are all possible bipartite decompositions of ρ , and [figure omitted; refer to PDF]

In [46, 106, 107, 117], it is shown that the upper and lower bounds of multipartite concurrence satisfy

[figure omitted; refer to PDF]

In fact one can obtain a more effective upper bound for multipartite concurrence. Let ρ=_∑iλi |_ψi ......_ψi |∈_{[Hamiltonian (script capital H)]1} [ecedil]7; _{[Hamiltonian (script capital H)]2} [ecedil]7; ... [ecedil]7; _{[Hamiltonian (script capital H)]N} , where |_ψi ... 's are the orthogonal pure states and _∑iλi =1 . We have

[figure omitted; refer to PDF] The right side of (3.49) gives a new upper bound of _CN (ρ) . Since [figure omitted; refer to PDF] the upper bound obtained in (3.49) is better than that in (3.48).

3.1.4. Bounds of Concurrence and Tangle

In [68], a lower bound for tangle defined in (3.8) has been derived as

[figure omitted; refer to PDF] where _||X||HS =Tr [X^X[dagger] ] denotes the Frobenius or Hilbert-Schmidt norm. Experimentally measurable lower and upper bounds for concurrence have been also given by Mintert et al. in [106, 107] and Zhang et al. in [46]:

[figure omitted; refer to PDF]

Since the convexity of ^C2 (ρ) , we have that τ(ρ)≥^C2 (ρ) always holds. For two-qubit quantum systems, tangle τ is always equal to the square of concurrence ^C2 [58, 113], as a decomposition {_pi ,|_ψi ...} achieving the minimum in (3.3) has the property that C(|_ψi ...)=C(|_ψj ...) ∀i,j . For higher dimensional systems we do not have similar relations. Thus it is meaningful to derive valid upper bound for tangle and lower bound for concurrence.

Theorem 3.9.

For any quantum state ρ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} , one has [118] [figure omitted; refer to PDF] [figure omitted; refer to PDF] where _ρA is the reduced matrix of ρ , and T(ρ) is the correlation matrix of ρ defined in (3.44).

Proof.

We assume that 1-Tr [^ρA2 ]≤1-Tr [^ρB2 ] for convenience. By the definition of τ , we have that for a pure state |ψ...,τ(|ψ...)=2(1-Tr [(^ρA|ψ...)2 ]) . Let ρ=_∑ipiρi be the optimal decomposition such that τ(ρ)=_∑ipi τ(_ρi ). We get [figure omitted; refer to PDF]

Note that, for pure state |ψ...∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} [68], [figure omitted; refer to PDF] Using the inequality a-b≥a-b for any a≥b , we get [figure omitted; refer to PDF] Now let ρ=_∑ipiρi be the optimal decomposition such that C(ρ)=_∑ipi C(_ρi ). We get [figure omitted; refer to PDF] which ends the proof.

The upper bound (3.53), together with the lower bounds (3.54), (3.43), (3.44), (3.51), and (3.52), can allow for estimations of entanglement for arbitrary quantum states. Moreover, since the upper bound is exactly the value of tangle for pure states, the upper bound can be a good estimation when the state is very weakly mixed.

3.2. Concurrence and Tangle of Two Entangled States Are Strictly Larger Than Those of One

In this subsection we show that although bound entangled states cannot be distilled, the concurrence and tangle of two entangled states will be always strictly larger than those of one, even if the two entangled states are both bound entangled.

Let ρ=_{∑ijklρij,kl} |ij......kl|∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} and σ=_{∑^{i[variant prime]j[variant prime]k[variant prime]l[variant prime]}σ^{i[variant prime]j[variant prime]} ,^{k[variant prime]l[variant prime]}} |^{i[variant prime]j[variant prime]} ......^{k[variant prime]l[variant prime]} |∈_{[Hamiltonian (script capital H)]^{A[variant prime]}} [ecedil]7;_{[Hamiltonian (script capital H)]^{B[variant prime]}} be two quantum states shared by subsystems A^{A[variant prime]} and B^{B[variant prime]} . We use ρ[ecedil]7;σ=_{∑ijkl,^{i[variant prime]j[variant prime]k[variant prime]l[variant prime]}ρij,klσ^{i[variant prime]j[variant prime]} ,^{k[variant prime]l[variant prime]}} |i^{i[variant prime]}_{...A^{A[variant prime]}} ...k^{k[variant prime]} |[ecedil]7;|j^{j[variant prime]}_{...B^{B[variant prime]}} ...l^{l[variant prime]} | to denote the state of the whole system.

Lemma 3.10.

For pure states |ψ...∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} and |[straight phi]...∈_{[Hamiltonian (script capital H)]^{A[variant prime]}} [ecedil]7;_{[Hamiltonian (script capital H)]^{B[variant prime]}} , the inequalities [figure omitted; refer to PDF] [figure omitted; refer to PDF] always hold, and "= " in the two inequalities hold if and only if at least one of {|ψ...,|[straight phi]...} is separable.

Proof.

Without loss of generality we assume that C(|ψ...)≥C(|[straight phi]...) . First note that [figure omitted; refer to PDF] Let ^ρA|ψ... =_∑iλi |i......i| and ^{ρA[variant prime] |[straight phi]...} =_∑jπj |j......j| be the spectral decomposition of ^ρA|ψ... and ^{ρA[variant prime] |[straight phi]...} , with _∑iλi =1 and _∑jπj =1 , respectively. By using (3.61) one obtains that [figure omitted; refer to PDF] while [figure omitted; refer to PDF]

Now using the definition of concurrence and the normalization conditions of _λi and _πj , one immediately gets [figure omitted; refer to PDF] If one of {|ψ...,|[straight phi]...} is separable, say |[straight phi]... , then the rank of ^{ρA[variant prime] |[straight phi]...} must be one, which means that there is only one item in the spectral decomposition in ^{ρA[variant prime] |[straight phi]...} . Using the normalization condition of _πj , we obtain Tr [(^{ρAA[variant prime] |ψ...[ecedil]7;|[straight phi]...)2} ]=Tr [(^ρA|ψ...)2 ] . Then inequality (3.64) becomes an equality.

On the other hand, if both |ψ... and |[straight phi]... are entangled (not separable), then there must be at least two items in the decomposition of their reduced density matrices ^ρA|ψ... and ^{ρA[variant prime] |[straight phi]...} , which means that Tr [(^{ρAA[variant prime] |ψ...[ecedil]7;|[straight phi]...)2} ] is strictly larger than Tr [(^ρA|ψ...)2 ] .

The inequality (3.60) also holds because, for pure quantum state ρ , τ(ρ)=^C2 (ρ) .

From the lemma, we have, for mixed states the following.

Theorem 3.11.

For any quantum states ρ∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} and σ∈_{[Hamiltonian (script capital H)]^{A[variant prime]}} [ecedil]7;_{[Hamiltonian (script capital H)]^{B[variant prime]}} , the inequalities [figure omitted; refer to PDF] always hold, and the "= '' in the two inequalities hold if and only if at least one of {ρ,σ} is separable, that is, if both ρ and σ are entangled (even if bound entangled), then C(ρ[ecedil]7;σ)>max {C(ρ),C(σ)} and τ(ρ[ecedil]7;σ)>max {τ(ρ),τ(σ)} always hold [118].

Proof.

We still assume that C(ρ)≥C(σ) for convenience. Let ρ=_∑ipiρi and σ=_∑jqjσj be the optimal decomposition such that C(ρ [ecedil]7; σ)=_∑ipiqj C(_ρi [ecedil]7; _σj ). By using the inequality obtained in Lemma 3.10, we have [figure omitted; refer to PDF]

Case 1.

Now let one of {ρ,σ} be separable, say σ , with ensemble representation σ=_∑jqjσj , where _∑jqj =1 , and _σj is the density matrix of separable pure state. Suppose that ρ=_∑ipiρi is the optimal decomposition of ρ such that C(ρ)=_∑ipi C(_ρi ) . Using Lemma 3.10, we have [figure omitted; refer to PDF] The inequalities (3.66) and (3.67) show that if σ is separable, then C(ρ[ecedil]7;σ)=C(ρ) .

Case 2.

If both ρ and σ are inseparable, that is, there is at least one pure state in the ensemble decomposition of ρ (and σ , resp.), using Lemma 3.10, then we have [figure omitted; refer to PDF]

The inequality for tangle τ can be proved in a similar way.

Remark 3.12.

In [119] it is shown that any entangled state ρ can enhance the teleportation power of another state σ . This holds even if the state ρ is bound entangled. But if ρ is bound entangled, then the corresponding σ must be free entangled (distillable). By Theorem 3.11, we can see that even if two entangled quantum states ρ and σ are bound entangled, their concurrence and tangle are strictly larger than those of one state.

3.3. Subadditivity of Concurrence and Tangle

We now give a proof of the subadditivity of concurrence and tangle, which illustrates that concurrence and tangle may be proper entanglement measurements.

Theorem 3.13.

Let ρ and σ be quantum states in _{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} , the one has [118] [figure omitted; refer to PDF]

Proof.

We first prove that the theorem holds for pure states, that is, for |ψ... and |[varphi]... in _{[Hamiltonian (script capital H)]A} [ecedil]7; _{[Hamiltonian (script capital H)]B} : [figure omitted; refer to PDF] Assume that ^ρA|ψ... =_∑iλi |i......i| and ^{ρA|[varphi]...} =_∑jπj |j......j| are the spectral decomposition of the reduced matrices ^ρA|ψ... and ^{ρA|[varphi]...} . One has [figure omitted; refer to PDF]

Now we prove that (3.69) holds for any mixed-quantum states ρ and σ . Let ρ=_∑ipiρi and σ=_∑jqjσj be the optimal decomposition such that C(ρ)=_∑ipi C(_ρi ) and C(σ)=_∑jqj C(_σj ) . We have [figure omitted; refer to PDF]

The inequality for τ can be derived in a similar way.

4. Fidelity of Teleportation and Distillation of Entanglement

Quantum teleportation is an important subject in quantum information processing. In terms of a classical communication channel and a quantum resource (a nonlocal entangled state like an EPR pair of particles), the teleportation protocol gives ways to transmit an unknown quantum state from a sender traditionally named "Alice" to a receiver "Bob" who are spatially separated. These teleportation processes can be viewed as quantum channels. The nature of a quantum channel is determined by the particular protocol and the state used as a teleportation resource. The standard teleportation protocol _T0 proposed by Bennett et al. in 1993 uses Bell measurements and Pauli rotations. When the maximally entangled pure state |[varphi]...=(1/n)^∑i=0n-1 |ii... is used as the quantum resource, it provides an ideal noiseless quantum channel ^{Λ_T0 (|[varphi]......[varphi]|)} (ρ)=ρ . However in realistic situation, instead of the pure maximally entangled states, Alice and Bob usually share a mixed entangled state due to the decoherence. Teleportation using mixed state as an entangled resource is, in general, equivalent to having a noisy quantum channel. An explicit expression for the output state of the quantum channel associated with the standard teleportation protocol _T0 with an arbitrary mixed-state resource has been obtained [120, 121].

It turns out that, by local quantum operations (including collective actions over all members of pairs in each lab) and classical communication (LOCC) between Alice and Bob, it is possible to obtain a number of pairs in nearly maximally entangled state |_ψ+ ... from many pairs of nonmaximally entangled states. Such a procedure proposed in [73-77] is called distillation. In [73] the authors give operational protocol to distill an entangled two-qubit state whose single fraction F , defined by F(ρ)=..._ψ+ |ρ|_ψ+ ... , is larger than 1/2 . The protocol is then generalized in [77] to distill any d -dimensional bipartite entangled quantum states with F(ρ)>1/d . It is shown that a quantum state ρ violating the reduction criterion can always be distilled. For such states, if their single fraction of entanglement F(ρ)=..._ψ+ |ρ|_ψ+ ... is greater than 1/d , one can distill these states directly by using the generalized distillation protocol, otherwise a proper filtering operation has to be used at first to transform ρ to another state ^{ρ[variant prime]} so that F(^{ρ[variant prime]} )>1/d .

4.1. Fidelity of Quantum Teleportation

Let [Hamiltonian (script capital H)] be a d -dimensional complex vector space with computational basis |i... , i=1,...,d . The fully entangled fraction (FEF) of a density matrix ρ∈[Hamiltonian (script capital H)][ecedil]7;[Hamiltonian (script capital H)] is defined by

[figure omitted; refer to PDF] under all unitary transformations U , where |_ψ+ ...=(1/d)^∑i=1d |ii... is the maximally entangled state and I is the corresponding identity matrix.

In [8, 9], the authors give an optimal teleportation protocol by using a mixed entangled quantum state. The optimal teleportation fidelity is given by [figure omitted; refer to PDF] which solely depends on the FEF of the entangled resource state ρ .

In fact the fully entangled fraction is tightly related to many quantum information processing such as dense coding [10], teleportation [5-7], entanglement swapping [14-18], and quantum cryptography (Bell inequalities) [11-13]. As the optimal fidelity of teleportation is given by FEF [8, 9], experimentally measurement of FEF can be also used to determine the entanglement of the nonlocal source used in teleportation. Thus an analytic formula for FEF is of great importance. In [122] an elegant formula of FEF for two-qubit system is derived analytically by using the method of Lagrange multipliers. For high-dimensional quantum states the analytical computation of FEF remains formidable and less results have been known. In the following we give an estimation on the values of FEF by giving some upper bounds of FEF.

Let _λi , i=1,...,^d2 -1 , be the generators of the SU(d) algebra. A bipartite state ρ∈[Hamiltonian (script capital H)][ecedil]7;[Hamiltonian (script capital H)] can be expressed as [figure omitted; refer to PDF] where _ri (ρ)=(1/2)Tr [ρ_λi (1)[ecedil]7;I] , _sj (ρ)=(1/2)Tr [ρI[ecedil]7;_λj (2)] , and _mij (ρ)=(1/4)Tr [ρ_λi (1)[ecedil]7;_λj (2)] . Let M(ρ) denote the correlation matrix with entries _mij (ρ) .

Theorem 4.1.

For any ρ∈[Hamiltonian (script capital H)][ecedil]7;[Hamiltonian (script capital H)] , the fully entangled fraction ...(ρ) satisfies [123] [figure omitted; refer to PDF] where ^MT stands for the transpose of M and _||M||KF =Tr [M^M[dagger] ] is the Ky Fan norm of M .

Proof.

First, we note that [figure omitted; refer to PDF] where _mij (_P+ )=(1/4)Tr [_P+λi [ecedil]7;_λj ] . By definition (4.1), one obtains [figure omitted; refer to PDF]

Since U_λi^U[dagger] is a traceless Hermitian operator, it can be expanded according to the SU(d) generators as [figure omitted; refer to PDF] Entries _Oij define a real (^d2 -1)×(^d2 -1) matrix O . From the completeness relation of SU(d) generators [figure omitted; refer to PDF] one can show that O is an orthonormal matrix. Using (4.7), we have [figure omitted; refer to PDF]

For the case d=2 , we can get an exact result from (4.4).

Corollary 4.2.

For two-qubit system, one has [figure omitted; refer to PDF] that is, the upper bound derived in Theorem 4.1 is exactly the FEF.

Proof.

We have shown in (4.7) that, given an arbitrary unitary U , one can always obtain an orthonormal matrix O . Now we show that in two-qubit case, for any 3×3 orthonormal matrix O , there always exits 2×2 unitary matrix U such that (4.7) holds.

For any vector t={_t1 ,_t2 ,_t3 } with unit norm, define an operator X≡^∑i=13_tiσi , where _σi 's are Pauli matrices. Given an orthonormal matrix O , one obtains a new operator ^{X[variant prime]} ≡^{∑i=13ti[variant prime]}_σi =^∑i,j=13_Oijtjσi .

X and ^{X[variant prime]} are both Hermitian traceless matrices. Their eigenvalues are given by the norms of the vectors t and ^{t[variant prime]} ={^{t1[variant prime]} ,^{t2[variant prime]} ,^{t3[variant prime]} } , respectively. As the norms are invariant under orthonormal transformations O , they have the same eigenvalues: ±^t12 +^t22 +^t32 . Thus there must be a unitary matrix U such that ^{X[variant prime]} =UX^U[dagger] . Hence the inequality in the proof of Theorem 4.1 becomes an equality. The upper bound (4.4) then becomes exact at this situation, which is in accord with the result in [122].

Remark 4.3.

The upper bound of FEF (4.4) and the FEF (4.10) depend on the correlation matrices M(ρ) and M(_P+ ) . They can be calculated directly according to a given set of SU(d) generators _λi , i=1,...,^d2 -1 . As an example, for d=3 , if we choose [figure omitted; refer to PDF] then we have [figure omitted; refer to PDF] Nevertheless the FEF and its upper bound do not depend on the choice of the SU(d) generators.

The usefulness of the bound depends on detailed states. In the following we give two new upper bounds, which is different from Theorem 4.1. These bounds work for different states.

Let h and g be n×n matrices such that h|j...=|(j+1)mod n... , g|j...=^ωj |j... , with ω=exp {-2iπ/n} . We can introduce ⁿ² linear-independent n×n matrices _Ust =^htgs , which satisfy

[figure omitted; refer to PDF] One can also check that {_Ust } satisfy the condition of bases of the unitary operators in the sense of [124], that is,

[figure omitted; refer to PDF] where _In×n is the n×n identity matrix. {_Ust } form a complete basis of n×n matrices, namely, for any n×n matrix W , W can be expressed as

[figure omitted; refer to PDF]

From {_Ust } , we can introduce the generalized Bell states

[figure omitted; refer to PDF] where |_Φst ... are all maximally entangled states and form a complete orthogonal normalized basis of _{[Hamiltonian (script capital H)]d} [ecedil]7;_{[Hamiltonian (script capital H)]d} .

Theorem 4.4.

For any quantum state ρ∈_{[Hamiltonian (script capital H)]d} [ecedil]7;_{[Hamiltonian (script capital H)]d} , the fully entangled fraction defined in (4.1) fulfills the following inequality: [figure omitted; refer to PDF] where _λj s are the eigenvalues of the real part of matrix M=(TiT-iTT) , T is a ^d2 ×^d2 matrix with entries _Tn,m =..._Φn |ρ|_Φm ... , and _Φj are the maximally entangled basis states defined in (4.16) [125].

Proof.

From (4.15), any d×d unitary matrix U can be represented by U=^∑k=1d2_zkUk , where _zk =(1/d)Tr [^Uk[dagger] U] . Define [figure omitted; refer to PDF] Then the unitary matrix U can be rewritten as U=^∑k=12d2_zk^{Uk[variant prime]} . The necessary condition for the unitary property of U implies that _∑k^xk2 =1 . Thus we have [figure omitted; refer to PDF] where _Mmn is defined in the theorem. One can deduce that [figure omitted; refer to PDF] from the hermiticity of ρ .

Taking into account the constraint with an undetermined Lagrange multiplier λ , we have [figure omitted; refer to PDF] Accounting to (4.20) we have the eigenvalue equation [figure omitted; refer to PDF]

Inserting (4.22) into (4.19) results in [figure omitted; refer to PDF] where _ηj =-_λj is the corresponding eigenvalues of the real part of the matrix M .

Example 4.5.

Horodecki gives a very interesting bound entangled state in [31] as [figure omitted; refer to PDF] One can easily compare the upper bound obtained in (4.17) and that in (4.4). From Figure 2 we see that, for 0≤a<0.572 , the upper bound in (4.17) is larger than that in (4.4). But for 0.572<a<1 the upper bound in (4.17) is always lower than that in (4.4), which means that the upper bound (4.17) is tighter than (4.4).

Figure 2: Upper bound of ...(ρ(a)) from (4.17) (solid line) and upper bound from (4.4) (dashed line) for state (4.24).

[figure omitted; refer to PDF]

In fact, we can drive another upper bound for FEF which will be very tight for weakly mixed-quantum states.

Theorem 4.6.

For any bipartite quantum state ρ∈_{[Hamiltonian (script capital H)]d} [ecedil]7;_{[Hamiltonian (script capital H)]d} , the following inequality holds [125]: [figure omitted; refer to PDF] where _ρA is the reduced matrix of ρ .

Proof.

Note that in [77] the authors have obtained the FEF for pure state |ψ... as [figure omitted; refer to PDF] where ^ρA|ψ... is the reduced matrix of |ψ......ψ| .

For mixed state ρ=_∑ipi^ρi , we have [figure omitted; refer to PDF] Let _λij be the real and nonnegative eigenvalues of the matrix _pi^ρAi . Recall that for any function F=_∑i (_∑j^xij2)1/2 subjected to the constraints _zj =_∑ixij with _xij being real and nonnegative, the inequality _∑j^zj2 ≤^F2 holds, from which it follows that [figure omitted; refer to PDF] which ends the proof.

4.2. Fully Entangled Fraction and Concurrence

The upper bound of FEF has also interesting relations to the entanglement measure concurrence. As shown in [122], the concurrence of a two-qubit quantum state has some kinds of relation with the optimal teleportation fidelity. For quantum state with high dimension, we have the similar relation between them too.

Theorem 4.7.

For any bipartite quantum state ρ∈_{[Hamiltonian (script capital H)]d} [ecedil]7;_{[Hamiltonian (script capital H)]d} , one has [118] [figure omitted; refer to PDF]

Proof.

In [126], the authors show that, for any pure state |ψ...∈_{[Hamiltonian (script capital H)]A} [ecedil]7;_{[Hamiltonian (script capital H)]B} , the following inequality holds: [figure omitted; refer to PDF] where [straight epsilon] denotes the set of d×d dimensional maximally entangled states.

Let ρ=_∑ipi |_[varphi]i ......_[varphi]i | be the optimal decomposition such that C(ρ)=_∑ipi C(|_ψi ...) . We have [figure omitted; refer to PDF] which ends the proof.

Inequality (4.29) has demonstrated the relation between the lower bound of concurrence and the fully entangled fraction (thus the optimal teleportation fidelity), that is, the fully entangled fraction of a quantum state ρ is limited by its concurrence.

We now consider tripartite case. Let _ρABC be a state of three-qubit systems denoted by A , B , and C . We study the upper bound of the FEF, ...(_ρAB ) , between qubits A and B , and its relations to the concurrence under bipartite partition AB and C . For convenience we normalize ...(_ρAB ) to be

[figure omitted; refer to PDF] Let C(_ρAB|"C ) denote the concurrence between subsystems AB and C .

Theorem 4.8.

For any triqubit state _ρABC , _...N (_ρAB ) satisfies [123] [figure omitted; refer to PDF]

Proof.

We first consider the case that _ρABC is pure, _ρABC =|ψ_...ABC ...ψ| . By using the Schmidt decomposition between qubits A,B , and C , |ψ_...ABC can be written as [figure omitted; refer to PDF] for some orthonormalized bases |_iAB ... , |_iC ... of subsystems AB , C , respectively. The reduced density matrix _ρAB has the form [figure omitted; refer to PDF] where Λ is a 4×4 diagonal matrix with diagonal elements {^η12 ,^η22 ,0,0} , U is a unitary matrix, and ^U* denotes the conjugation of U .

The FEF of the two-qubit state _ρAB can be calculated by using formula (4.10) or the one in [122]. Let [figure omitted; refer to PDF] be the 4×4 matrix constituted by the four Bell bases. The FEF of _ρAB can be written as [figure omitted; refer to PDF] where _ηmax (X) stands for the maximal eigenvalues of the matrix X .

For pure state (4.34) in bipartite partition AB and C , we have [figure omitted; refer to PDF] From (4.32), (4.37), and (4.38) we get [figure omitted; refer to PDF]

We now prove that the above inequality (4.39) also holds for mixed state _ρABC . Let _ρABC =_∑ipi |_ψi...ABC ..._ψi | be the optimal decomposition of _ρABC such that C(_ρAB|C )=_∑ipi C(|_ψi ..._)AB|C . We have [figure omitted; refer to PDF] where ^ρAB|Ci =|_ψi...ABC ..._ψi | and ^ρABi =T_rC [^ρAB|Ci ] .

From Theorem 4.8 we see that the FEF of qubits A and B are bounded by the concurrence between qubits A , B , and qubit C . The upper bound of FEF for _ρAB decreases when the entanglement between qubits A,B , and C increases. As an example, we consider the generalized W state defined by |^{W[variant prime]} ...=α|100...+β|010...+γ|001... , |α^|2 +|β^|2 +|γ^|2 =1 . The reduced density matrix is given by [figure omitted; refer to PDF] The FEF of ^{ρABW[variant prime]} is given by [figure omitted; refer to PDF] while the concurrence of |^{W[variant prime]} ... has the form _CAB|C (|^{W[variant prime]} ...)=2|γ||α^|2 +|β^|2 . We see that (4.33) always holds. In particular for |α|=|β| and |γ|≤2/2 , the inequality (4.33) is saturated (see Figure 3).

Figure 3: _...N (^{ρABW[variant prime]} ) (dashed line) and upper bound 1-^C2 (|^{W[variant prime]}_...AB|"C ) (solid line) of state |^{W[variant prime]}_...AB|"C at |α|=|β| .

[figure omitted; refer to PDF]

4.3. Improvement of Entanglement Distillation Protocol

The upper bound can give rise to not only an estimation of the fidelity in quantum information processing such as teleportation, but also an interesting application in entanglement distillation of quantum states. In [77] a generalized distillation protocol has been presented. It is shown that a quantum state ρ violating the reduction criterion can always be distilled. For such states if their single fraction of entanglement F(ρ)=..._ψ+ |ρ|_ψ+ ... is greater than 1/d , then one can distill these states directly by using the generalized distillation protocol. If the FEF (the largest value of single fraction of entanglement under local unitary transformations) is less than or equal to 1/d , then a proper filtering operation has to be used at first to transform ρ to another state ^{ρ[variant prime]} so that F(^{ρ[variant prime]} )>1/d . For d=2 , one can compute FEF analytically according to the corollary. For d≥3 our upper bound (4.4) can supply a necessary condition in the distillation.

Theorem 4.9.

For an entangled state ρ∈[Hamiltonian (script capital H)][ecedil]7;[Hamiltonian (script capital H)] violating the reduction criterion, if the upper bound (4.4) is less than or equal to 1/d , then the filtering operation has to be applied before using the generalized distillation protocol [123].

As an example, we consider a 3×3 state

[figure omitted; refer to PDF] where σ=(x|0......0|+(1-x)|1......1|)[ecedil]7;(x|0......0|+(1-x)|1......1|) . It is direct to verify that ρ violates the reduction criterion for 0≤x≤1 , as (_ρ1 [ecedil]7;I)-ρ has a negative eigenvalue -2/27 . Therefore the state is distillable. From Figure 4, we see that for 0≤x<0.0722 and 0.9278<x≤1 , the fidelity is already greater than 1/3 ; thus the generalized distillation protocol can be applied without the filtering operation. However for 0.1188≤x≤0.8811 , even the upper bound of the fully entangled fraction is less than or equal to 1/3 ; hence the filtering operation has to be applied first, before using the generalized distillation protocol.

Figure 4: Upper bound of ...(ρ)-1/3 from (4.4) (solid line) and fidelity F(ρ)-1/3 (dashed line) for state (4.43).

[figure omitted; refer to PDF]

Moreover, the lower bounds of concurrence can be also used to study the distillability of quantum states. Based on the positive partial transpose (PPT) criterion, a necessary and sufficient condition for the distillability was proposed in [127], which is not operational in general. An alternative distillability criterion based on the bound _τ2 in (3.19) can be obtained to improve the operationality.

Theorem 4.10.

A bipartite quantum state ρ is distillable if and only if _τ2 (^{ρ[ecedil]7;N} )>0 for some number N [67].

Proof.

It was shown in [127] that a density matrix ρ is distillable if and only if there are some projectors P , Q that map high-dimensional spaces to two-dimensional ones and some number N such that the state P[ecedil]7;Q^{ρ[ecedil]7;N} P[ecedil]7;Q is entangled [127]. Thus if _τ2 (^{ρ[ecedil]7;N} )>0 , then there exists one submatrix of matrix ^{ρ[ecedil]7;N} , similar to (3.20), which has nonzero _τ2 and is entangled in a 2[ecedil]7;2 space; hence ρ is distillable.

Corollary 4.11.

(a) The lower bound _τ2 (ρ)>0 is a sufficient condition for the distillability of any bipartite state ρ .

(b) The lower bound _τ2 (ρ)=0 is a necessary condition for separability of any bipartite state ρ .

Remark 4.12.

Corollary 4.11 directly follows from Theorem 4.10 and this case is referred to as one distillable [128]. The problem of whether non-PPT (NPPT) nondistillable states exist is studied numerically in [128, 129]. By using Theorem 4.10, although it seems impossible to solve the problem completely, it is easy to judge the distillability of a state under condition that is one distillable.

The lower bound _τ2 , PPT criterion, separability, and distillability for any bipartite quantum state ρ have the following relations: if _τ2 (ρ)>0 , then ρ is entangled. If ρ is separable, then it is PPT. If _τ2 (ρ)>0 , then ρ is distillable. If ρ is distillable, then it is NPPT. From the last two propositions it follows that if ρ is PPT, then _τ2 (ρ)=0 , that is, if _τ2 (ρ)>0 , then ρ is NPPT.

Theorem 4.13.

For any pure tripartite state |[varphi]_...ABC in arbitrary d[ecedil]7;d[ecedil]7;d dimensional spaces, bound _τ2 satisfies [67] [figure omitted; refer to PDF] where _ρAB =_{Tr C} (|[varphi]_...ABC ...[varphi]|) , _ρAC =_{Tr B} (|[varphi]_...ABC ...[varphi]|) , and _ρA:BC =_{Tr BC} (|[varphi]_...ABC ...[varphi]|) .

Proof.

Since ^Cmn2 ≤^{(λmn(1) )2} ≤^{∑i=14(λmn(i) )2} =Tr (ρ_ρ...mn ) , one can derive the inequality [figure omitted; refer to PDF] where D=d(d-1)/2 . Note that _∑lk Tr [_ρAB (_ρ...AB)lk ]≤1-Tr ^ρA2 -Tr ^ρB2 +Tr ^ρC2 and _∑pq Tr [_ρAC (_ρ...AC)pq ]≤1-Tr ^ρA2 +Tr ^ρB2 -Tr ^ρC2 , where l,pk,q,=1,...,D . By using the similar analysis in [113], one has that the right-hand side of (4.45) is equal to 2(1-Tr ^ρA2 )=^C2 (_ρA:BC ) . Taking into account that _τ2 (_ρA:BC )=^...9E;2 (_ρA:BC ) for a pure state, one obtains inequality (4.44).

Generally for any pure multipartite quantum state _{ρAB1B2 ...bn} , one has the following monogamy inequality: [figure omitted; refer to PDF]

5. Summary and Conclusion

We have introduced some recent results on three aspects in quantum information theory. The first one is the separability of quantum states. New criteria to detect more entanglements have been discussed. The normal forms of quantum states have been also studied, which helps in investigating the separability of quantum states. Moreover, since many kinds of quantum states can be transformed into the same normal forms, quantum states can be classified in terms of the normal forms. For the well-known entanglement measure concurrence, we have discussed the tight lower and upper bounds. It turns out that, although one cannot distill a singlet from many pairs of bound entangled states, the concurrence and tangle of two entangled quantum states are always larger than those of one, even if both of two entangled quantum states are bound entangled. Related to the optimal teleportation fidelity, upper bounds for the fully entangled fraction have been studied, which can be used to improve the distillation protocol. Interesting relations between fully entangled fraction and concurrence have been also introduced. All these related problems in the theory of quantum entanglement have not been completely solved yet. Many problems remain open concerning the physical properties and mathematical structures of quantum entanglement, and the applications of entangled states in information processing.

References

[1] A. Einstein, B. Podolsky, N. Rosen, "Can quantum-mechanical description of physical reality be considered complete?," Physical Review , vol. 47, no. 10, pp. 777-780, 1935.

[2] A. Peres Quantum Mechanics: Concepts and Methods , Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.

[3] M. A. Nielsen, I. L. Chuang Quantum Computation and Quantum Information , pp. xxvi+676, Cambridge University Press, Cambridge, UK, 2000.

[4] D. P. Di Vincenzo, "Quantum computation," Science , vol. 270, pp. 255-261, 1995.

[5] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters, "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels," Physical Review Letters , vol. 70, no. 13, pp. 1895-1899, 1993.

[6] S. Albeberio, S. M. Fei, "Teleportation of general finite-dimensional quantum systems," Physics Letters A , vol. 276, pp. 8-11, 2000.

[7] G. M. D'Ariano, P. Lo Presti, M. F. Sacchi, "Bell measurements and observables," Physics Letters A , vol. 272, no. 1-2, pp. 32-38, 2000., [email protected]

[8] M. Horodecki, P. Horodecki, R. Horodecki, "General teleportation channel, singlet fraction, and quasidistillation," Physical Review A , vol. 60, no. 3, pp. 1888-1898, 1999., [email protected]; [email protected]; [email protected]

[9] S. Albeverio, S.-M. Fei, W.-L. Yang, "Optimal teleportation based on bell measurements," Physical Review A , vol. 66, no. 1, 2002.

[10] C. H. Bennett, S. J. Wiesner, "Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states," Physical Review Letters , vol. 69, no. 20, pp. 2881-2884, 1992.

[11] A. K. Ekert, "Quantum cryptography based on Bell's theorem," Physical Review Letters , vol. 67, no. 6, pp. 661-663, 1991.

[12] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, "Quantum privacy amplification and the security of quantum cryptography over noisy channels," Physical Review Letters , vol. 77, no. 13, pp. 2818-2821, 1996.

[13] C. A. Fuchs, N. Gisin, R. B. Griffiths, C.-S. Niu, A. Peres, "Optimal eavesdropping in quantum cryptography. I. Information bound and optimal strategy," Physical Review A , vol. 56, no. 2, pp. 1163-1172, 1997.

[14] M. Zukowski, A. Zeilinger, M. A. Horne, A. K. Ekert, ""Event-ready-detectors" Bell experiment via entanglement swapping," Physical Review Letters , vol. 71, no. 26, pp. 4287-4290, 1993.

[15] S. Bose, V. Vedral, P. L. Knight, "Multiparticle generalization of entanglement swapping," Physical Review A , vol. 57, no. 2, pp. 822-829, 1998.

[16] S. Bose, V. Vedral, P. L. Knight, "Purification via entanglement swapping and conserved entanglement," Physical Review A , vol. 60, no. 1, pp. 194-197, 1999.

[17] B.-S. Shi, Y.-K. Jiang, G.-C. Guo, "Optimal entanglement purification via entanglement swapping," Physical Review A , vol. 62, no. 5, 2000.

[18] L. Hardy, D. D. Song, "Entanglement-swapping chains for general pure states," Physical Review A , vol. 62, no. 5, 2000.

[19] C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, W. K. Wootters, "Remote state preparation," Physical Review Letters , vol. 87, no. 7, 2001.

[20] B.-S. Shi, A. Tomita, "Remote state preparation of an entangled state," Journal of Optics B , vol. 4, no. 6, pp. 380-382, 2002.

[21] J. M Liu, Y. Z. Wang, "Probabilistic remote state preparation by W states," Chinese Physics , vol. 13, pp. 147-152, 2004.

[22] D. W. Leung, P. W. Shor, "Oblivious remote state preparation," Physical Review Letters , vol. 90, no. 12, 2003.

[23] D. W. Berry, B. C. Sanders, "Optimal remote state preparation," Physical Review Letters , vol. 90, no. 5, 2003.

[24] M.-Y. Ye, Y.-S. Zhang, G.-C. Guo, "Faithful remote state preparation using finite classical bits and a nonmaximally entangled state," Physical Review A , vol. 69, no. 2, 2004.

[25] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, "Quantum entanglement," Reviews of Modern Physics , vol. 81, no. 2, pp. 865-942, 2009.

[26] O. Gühne, G. Tóth, "Entanglement detection," Physics Reports , vol. 474, no. 1-6, pp. 1-75, 2009.

[27] S. Albeverio, S.-M. Fei, "A note on invariants and entanglements," Journal of Optics B , vol. 3, no. 4, pp. 223-227, 2001.

[28] R. F. Werner, "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model," Physical Review A , vol. 40, no. 8, pp. 4277-4281, 1989.

[29] A. Peres, "Separability criterion for density matrices," Physical Review Letters , vol. 77, no. 8, pp. 1413-1415, 1996.

[30] M. Horodecki, P. Horodecki, R. Horodecki, "Separability of mixed states: necessary and sufficient conditions," Physics Letters A , vol. 223, no. 1-2, pp. 1-8, 1996.

[31] P. Horodecki, "Separability criterion and inseparable mixed states with positive partial transposition," Physics Letters A , vol. 232, no. 5, pp. 333-339, 1997.

[32] O. Rudolph, "Some properties of the computable cross-norm criterion for separability," Physical Review A , vol. 67, no. 3, 2003.

[33] K. Chen, L.-A. Wu, "A matrix realignment method for recognizing entanglement," Quantum Information and Computation , vol. 3, no. 3, pp. 193-202, 2003.

[34] M. Horodecki, P. Horodecki, R. Horodecki, "Separability of mixed quantum states: linear contractions and permutation criteria," Open Systems & Information Dynamics , vol. 13, no. 1, pp. 103-111, 2006.

[35] K. Chen, L.-A. Wu, "The generalized partial transposition criterion for separability of multipartite quantum states," Physics Letters A , vol. 306, no. 1, pp. 14-20, 2002.

[36] O. Rudolph, "Some properties of the computable cross-norm criterion for separability," Physical Review A , vol. 67, no. 3, 2003.

[37] K. Chen, L.-A. Wu, "Test for entanglement using physically observable witness operators and positive maps," Physical Review A , vol. 69, no. 2, 2004., [email protected]

[38] L. Clarisse, P. Wocjan, "On independent permutation separability criteria," Quantum Information & Computation , vol. 6, no. 3, pp. 277-288, 2006.

[39] S. Albeverio, K. Chen, S.-M. Fei, "Generalized reduction criterion for separability of quantum states," Physical Review A , vol. 68, no. 6, 2003.

[40] M. Horodecki, P. Horodecki, "Reduction criterion of separability and limits for a class of distillation protocols," Physical Review A , vol. 59, no. 6, pp. 4206-4216, 1999.

[41] N. J. Cerf, C. Adami, R. M. Gingrich, "Reduction criterion for separability," Physical Review A , vol. 60, no. 2, pp. 898-909, 1999.

[42] M. A. Nielsen, J. Kempe, "Separable states are more disordered globally than locally," Physical Review Letters , vol. 86, no. 22, pp. 5184-5187, 2001.

[43] H. F. Hofmann, S. Takeuchi, "Violation of local uncertainty relations as a signature of entanglement," Physical Review A , vol. 68, no. 3, 2003.

[44] O. Gühne, M. Mechler, G. Tóth, P. Adam, "Entanglement criteria based on local uncertainty relations are strictly stronger than the computable cross norm criterion," Physical Review A , vol. 74, no. 1, 2006.

[45] O. Gühne, "Characterizing entanglement via uncertainty relations," Physical Review Letters , vol. 92, no. 11, 2004.

[46] C.-J. Zhang, Y.-S. Zhang, S. Zhang, G.-C. Guo, "Optimal entanglement witnesses based on local orthogonal observables," Physical Review A , vol. 76, no. 1, 2007.

[47] J. I. De Vicente, "Separability criteria based on the Bloch representation of density matrices," Quantum Information & Computation , vol. 7, no. 7, pp. 624-638, 2007.

[48] J. I. De Vicente, "Further results on entanglement detection and quantification from the correlation matrix criterion," Journal of Physics A , vol. 41, no. 6, 2008., [email protected]

[49] O. Gühne, P. Hyllus, O. Gittsovich, J. Eisert, "Covariance matrices and the separability problem," Physical Review Letters , vol. 99, no. 13, 2007.

[50] O. Gittsovich, O. Gühne, P. Hyllus, J. Eisert, "Unifying several separability conditions using the covariance matrix criterion," Physical Review A , vol. 78, no. 5, 2008.

[51] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin, W. K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Physical Review Letters , vol. 78, pp. 2031, 1996.

[52] D. Bruß, "Characterizing entanglement," Journal of Mathematical Physics , vol. 43, no. 9, pp. 4237-4251, 2002.

[53] W. K. Wootters, "Entanglement of formation of an arbitrary state of two qubits," Physical Review Letters , vol. 80, no. 10, pp. 2245-2248, 1998.

[54] B. M. Terhal, K. G. H. Vollbrecht, "Entanglement of formation for isotropic states," Physical Review Letters , vol. 85, no. 12, pp. 2625-2628, 2000.

[55] S.-M. Fei, J. Jost, X. Li-Jost, G.-F. Wang, "Entanglement of formation for a class of quantum states," Physics Letters A , vol. 310, no. 5-6, pp. 333-338, 2003.

[56] S.-M. Fei, X. Li-Jost, "A class of special matrices and quantum entanglement," Reports on Mathematical Physics , vol. 53, no. 2, pp. 195-210, 2004.

[57] S.-M. Fei, Z.-X. Wang, H. Zhao, "A note on entanglement of formation and generalized concurrence," Physics Letters A , vol. 329, no. 6, pp. 414-419, 2004.

[58] P. Rungta, C. M. Caves, "Concurrence-based entanglement measures for isotropic states," Physical Review A , vol. 67, no. 1, 2003.

[59] F. Mintert, M. Kus, A. Buchleitner, "Concurrence of mixed bipartite quantum states in arbitrary dimensions," Physical Review Letters , vol. 92, no. 16, 2004.

[60] K. Chen, S. Albeverio, S.-M. Fei, "Entanglement of formation of bipartite quantum states," Physical Review Letters , vol. 95, no. 21, 2005.

[61] K. Chen, S. Albeverio, S.-M. Fei, "Concurrence of arbitrary dimensional bipartite quantum states," Physical Review Letters , vol. 95, no. 4, 2005.

[62] H.-P. Breuer, "Separability criteria and bounds for entanglement measures," Journal of Physics A , vol. 39, no. 38, pp. 11847-11860, 2006.

[63] H.-P. Breuer, "Optimal entanglement criterion for mixed quantum states," Physical Review Letters , vol. 97, no. 8, 2006.

[64] J. I. De Vicente, "Lower bounds on concurrence and separability conditions," Physical Review A , vol. 75, no. 5, 2007.

[65] X.-H. Gao, S.-M. Fei, K. Wu, "Lower bounds of concurrence for tripartite quantum systems," Physical Review A , vol. 74, no. 5, 2006.

[66] E. Gerjuoy, "Lower bound on entanglement of formation for the qubit-qudit system," Physical Review A , vol. 67, no. 5, 2003.

[67] Y.-C. Ou, H. Fan, S.-M. Fei, "Proper monogamy inequality for arbitrary pure quantum states," Physical Review A , vol. 78, no. 1, 2008.

[68] J. I. De Vicente, "Further results on entanglement detection and quantification from the correlation matrix criterion," Journal of Physics A , vol. 41, no. 6, 2008., [email protected]

[69] F. Mintert, M. Kus, A. Buchleitner, "Concurrence of mixed multipartite quantum states," Physical Review Letters , vol. 95, no. 26, 2005.

[70] S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F. Mintert, A. Buchleitner, "Experimental determination of entanglement with a single measurement," Nature , vol. 440, no. 7087, pp. 1022-1024, 2006., [email protected]

[71] S. P. Walborn, P. H. S. Ribeiro, L. Davidovich, F. Mintert, A. Buchleitner, "Experimental determination of entanglement by a projective measurement," Physical Review A , vol. 75, no. 3, 2007., [email protected]

[72] S.-M. Fei, M.-J. Zhao, K. Chen, Z.-X. Wang, "Experimental determination of entanglement for arbitrary pure states," Physical Review A , vol. 80, no. 3, 2009.

[73] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Physical Review Letters , vol. 76, no. 5, pp. 722-725, 1996.

[74] M. Horodecki, P. Horodecki, R. Horodecki, "Inseparable two spin-1/2 density matrices can be distilled to a singlet form," Physical Review Letters , vol. 78, no. 4, pp. 574-577, 1997.

[75] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, "Quantum privacy amplification and the security of quantum cryptography over noisy channels," Physical Review Letters , vol. 77, no. 13, pp. 2818-2821, 1996.

[76] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, W. K. Wootters, "Mixed-state entanglement and quantum error correction," Physical Review A , vol. 54, no. 5, pp. 3824-3851, 1996.

[77] M. Horodecki, P. Horodecki, "Reduction criterion of separability and limits for a class of distillation protocols," Physical Review A , vol. 59, no. 6, pp. 4206-4216, 1999.

[78] J. Priskill Quantum Information and Quantum Computation , California Institute of Technology, 1998.

[79] A. Acin, A. Andrianov, L. Costa, E. Jane, J. I. Latorre, R. Tarrach, "Generalized Schmidt decomposition and classification of three-quantum-bit states," Physical Review Letters , vol. 85, no. 7, pp. 1560-1563, 2000.

[80] A. Sanpera, R. Tarrach, G. Vidal, "Local description of quantum inseparability," Physical Review A , vol. 58, no. 2, pp. 826-830, 1998.

[81] N. J. Cerf, C. Adami, R. M. Gingrich, "Reduction criterion for separability," Physical Review A , vol. 60, no. 2, pp. 898-909, 1999.

[82] M. Horodecki, P. Horodecki Physical Review A , vol. 54, pp. 2406, 1999.

[83] O. Rudolph, "On the cross norm criterion for separability," Journal of Physics A , vol. 36, no. 21, pp. 5825, 2003.

[84] F. T. Hioe, J. H. Eberly, "N-level coherence vector and higher conservation laws in quantum optics and quantum mechanics," Physical Review Letters , vol. 47, no. 12, pp. 838-841, 1981.

[85] W. Greiner, B. Muller Quantum Mechanics: Symmetries , Springer, Berlin, Germany, 1989.

[86] J. E. Harriman, "Geometry of density matrices. I. Definitions, N matrices and 1 matrices," Physical Review A , vol. 17, no. 4, pp. 1249-1256, 1978.

[87] M. Li, S.-M. Fei, Z.-X. Wang, "Separability of tripartite quantum system," International Journal of Quantum Information , vol. 6, no. 4, pp. 859-866, 2008.

[88] S. Wu, J. Anandan, "Some aspects ot separability," Physics Letters A , vol. 297, no. 1-2, pp. 4-8, 2002.

[89] O. Rudolph, "Further results on the cross norm criterion for separability," Quantum Information Processing , vol. 4, no. 3, pp. 219-239, 2005.

[90] M. Li, S.-M. Fei, Z.-X. Wang, "Separability and entanglement of quantum states based on covariance matrices," Journal of Physics A , vol. 41, no. 20, 2008.

[91] R. A. Horn, C. R. Johnson Matrix Analysis , pp. xiii+561, Cambridge University Press, Cambridge, UK, 1985.

[92] S. Yu, N. Liu, "Entanglement detection by local orthogonal observables," Physical Review Letters , vol. 95, no. 15, 2005.

[93] F. Verstraete, J. Dehaene, B. De Moor, "Normal forms and entanglement measures for multipartite quantum states," Physical Review A , vol. 68, no. 1, 2003.

[94] J. M. Leinaas, J. Myrheim, E. Ovrum, "Geometrical aspects of entanglement," Physical Review A , vol. 74, no. 1, 2006.

[95] F. Verstraete , Ph. K. thesis, Katholieke Universiteit Leuven, 2002.

[96] J. I. De Vicente, "Separability criteria based on the bloch representation of density matrices," Quantum Information and Computation , vol. 7, no. 7, pp. 624-638, 2007.

[97] A. S. M. Hassan, P. S. Joag, "Separability criterion for multipartite quantum states based on the bloch representation of density matrices," Quantum Information and Computation , vol. 8, no. 8-9, pp. 773-790, 2008.

[98] M. Li, S.-M. Fei, Z.-X. Wang, "A note on normal forms of quantum states and separability," Communications in Theoretical Physics , vol. 50, no. 6, pp. 1307-1311, 2008.

[99] G. Vidal, D. Jonathan, M. A. Nielsen, "Approximate transformations and robust manipulation of bipartite pure-state entanglement," Physical Review A , vol. 62, no. 1, 2000.

[100] V. Vedral, M. B. Plenio, "Entanglement measures and purification procedures," Physical Review A , vol. 57, no. 3, pp. 1619-1633, 1998.

[101] S. Hill, W. K. Wootters, "Entanglement of a pair of quantum bits," Physical Review Letters , vol. 78, no. 26, pp. 5022-5025, 1997.

[102] A. Osterloh, L. Amico, G. Falci, R. Fazio, "Scaling of entanglement close to a quantum phase transition," Nature , vol. 416, no. 6881, pp. 608-610, 2002.

[103] L.-A. Wu, M. S. Sarandy, D. A. Lidar, "Quantum phase transitions and bipartite entanglement," Physical Review Letters , vol. 93, no. 25, 2004.

[104] S. Ghosh, T. F. Rosenbaum, G. Aeppli, S. N. Coppersmith, "Entangled quantum state of magnetic dipoles," Nature , vol. 425, no. 6953, pp. 48-51, 2003., [email protected]

[105] V. Vedral, "Entanglement hits the big time," Nature , vol. 425, no. 6953, pp. 28-29, 2003., [email protected]

[106] F. Mintert, M. Kus, A. Buchleitner, "Concurrence of mixed bipartite quantum states in arbitrary dimensions," Physical Review Letters , vol. 92, no. 16, 2004.

[107] F. Mintert , Ph.D. thesis, Munich University, Munich, Germany, 2004.

[108] P. Rungta, V. Buzek, C. M. Caves, M. Hillery, G. J. Milburn, "Universal state inversion and concurrence in arbitrary dimensions," Physical Review A , vol. 64, no. 4, 2001.

[109] A. Uhlmann, "Fidelity and concurrence of conjugated states," Physical Review A , vol. 62, no. 3, 2000.

[110] L. Aolita, F. Mintert, "Measuring multipartite concurrence with a single factorizable observable," Physical Review Letters , vol. 97, no. 5, 2006., [email protected]

[111] A. R. R. Carvalho, F. Mintert, A. Buchleitner, "Decoherence and multipartite entanglement," Physical Review Letters , vol. 93, no. 23, 2004.

[112] M. Li, S.-M. Fei, Z.-X. Wang, "A lower bound of concurrence for multipartite quantum states," Journal of Physics A , vol. 42, no. 14, 2009.

[113] V. Coffman, J. Kundu, W. K. Wootters, "Distributed entanglement," Physical Review A , vol. 61, no. 5, 2000.

[114] M. Zukowski, A. Zeilinger, M. A. Horne, A. K. Ekert, ""Event-ready-detectors" Bell experiment via entanglement swapping," Physical Review Letters , vol. 71, no. 26, pp. 4287-4290, 1993.

[115] J.-M. Cai, Z.-W. Zhou, S. Zhang, G.-C. Guo, "Compatibility conditions from multipartite entanglement measures," Physical Review A , vol. 75, no. 5, 2007.

[116] M. Li, S.-M. Fei, X. Li-Jost, Z.-X. Wang Reports on Mathematical Physics . In press

[117] L. Aolita, A. Buchleitner, F. Mintert, "Scalable method to estimate experimentally the entanglement of multipartite systems," Physical Review A , vol. 78, no. 2, 2008.

[118] M. Li, S.-M. Fei, X. Li-Jost, Z.-X. Wang Chinese Physics B . In press

[119] L. Masanes, "All bipartite entangled states are useful for information processing," Physical Review Letters , vol. 96, no. 15, 2006.

[120] G. Bowen, S. Bose, "Teleportation as a depolarizing quantum channel, relative entropy, and classical capacity," Physical Review Letters , vol. 87, no. 26, 2001.

[121] S. Albeverio, S.-M. Fei, W.-L. Yang, "Teleportation with an arbitrary mixed resource as a trace-preserving quantum channel," Communications in Theoretical Physics , vol. 38, no. 3, pp. 301-304, 2002.

[122] J. Grondalski, D. M. Etlinger, D. F. V. James, "The fully entangled fraction as an inclusive measure of entanglement applications," Physics Letters A , vol. 300, no. 6, pp. 573-580, 2002.

[123] M. Li, S.-M. Fei, Z.-X. Wang, "Upper bound of the fully entangled fraction," Physical Review A , vol. 78, no. 3, 2008.

[124] R. F. Werner, "All teleportation and dense coding schemes," Journal of Physics A , vol. 34, no. 35, pp. 7081-7094, 2001.

[125] R.-J. Gu, M. Li, S.-M. Fei, X.-Q. Li-Jost, "On estimation of fully entangled fraction," Communications in Theoretical Physics , vol. 53, no. 2, pp. 265-268, 2010.

[126] C.-S. Yu, X. X. Yi, H.-S. Song, "Evolution of entanglement for quantum mixed states," Physical Review A , vol. 78, no. 6, 2008.

[127] M. Horodecki, P. Horodecki, R. Horodecki, "Mixed-state entanglement and distillation: is there a "bound" entanglement in nature?," Physical Review Letters , vol. 80, no. 24, pp. 5239-5242, 1998.

[128] W. Du, J. I. Cirac, M. Lewenstein, D. Bruß, "Distillability and partial transposition in bipartite systems," Physical Review A , vol. 61, no. 6, 2000.

[129] D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, A. V. Thapliyal, "Evidence for bound entangled states with negative partial transpose," Physical Review A , vol. 61, no. 6, 2000.