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

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.

It has become clear that entanglement is not only the subject of philosophical debates, but also a new quantum resource for tasks which can not 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 [14,15]. 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 can we 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 [16] can be used to judge if the state is separable or not. In addition separable states must satisfy all possible Bell inequalities [17].
For mixed states we still have no general criterion. The well-known PPT (partial positive transposition) criterion was proposed by Peres in 1996 [18]. 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 Horodeckis [19] 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 [20] 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 [21,22]. It demonstrates a remarkable ability to detect many bound entangled states and even genuinely tripartite entanglement [23]. Considerable efforts have been made in finding stronger variants and multipartite generalizations for this criterion [24,25].
It was shown that PPT criterion and realignment criterion are equivalent to the permutations of the density matrix's indices [23]. Another important criterion for separability is the reduction criterion [26,27]. This criterion is equivalent to the PPT criterion for 2 × N composite systems. Although it is generally weaker than the PPT, the reduction criteria has tight relation to the distillation of quantum states.
There are also some other necessary criteria for separability. Nielsen et al. [28] presented a necessary criterion called majorization: the decreasing ordered vector of the eigenvalues for ρ is majorized by that of ρ A 1 or ρ A 2 alone for a separable state. i.e. if a state ρ is separable, Here λ ↓ ρ denotes the decreasing ordered vector of the eigenvalues of ρ. A d-dimensional vector x ↓ is majorized by y ↓ , x ↓ ≺ y ↓ , if k j=1 x ↓ j ≤ k j=1 y ↓ j for k = 1, . . . , d − 1 and the equality holds for k = d. Zeros are appended to the vectors λ ↓ ρ A 1 ,A 2 such that their dimensions equal to the one of λ ↓ ρ .
In Ref. [20], another necessary criterion called range criterion was given. If a bipartite state ρ acting on the space H A ⊗H B is separable, then there exists a family of product vectors ψ i ⊗ φ i such that: (i) they span the range of ρ; (ii) the vector span the range of ρ T B , where * denotes complex conjugation in the basis in which partial transposition was performed, ρ T B is the partially transposed matrix of ρ with respect to the subspace B. In particular, any of the vectors ψ i ⊗ φ * i belongs to the range of ρ.
Recently, some elegant results for the separability problem have been derived.
In [29,30,31], a separability criteria based on the local uncertainty relations (LUR) was obtained. The authors show that for any separable state ρ ∈ H A ⊗ H B , where G A k or G B k are arbitary local orthogonal and normalized operators (LOOs) in H A ⊗ 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 [32] by choosing the optimal LOOs. In [33] 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 [34], i.e. 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 [35]. It has been shown that the LUR criterion, including the optimized one, can be derived from the covariance matrix criterion [36].
(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 [37]. A good entanglement measure has to fulfill some conditions [38]. For bipartite quantum systems, we have several good entanglement measures such as Entanglement of Formation(EOF), Concurrence, Tangle ctc. For two-quibt 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 [39]. 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 [40,41,42,43,44].
The first analytic lower bound of concurrence for arbitrary dimensional bipartite quantum states was derived by Mintert et al. in [45]. 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 [46,47]. These bounds are exact for some special classes of states and can be used to detect many bound entangled states. In [48] another lower bound on EOF for bipartite states has been presented from a new separability criterion [49].
A lower bound of concurrence based on local uncertainty relations (LURs) criterion is derived in [50]. This bound is further optimized in [32]. The lower bound of concurrence for tripartite systems has been studied in [51]. In [52,53] the authors presented lower bounds of concurrence for bipartite systems by considering the "twoqubit" 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 [54].
In [55] Mintert et al. proposed an experimental method to measure the concurrence directly by using joint measurements on two copies of a pure state. Then S. P. Walborn et al. presented an experimental determination of concurrence for two-qubit states [56], where only one-setting measurement is needed, but two copies of the state have to be prepared in every measurement. In [57] another way of experimental determination of concurrence for two-qubit and multi-qubit 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 [56], 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.
We also show that the concurrence and tangle of two entangled quantum states will be always larger than that of one, even both 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 qubits 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 helps 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.
A multipartite quantum mixed state ρ 12...N ∈ H 1 ⊗ H 2 ⊗ · · · ⊗ H N is said to be fully separable if it can be written as where ρ i 1 , ρ i 2 . . . , ρ i N are the reduced density matrices with respect to the systems 1, 2, . . . , N respectively, q i > 0 and i q i = 1. This is equivalent to the condition where |ψ 1 i , |φ 2 i , . . . , |µ N i are normalized pure states of systems 1, 2, . . . , N respectively, p i > 0 and i p i = 1.
For pure states, the definition (2.1) itself is an operational separability criterion. In particular, for bipartite case, there are Schmidt decompositions: For multipartite pure states, one has no such Schmidt decomposition. In [63] it has been verified that any pure three-qubit state |Ψ can be uniquely written as For mixed states it is generally very hard to verify if a decomposition like (2.2) exists. For a given generic separable density matrix, it is also not easy to find the decomposition (2.2) in detail.

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.

Partial positive transpose criterion
The positive partial transpose (PPT) criterion provided by Peres [18] says that if a bipartite state ρ AB ∈ H A ⊗ H B is separable, then the new matrix ρ T B AB with matrix elements defined in some fixed product basis as: is also a density matrix (i.e. has nonnegative spectrum). The operation T B , 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 [64].
Afterwards the Horodeckis showed that the Peres' criterion is also sufficient for 2 × 2 and 2 × 3 bipartite systems [19]. 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 [20]. States of this kind are said to be bound entangled (BE).

Reduced density matrix criterion
Cerf et al. [65] and Horodecki [66] independently, introduced a map Γ : , which gives rise to a simple necessary condition for separability in arbitrary dimensions, called the reduction criterion: If 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.

Realignment criterion
There is yet another class of criteria based on linear contractions on product states.
They stem from the new criterion discovered in [67,22] called computable cross norm (CCN) criterion or matrix realignment criterion which is operational and independent on PPT test [18]. If a state ρ AB is separable then the realigned matrix R(ρ) with elements R(ρ) ij,kl = ρ ik,jl has trace norm not greater than one, Quite remarkably, the realignment criterion can detect some PPT entangled (bound entangled) states [67,22] and can be used for construction of some nondecomposable maps. It also provides nice lower bound for concurrence [47].

Criteria based on Bloch representations
Any Hermitian operator on an N-dimensional Hilbert space H can be expressed according to the generators of the special unitary group SU(N) [68]. The generators of SU(N) can be introduced according to the transition-projection operators P jk = |j k|, where |i , i = 1, ..., N, are the orthonormal eigenstates of a linear Hermitian operator on H. Set ω l = − 2 l(l + 1) (P 11 + P 22 + · · · + P ll − lP l+1,l+1 ), which satisfy the relations and thus generate the SU(N) [69].
Any Hermitian operator ρ in H can be represented in terms of these generators of SU(N), 5) where I N is a unit matrix and r = (r 1 , r 2 , · · · r N 2 −1 ) ∈ R N 2 −1 . 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(R N 2 −1 ).
A matrix of the form (2.5) 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(R N 2 −1 ) is a subset of the ball D R (R N 2 −1 ) of radius R = 2(1 − 1 N ), which is the minimum ball containing it, and that the ball [70], that is, Let the dimensions of systems A, B and C be d A = N 1 , d B = N 2 and d C = N 3 respectively. Any tripartite quantum states ρ ABC ∈ H A ⊗ H B ⊗ H C can be written as: where λ i (1), λ j (2) are the generators of SU(N 1 ) and SU(N 2 ); M i , M j and M ij are operators of H C .
A separable tripartite state ρ ABC can be written as From (2.5) it can also be represented as: i · · · , a Comparing (2.6) with (2.8), we have For any (N 2 where T is a transformation acting on an (N 2 Using R we define a new operator γ R , [Proof] From (2.9) and (2.11) we get A straightforward calculation gives rise to .
For tripartite case, we take the following 3 × 3 × 3 mixed state as an example: we have that ρ is entangled for 0.6248 < p ≤ 1. In fact the criterion for 2 × N systems [71] is equivalent to the PPT criterion [72]. Similarly theorem 2.3 is also equivalent to the PPT criterion for 2 × 2 × N systems.

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 [35]. Let H B d ) such that they form an orthonormal normalized basis of the observable space, satisfying Tr[A k A l ] = δ k,l (resp. Tr[B k B l ] = δ k,l ). Consider the total set {M k } = {A k ⊗ I, I ⊗ B k }. It can be proven that [30], The covariance matrix γ is defined with entries which has a block structure [35]: Such covariance matrix has a concavity property: for a mixed density matrix ρ = k p k ρ k with p k ≥ 0 and For a bipartite product state ρ AB = ρ A ⊗ ρ B , C in (2.14) is zero. Generally if ρ AB is separable, then there exist states |a k a k | on H A d , |b k b k | on H B d and p k such that For a separable bipartite state, it has been shown that [35] [32] that if ρ AB is separable, then . (2.17) From the covariance matrix approach, we can also get an alternative criterion.
From (2.14) and (2.15) we have that if ρ AB is separable, then Hence all the 2 × 2 minor submatrices of X must be positive. Namely one has Summing over all i, j and using (2.12), we get That is   The separability criteria based on covariance matrix approach can be generalized to multipartite systems. We first consider the tripartite case, The covariance matrix defined by (2.13) has then the following block structure: Theorem 2.4 If ρ ABC is fully separable, then

23)
and 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 Summing over all i, j and using (2.12), we get which proves (2.21). (2.22) and (2.23) can be similarly proved.
Let D = U † ΛV be the singular value decomposition of D. Make a transformation of the orthonormal normalized basis of the local orthonormal observable space: In the new basis we have which proves (2.24). (2.25) and (2.26) can similarly treated.
We consider now the case that ρ ABC is bi-partite separable.
Theorem 2.5 If ρ ABC is a bi-partite separable state with respect to the bipartite partition of the sub-systems A and BC (resp. AB and C; resp. AC and B), then [Proof] We prove the case that ρ ABC is bi-partite 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 (2.20) . By using the concavity of covariance matrix we have Accounting to the method used in proving Theorem 2, we get ( Then the covariance matrix of ρ can be written as Then for a fully separable multipartite state ρ = From which we have the following separability criterion for multipartite systems: must be fulfilled for any i = j.

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) [74] for both bipartite and multipartite systems can be also constructed.
is mapped to the following form under local filtering transformations [75]: where F A/B ∈ GL(M/N, C) 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 [76], where ξ i ≥ 0, G A i and G B i are some traceless orthogonal observables. The matrices F A and F B can be obtained by minimizing the function . Then by iterations one can get the optimal A and B. In particular, there is a matlab code available in [77].
For bipartite separable states ρ, the CM separability criterion [78] says that As the filtering transformation does not change the separability of a state, one can study the separability ofρ instead of ρ. Under the normal form (2.37) the criterion (2.39) becomes In [30] a separability criterion based on local uncertainty relation (LUR) has been obtained. It says that for any separable state ρ, s are LOOs such as the normalized generators of SU(M/N) and G A k = 0 for k = M 2 + 1, · · · , N 2 . The criterion is shown to be strictly stronger than the realignment criterion [47]. Under the normal form (2.37) criterion (2.41) becomes holds for any M and N, from (2.40) and (2.42) 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 λ α k appears at the µ k th position and The generalized CM criterion says that: if ρ in (2.43) is fully separable, then The KF norm is defined by The criterion (2.44) can be improved by investigating the normal form of (2.43).
Theorem 2.7 By filtering transformations of the form where F i ∈ GL(d i , C), i = 1, 2, · · · N, followed by normalization, any strictly positive state ρ can be transformed into a normal form (2.46) [Proof] Let D 1 , D 2 , · · · , D N be the sets of density matrices of the N subsystems. The cartesian product D 1 × D 2 × · · · × D N consisting of all product density matrices ρ 1 ⊗ ρ 2 ⊗ · · · ⊗ ρ N with normalization Tr[ρ i ] = 1, i = 1, 2, · · · , N, is a compact set of matrices on the full Hilbert space H. For the given density matrix ρ we define the following function of ρ i The function is well-defined on the interior of As ρ is assumed to be strictly positive, we have Tr It follows that f → ∞ on the boundary of D 1 × D 2 × · · · × D N where at least one of the ρ i s satisfies that det ρ i = 0. It follows further that f has a positive minimum on the interior of D 1 × D 2 × · · · × D N with the minimum value attained for at least one product density matrix τ 1 ⊗ τ 2 ⊗ · · · ⊗ τ N with det τ i > 0, i = 1, 2, · · · , N. Any positive density matrix τ i with det τ i > 0 can be factorized in terms of Hermitian matrices F i as We see that when F † i ρ i F i = τ i , f has a minimum and Since f is stationary under infinitesimal variations about the minimum it follows that subjected to the constraint det(I d i + δρ i ) = 1, which is equivalent to Tr[δρ i ] = 0, i = 1, 2, · · · , N, using det(e A ) = e Tr[A] for a given matrix A. Thus, δρ i can be represented by the SU generators, any α k and µ k . Hence the terms proportional to λ Corollary 2.7 The normal form of a product state in H must be proportional to the identity.
[Proof] Let ρ be such a state. From (2.46), we get that Therefore for a product state ρ we have As an example for separability of multipartite states in terms of their normal forms (2.46), we consider the PPT entangled edge state [63] mixed with noises: Select a = 2, b = 3, and c = 0.6. Using the criterion in [79] we get that ρ p is entangled for 0.92744 < p ≤ 1. But after transforming ρ p to its normal form (2.46), the criterion can detect entanglement for 0.90285 < p ≤ 1.
Here we indicate that the filtering transformation does not change the PPT property.
For any vector |ψ , we have . ρ T B ≥ 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.

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 [74] is an observable of the composite system such that (i) nonnegative expectation values in all separable states, (ii) at least one negative eigenvalue (can recognizes at least one entangled state). Consider bipartite systems in Theorem 2.8 For any LOOs G A k and G B k , and (2.50) [Proof] Let ρ = where we have used Tr[RT ] ≤ ||T || KF for any unitary R in the first inequality and the CM criterion in the second inequality.
where the CM criterion has been used in the last step.
As the CM criterion can be generalized to multipartite form [79], we can also define entanglement witness for multipartite system in

Concurrence and Tangle
In this section, we focus on two important measures: concurrence and tangle (see, [80]). An elegant formula for concurrence of two-qubit states is derived analytically by Wootters [39,81]. This quantity has recently been shown to play an essential role in describing quantum phase transition in various interacting quantum manybody systems [82] and may affect macroscopic properties of solids significantly [83]. Furthermore, concurrence also provides an estimation [84] for the entanglement of formation (EOF) [61], which quantifies the required minimally physical resources to prepare a quantum state.
Let H A (resp. 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 H A ⊗ H B is of the form where a ij ∈ C satisfy the normalization M i=1 N j=1 a ij a * ij = 1. The concurrence of (3.52) is defined by [85,16] where ρ A = Tr B [|ψ ψ|]. The definition is extended to general mixed states ρ = i p i |ψ i ψ i | by the convex roof, For two qubits systems, the concurrence of |Ψ is given by: where |Ψ = σ y ⊗ σ y |Ψ * , |Ψ * is the complex conjugate of |Ψ , σ y is the Pauli For a mixed two-qubit quantum state ρ, the entanglement of formation E(ρ) has a simple relation with the concurrence [39,81] where the λ i s are the eigenvalues, in decreasing order, of the Hermitian matrix √ ρ ρ √ ρ and ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ).
Another entanglement measure called tangle is defined by for a pure state |ψ . For mixed state ρ = i p i |ψ i ψ i |, the definition is given by For multipartite state |ψ ∈ H 1 ⊗ H 2 ⊗ · · · ⊗ H N , dimH i = d i , i = 1, ..., N, the concurrence of |ψ is defined by [86] where α labels all different reduced density matrices.
Up to constant factor (3.59) can be also expressed in another way. Let H denotes a d-dimensional vector space with basis |i , i = 1, 2, ..., d. An N-partite pure state in H ⊗ · · · ⊗ H is generally of the form, (3.60) Let α and α ′ (resp.β and β ′ ) be subsets of the subindices of a, associated to the same sub Hilbert spaces but with different summing indices. α (or α ′ ) and β (or β ′ ) span the whole space of the given sub-indix of a. The generalized concurrence of |Ψ is then given by [16]  For a mixed multipartite quantum state, the corresponding concurrence is given by the convex roof:

Lower bound of concurrence from covariance matrix criterion
In [47] a lower bound of C(ρ) has been obtained, where T A 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 [50], where G A i and G B i are any set of local orthonormal observables, Bound (3.64) again depends on the choice of the local orthonormal observables. This bound can be optimized, in the sense that a local orthonormal observableindependent up bound of the right hand side of (3.64) can be obtained. . (3.65) [Proof] The other orthonormal normalized basis of the local orthonormal observable space can be obtained from A i and B i by unitary transformations U and V : Select U and V so that C = U † ΛV is the singular value decomposition of C. Then the new observables can be written as Substituting above relation to (3.64) one gets (3.65).

Bound (3.65) 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 [32] 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 [61], where I 9 is the 9 × 9 identity matrix, |ξ 0 = 1 . We simply choose the local orthonormal observables to be the normalized generators of SU(3). Formula (3.63) gives C(ρ) ≥ 0.050. Formula (3.64) gives C(ρ) ≥ 0.052 [50], while formula (3.65) yields a better lower bound C(ρ) ≥ 0.0555.

Lower bound of concurrence from "two-qubit" decomposition
In [53] the authors derived an analytical lower bound of concurrence for arbitrary bipartite quantum states by decomposing the joint Hilbert space into many 2 ⊗ 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 τ 2 of concurrence for bipartite states has been obtained in [53]. For a bipartite quantum state ρ in H ⊗ H, the concurrence C(ρ) satisfies where C mn (ρ) = max{0, λ mn the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρ ρ mn with ρ mn = (L m ⊗ L n )ρ * (L m ⊗ L n ), L m and L n are the generators of SO(d).

68)
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 ̺ is not normalized but positive semidefinite. C mn are just the concurrences of these states (3.68).
The bound τ 2 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.

(3) Lower bound and separability
An N-partite quantum state ρ is fully separable if and only if there exist p i with p i ≥ 0, It is easily verified that for a fully separable multipartite state ρ, τ N (ρ) = 0.
Thus τ N (ρ) > 0 indicates that there must be some kinds of entanglement inside the quantum state, which shows that the lower bound τ N (ρ) can be used to recognize entanglement.
(4) Relation between lower bounds of bi-and tripartite concurrence τ 3 is basically different from τ 2 as τ 3 characterizes also genuine tripartite entanglement that can not be described by bipartite decompositions. Nevertheless, there are interesting relations between them.
For a mixed multipartite quantum state, ρ = i p i |ψ i ψ i | ∈ H 1 ⊗H 2 ⊗· · ·⊗H N , the corresponding concurrence of (3.80) and (3.59) are then given by the convex roof: and (3.62). We now investigate the relation between these two kinds of concurrences. (3.82) where ρ A/B = Tr B/A [ρ] be the reduced density matrices of ρ.
Theorem 3.5 For a multipartite quantum state ρ ∈ H 1 ⊗H 2 ⊗· · ·⊗H N with N ≥ 3, the following inequality holds, where the maximum is taken over all kinds of bipartite concurrence.
Let ρ = i p i |ψ i ψ i | attain the minimal decomposition of the multipartite concurrence. One has Corollary 3.5 For a tripartite quantum state ρ ∈ H 1 ⊗ H 2 ⊗ H 3 , the following inequality holds: where the maximum is taken over all kinds of bipartite concurrence.
In [50,32], from the separability criteria related to local uncertainty relation, covariance matrix and correlation matrix, the following lower bounds for bipartite concurrence are obtained: and (3.86) where the entries of the matrix C, Now we consider a multipartite quantum state ρ ∈ H 1 ⊗ H 2 ⊗ · · · ⊗ H N as a bipartite state belonging to H A ⊗ H B with the dimensions of the subsystems A and B being d A = d s 1 d s 2 · · · d sm and d B = d s m+1 d s m+2 · · · d s N respectively. By using the corollary, (3.63), (3.85) and (3.86) one has the following lower bound: Theorem 3.6 For any N-partite quantum state ρ, − 1) , ρ i s are all possible bipartite decompositions of ρ, and In [32,84,90], it is shown that the upper and lower bound of multipartite concurrence satisfy .

(3.88)
In fact one can obtain a more effective upper bound for multi-partite concurrence.
where |ψ i s are the orthogonal pure states and i λ i = 1. We have The right side of (3.89) gives a new upper bound of C N (ρ). Since the upper bound obtained in (3.89) is better than that in (3.88).

Bounds of concurrence and tangle
In [54], a lower bound for tangle defined in (3.58) has been derived: where ||X|| HS = Tr[XX † ] denotes the Frobenius or Hilbert-Schmidt norm. Experimentally measurable lower and upper bounds for concurrence have been also given by Mintert and Zhang et.al. in [84,32]: Since the convexity of C 2 (ρ), we have that τ (ρ) ≥ C 2 (ρ) always holds. For two qubits quantum systems, tangle τ is always equal to the square of concurrence C 2 [44,87], as a decomposition {p i , |ψ i } achieving the minimum in Eq. (3.54) has the property that C(|ψ i ) = C(|ψ j ) ∀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. where ρ A is the reduced matrix of ρ, and T (ρ) is the correlation matrix of ρ defined in (3.86).
We get which ends the proof.

Concurrence and tangle of two entangled states are strictly larger than that of one
In this subsection we show that although bound entangled states can not be distilled, the concurrence and tangle of two entangled states will be always strictly larger than that of one, even the two entangled states are both bound entangled.
two quantum states shared by subsystems AA ′ and BB ′ . We use [Proof] Without loss of generality we assume C(|ψ ) ≥ C(|ϕ ). Fist note that Let ρ |ψ A = i λ i |i i| and ρ |ϕ A ′ = j π j |j j| be the spectral decomposition of ρ |ψ A and ρ |ϕ A ′ , with i λ i = 1 and j π j = 1 respectively. By using (3.99) one obtains that Now using the definition of concurrence and the normalization conditions of λ i and π j one immediately gets If one of {|ψ , |ϕ } is separable, say |ϕ , then the rank of ρ |ϕ A ′ must be one, which means that there is only one item in the spectral decomposition in ρ The inequality (3.98) also holds because that for pure quantum state ρ, τ (ρ) = C 2 (ρ).
From the lemma, we have, for mixed states, [Proof] We still assume C(ρ) ≥ C(σ) for convenience. Let ρ = i p i ρ i and σ = j q j σ j be the optimal decomposition such that C(ρ ⊗ σ) = i p i q j C(ρ i ⊗ σ j ). By using the inequality obtained in lemma 3.2 we have (3.105) Case 1: Now let one of {ρ, σ} be separable, say σ, with ensemble representation σ = j q j σ j , where j q j = 1 and σ j is the density matrix of separable pure state.
Case 2: If both ρ and σ are inseparable, i.e. there is at least one pure state in the ensemble decomposition of ρ (and σ respectively), using lemma 3.2 we have (3.107) The inequality for tangle τ can be proved in a similar way.
Remark : In [91] 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, the corresponding σ must be free entangled (distillable). By theorem 3.8, we can see that even two entangled quantum states ρ and σ are bound entangled, their concurrence and tangle are strictly larger than that of one state.

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 T 0 proposed by Bennett et.al in 1993 uses Bell measurements and Pauli rotations. When the maximally entangled pure state |φ >= 1 √ n n−1 i=0 |ii > is used as the quantum resource, it provides an ideal noiseless quantum channel Λ (|φ><φ|) T 0 (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 T 0 with an arbitrary mixed state resource has been obtained [92,93].
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 non-maximally entangled states. Such a procedure proposed in [58,59,60,61,62] is called distillation. In [58] 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 [62] 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 ′ so that F (ρ ′ ) > 1 d .

Fidelity of quantum teleportation
Let 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 ∈ H ⊗ H is defined by |ii is the maximally entangled state and I is the corresponding identity matrix.
In [6], the authors give a optimal teleportation protocol by using a mixed entangled quantum state. The optimal teleportation fidelity is given by which solely depends 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 [7], teleportation [5], entanglement swapping [9], and quantum cryptography (Bell inequalities) [8]. As the optimal fidelity of teleportation is given by F EF [6], experimentally measurement of FEF can be also used to determine the entanglement of the non-local source used in teleportation.
Thus an analytic formula for FEF is of great importance. In [94] 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 λ i , i = 1, ..., d 2 − 1, be the generators of the SU(d) algebra. A bipartite state ρ ∈ H ⊗ H can be expressed as (2)]. Let M(ρ) denote the correlation matrix with entries m ij (ρ). [Proof] First, we note that where m ij (P + ) = 1 4 Tr[P + λ i ⊗ λ j ]. By definition (4.112), one obtains Since Uλ i U † is a traceless Hermitian operator, it can be expanded according to the SU(d) generators, For the case d = 2, we can get an exact result from (4.115): i.e. the upper bound derived in Theorem 4.1 is exactly the F EF .
[Proof] We have shown in (4.116) 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 (4.116) holds.
For any vector t = {t 1 , t 2 , t 3 } with unit norm, define an operator X ≡ where σ i s are Pauli matrices. Given an orthonormal matrix O one obtains a new X and X ′ are both hermitian traceless matrices. Their eigenvalues are given by the norms of the vectors t and t ′ = {t ′ 1 , t ′ 2 , t ′ 3 } respectively. As the norms are invariant under orthonormal transformations O, they have the same eigenvalues: ± t 2 1 + t 2 2 + t 2 3 . Thus there must be a unitary matrix U such that X ′ = UXU † .
Hence the inequality in the proof of Theorem 4.1 becomes an equality. The upper bound (4.115) then becomes exact at this situation, which is in accord with the result in [94].
Remark : The upper bound of FEF (4.115) and the FEF (4.118) depend on the correlation matrices M(ρ) and M(P + ). They can be calculated directly according to a given set of SU(d) generators λ i , i = 1, ..., d 2 − 1. As an example, for d = 3, if we 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 >= ω j |j >, with ω = exp{ −2iπ n }. We can introduce n 2 linear-independent n × n-matrices U st = h t g s , which satisfy (4.120) One can also check that {U st } satisfy the condition of bases of the unitary operators in the sense of [95], i.e.
where λ j s are the eigenvalues of the real part of matrix M = T iT −iT T , T is a d 2 × d 2 matrix with entries T n,m = Φ n |ρ|Φ m and Φ j are the maximally entangled basis states defined in (4.123).
[Proof] From (4.122), any d × d unitary matrix U can be represented by U = x m x n M mn , (4.126) where M mn is defined in the theorem. One can deduce that from the hermiticity of ρ.
Taking into account the constraint with an undetermined Lagrange multiplier λ, Accounting to (4.127) we have the eigenvalue equation where η j = −λ j is the corresponding eigenvalues of the real part of the matrix M. Example: Horodecki gives a very interesting bound entangled state in [20], (4.131) One can easily compare the upper bound obtained in (4.124) and that in (4.115). From Fig. 2 we see that for 0 ≤ a < 0.572, the upper bound in (4.124) is larger than that in (4.115). But for 0.572 < a < 1 the upper bound in (4.124) is always lower than that in (4.115), which means the upper bound (4.124) is tighter than (4.115).
In fact, we can drive another upper bound for FEF which will be very tight for weakly mixed quantum states.

Theorem 4.3
For any bipartite quantum state ρ ∈ H d ⊗H d , the following inequality holds:

132)
where ρ A is the reduced matrix of ρ.
[Proof] Note that in [62] the authors have obtained the FEF for pure state |ψ , where ρ |ψ A is the reduced matrix of |ψ ψ|. For mixed state ρ = i p i ρ i , we have Let λ ij be the real and nonnegative eigenvalues of the matrix p i ρ i A . Recall that for any function F = i ( j x 2 ij ) 1 2 subjected to the constraints z j = i x ij with x ij being real and nonnegative, the inequality j z 2 j ≤ F 2 holds, from which it follows that which ends the proof.

Fully entangled fraction and concurrence
The upper bound of F EF has also interesting relations to the entanglement measure concurrence. As shown in [94], 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. (4.136) [Proof] In [96], the authors show that for any pure state |ψ ∈ H A ⊗ H B , the following inequality holds: where ε denotes the set of d × d-dimensional maximally entangled states.
Let ρ = i p i |φ i φ i | be the optimal decomposition such that C(ρ) = i p i C(|ψ i ).
We have which ends the proof.
The inequality (4.136) has demonstrated the relation between the lower bound of concurrence and the fully entangled fraction (thus the optimal teleportation fidelity), i.e. the fully entangled fraction of a quantum state ρ is limited by it's concurrence.
We now consider tripartite case. Let ρ ABC be a state of three-qubit systems denoted by A, B and C. We study the upper bound of the F EF , F (ρ AB ), between qubits A and B, and its relations to the concurrence under bipartite partition AB and C. For convenience we normalize F (ρ AB ) to be Let C(ρ AB|C ) denote the concurrence between subsystems AB and C.
Theorem 4.5 For any triqubit state ρ ABC , F N (ρ AB ) satisfies (4.139) [Proof] We first consider the case that ρ ABC is pure, ρ ABC = |ψ ABC ψ|. By using the Schmidt decomposition between qubits A, B and C, |ψ ABC can be written as: for some othonormalized bases |i AB , |i C of subsystems AB, C respectively. The reduced density matrix ρ AB has the form where Λ is a 4 × 4 diagonal matrix with diagonal elements {η 2 1 , η 2 2 , 0, 0}, U is a unitary matrix and U * denotes the conjugation of U.
The F EF of the two-qubit state ρ AB can be calculated by using formula (4.118) or the one in [94]. Let be the 4 × 4 matrix constituted by the four Bell bases. The F EF of ρ AB can be written as where η max (X) stands for the maximal eigenvalues of the matrix X.  We now prove that the above inequality (4.143) also holds for mixed state ρ ABC . Let ρ ABC = i p i |ψ i ABC ψ i | be the optimal decomposition of ρ ABC such that C(ρ AB|C ) = i p i C(|ψ i ) AB|C . We have where ρ i AB|C = |ψ i ABC ψ i | and ρ i AB = Tr C [ρ i AB|C ]. From Theorem 4.5 we see that the F EF of quibts A and B are bounded by the concurrence between qubits A, B and qubit C. The upper bound of F EF for ρ AB decreases when the entanglement between qubits A, B and C increases. As an example, we consider the generalized W state defined by |W ′ = α|100 + β|010 + γ|001 , |α| 2 + |β| 2 + |γ| 2 = 1. The reduced density matrix is given by The F EF of ρ W ′ AB is given by While the concurrence of |W ′ has the from C AB|C (|W ′ ) = 2|γ| |α| 2 + |β| 2 . We see that (4.139) always holds. In particular for |α| = |β| and |γ| ≤ √ 2 2 , the inequality (4.139) is saturated (see Fig. 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 [62] 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 F EF (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 ′ so that F (ρ ′ ) > 1 d . For d = 2, one can compute F EF analytically according to the corollary. For d ≥ 3 our upper bound (4.115) can supply a necessary condition in the distillation: if the upper bound (4.115) is less than or equal to 1 d , then the filtering operation has to be applied before using the generalized distillation protocol.
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.
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 [97], which is not operational in general. An alternative distillability criterion based on the bound τ 2 in (3.67) can be obtained to improve the operationality. [Proof] It was shown in [97] that a density matrix ρ id 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 ⊗ Qρ ⊗N P ⊗ Q is entangled [97]. Thus if τ 2 (ρ ⊗N ) > 0, there exists one submatrix of matrix ρ ⊗N , similar to Eq. (3.68), which has nonzero τ 2 and is entangled in a 2 ⊗ 2 space, hence ρ is distillable.
Corollary 4.7 The lower bound τ 2 (ρ) > 0 is a sufficient condition for the distillability of any bipartite state ρ.
Corollary 4.7 The lower bound τ 2 (ρ) = 0 is a necessary condition for separability of any bipartite state ρ.
Remark: Corollary 4.7 directly follows from Theorem 4.7 and this case is referred to as one-distillable [98]. The problem of whether non-PPT (NPPT) nondistillable states exist is studied numerically in [98,99]. By using Theorem 4.7, although it seems impossible to solve the problem completely, it is easy to judge the distillability of a state under condition that it is one-distillable.

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 form 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 can not distill a singlet from many pairs of bound entangled states, the concurrence and tangle of two entangled quantum states are always larger than that of one, even both 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.