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.

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 [59], dense coding [10], quantum cryptographic schemes [1113], entanglement swapping [1418], and remote states preparation (RSP) [1924]. 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 and 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 and 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 [3539]. 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 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 is majorized by that of or alone for a separable state. That is, if a state is separable, then , . Here denotes the decreasing ordered vector of the eigenvalues of . A -dimensional vector is majorized by , , if for and the equality holds for . Zeros are appended to the vectors such that their dimensions are equal to the one of .

In [31], another necessary criterion called range criterion was given. If a bipartite state acting on the space is separable, then there exists a family of product vectors such that (i) they span the range of ; (ii) the vector spans the range of , where denotes complex conjugation in the basis in which partial transposition was performed and is the partially transposed matrix of with respect to the subspace . In particular, any of the vectors belongs to the range of .

Recently, some elegant results for the separability problem have been derived. In [4345], a separability criteria based on the local uncertainty relations (LURs) was obtained. The authors show that, for any separable state , where or are arbitrary local orthogonal and normalized operators (LOOs) in . 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 [5458].

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 [57], 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 [7376]. 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 is said to be fully separable if it can be written as

where and are reduced density matrices defined as , . This is equivalent to the condition where .

A multipartite quantum mixed state is said to be fully separable if it can be written as

where are the reduced density matrices with respect to the systems , respectively, , and . This is equivalent to the condition where are normalized pure states of systems , respectively, , and .

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 is a pure state of a composite system, , then there exist orthonormal states for system and orthonormal states for system such that where are nonnegative real numbers satisfying , known as Schmidt coefficients.

and are called Schmidt bases with respect to and . The number of nonzero values is called Schmidt number, also known as Schmidt rank, which is invariant under unitary transformations on system or system . For a bipartite pure state , is separable if and only if the Schmidt number of is one.

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

with normalization condition , where , . 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 is separable, then the new matrix with matrix elements defined in some fixed product basis as is also a density matrix (i.e., it has nonnegative spectrum). The operation , called partial transpose, just corresponds to the transposition of the indices with respect to the second subsystem . It has an interpretation as a partial time reversal [80].

Afterwards Horodecki et al. showed that Peres' criterion is also sufficient for and 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 and 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 ), which gives rise to a simple necessary condition for separability in arbitrary dimensions, called the reduction criterion. If is separable, then where , . This criterion is simply equivalent to the P-H criterion for composite systems. It is also sufficient for and 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 is separable, then the realigned matrix with elements has trace norm not greater than one: 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 -dimensional Hilbert space can be expressed according to the generators of the special unitary group [84]. The generators of can be introduced according to the transition-projection operators , where , , are the orthonormal eigenstates of a linear Hermitian operator on . Set where and . We get a set of operators which satisfies the relations and thus generate the [85].

Any Hermitian operator in can be represented in terms of these generators of as

where is a unit matrix and and is called Bloch vector. The set of all the Bloch vectors that constitute a density operator is known as the Bloch vector space .

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 is a subset of the ball of radius , which is the minimum ball containing it, and that the ball of radius is included in [86], that is,

Let the dimensions of systems , and be , and , respectively. Any tripartite quantum states can be written as

where , are the generators of and ; , and are operators of .

Theorem 2.2. Let , and , . For a tripartite quantum state with representation (2.15), one has [87]

Proof. Since , and , , we have that and are positive Hermitian operators. Let . Then and . The partial trace of over (and ) should be also positive. Hence

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

Corollary 2.3. Let , which can be generally written as , then, for any with , .

A separable tripartite state can be written as From (2.13) it can also be represented as

where and are real vectors on the Bloch sphere satisfying and .

Comparing (2.15) with (2.19), we have

For any real matrix and real matrix satisfying and , we define a new matrix

where is a transformation acting on an matrix by

Using , we define a new operator :

where , and .

Theorem 2.4. If is separable, then [87] .

Proof. From (2.20) and (2.23) we get A straightforward calculation gives rise to As and , we get Therefore is still a density operator, that is, .

Theorem 2.4 gives a necessary separability criterion for general tripartite systems. The result can be also applied to bipartite systems. Let , . For any real matrix satisfying and any state , we define where .

Corollary 2.5. For , if there exists an with such that , then must be entangled.

For systems, the above corollary is reduced to the results in [88]. As an example we consider the istropic states If we choose to be , we get that is entangled for .

For tripartite case, we take the following mixed state as an example: where . Taking , we have that is entangled for .

In fact the criterion for systems [88] is equivalent to the PPT criterion [89]. Similarly Theorem 2.4 is also equivalent to the PPT criterion for 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 and be -dimensional complex vector spaces and a bipartite quantum state in . Let (resp., ) be observables on (resp., ) such that they form an orthonormal normalized basis of the observable space, satisfying (resp., ). Consider the total set . It can be proven that [44]

The covariance matrix is defined with entries

which has a block structure [49]

where , and . Such covariance matrix has a concavity property: for a mixed density matrix with and , one has .

For a bipartite product state , in (2.32) is zero. Generally if is separable, then there exist states on on and such that

where , .

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

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

From the covariance matrix approach, we can also get an alternative criterion. From (2.32) and (2.33) we have that if is separable, then Hence all the minor submatrices of must be positive. Namely, one has that is, . Summing over all , and using (2.30), we get

That is,

where stands for the Euclid norm of , that is, .

Formulae (2.35) and (2.39) are independent and could be complement. When (2.39) can recognize the entanglement but (2.35) cannot. When (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 . Take observables on , respectively, on , respectively, on . Set . The covariance matrix defined by (2.31) has then the following block structure:

where , , , , , and .

Theorem 2.6. If is fully separable, then [90]

Proof. For a tripartite product state , and in (2.42) are zero. If is fully separable, then there exist states in , in , and in , and such that , where , , and , that is, Thus all the minor submatrices of must be positive. Selecting one with two rows and columns from the first two block rows and columns of , we have that is, . Summing over all , and using (2.30), we get which proves (2.43). Equations (2.44) and (2.45) can be similarly proved.
From (2.50) we also have . Therefore Note that . By using that for any matrix and any unitary [91], we have .
Let be the singular value decomposition of . Make a transformation of the orthonormal normalized basis of the local orthonormal observable space and . In the new basis we have Then (2.52) becomes which proves (2.46). Equations (2.47) and (2.48) can be similarly treated.

We consider now the case that is bipartite separable.

Theorem 2.7. If is a bipartite separable state with respect to the bipartite partition of the sub-systems and (resp., and ; resp., and ), 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 is bipartite separable with respect to the system and systems partition. The other cases can be similarly treated. In this case the matrices and in the covariance matrix (2.42) are zero. takes the form . Define , . has a form where and , . By using the concavity of covariance matrix we have Accounting to the method used in proving Theorem , 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 . Define , where , () are the normalized generators of satisfying and acting on the th system , . Denote as the set of all . Then the covariance matrix of can be written as

where and for .

For a product state , , , in (2.57) are zero matrices. Define Then for a fully separable multipartite state , one has from which we have the following separability criterion for multipartite systems.

Theorem 2.9. If a state is fully separable, then the following inequalities must be fulfilled for any [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, with , , and is mapped to the following form under local filtering transformations [93]:

where 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 into a normal form [94]:

where and and are some traceless orthogonal observables. The matrices and can be obtained by minimizing the function where and . In fact, one can choose , and , where . Then by iterations one can get the optimal and . In particular, there is a matlab code available in [95].

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

where is an matrix with , stands for the trace norm of , s are the generators of and have been chosen to be normalized, and .

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

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

where s are LOOs such as the normalized generators of and for . The criterion is shown to be strictly stronger than the realignment criterion [61]. Under the normal form (2.62) criterion (2.66) becomes that is,

As holds for any and , 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 be a strictly positive density matrix in and . can be generally expressed in terms of the generators [97] as

where with appears at the th position and

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

for . The KF norm is defined by where is a kind of matrix unfolding of .

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

Theorem 2.10. By filtering transformations of the form where , followed by normalization, any strictly positive state can be transformed into a normal form [98]:

Proof. Let be the sets of density matrices of the subsystems. The cartesian product consisting of all product density matrices with normalization , , is a compact set of matrices on the full Hilbert space . For the given density matrix we define the following function of : The function is well defined on the interior of where . As is assumed to be strictly positive, we have . Since is compact, we have with a lower bound depending on .
It follows that on the boundary of where at least one of the s satisfies . It follows further that has a positive minimum on the interior of with the minimum value attained for at least one product density matrix with , . Any positive density matrix with can be factorized in terms of Hermitian matrices as where . Denote that , so that . Set and define
We see that, when , has a minimum and
Since is stationary under infinitesimal variations about the minimum, it follows that for all infinitesimal variations subjected to the constraint , which is equivalent to , , using for a given matrix . Thus, can be represented by the generators, . It follows that for any and . Hence the terms proportional to in (2.69) disappear.

Corollary 2.11. The normal form of a product state in must be proportional to the identity.

Proof. Let be such a state. From (2.74), we get that Therefore for a product state