1 Introduction

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

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

Ming Li, Shao-Ming Fei and Xianqing Li-Jost

College of Mathematics and Computational Science, China University of Petroleum, 257061 Dongying, China

Department of Mathematics, Capital Normal University, 100037 Beijing, China

Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germnay

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 [5, 6], dense coding [7], quantum cryptographic schemes [8], entanglement swapping [9] and remote states preparation (RSP) [10, 11, 12, 13]. 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 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 and 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 and 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 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 is majorized by that of or alone for a separable state. i.e. 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 equal to the one of .

In Ref. [20], 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 span the range of , where denotes complex conjugation in the basis in which partial transposition was performed, 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 [29, 30, 31], a separability criteria based on the local uncertainty relations (LUR) was obtained. The authors show that for any separable state ,

 1−∑k⟨GAk⊗GBk⟩−12⟨GAk⊗I−I⊗GBk⟩2≥0,

where or are arbitary 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 [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].

1. 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 “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 [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.

1. 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], 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 [6]. 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 [58, 59, 60, 61]. 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 [58] in 1996. The protocol is then generalized to distill any bipartite quantum state with higher dimension by Horodeckis in 1999 [62]. It is proven that a quantum state can be always distilled if it violates the reduced matrix separability criterion [62].

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 criterions 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 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.

## 2 Separability criteria and normal form

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

 ρ12…N=ρ1⊗ρ2⊗⋯⊗ρN, (2.1)

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

 ρ12…N=|ψ1⟩⟨ψ1|⊗|ϕ2⟩⟨ϕ2|⊗⋯⊗|μN⟩⟨μN|,

where .

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

 ρ12…N=∑iqiρi1⊗ρi2⊗⋯⊗ρiN, (2.2)

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

 ρ12…N=∑ipi|ψ1i⟩⟨ψ1i|⊗|ϕ2i⟩⟨ϕ2i|⊗⋯⊗|μNi⟩⟨μNi|,

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

(Schmidt decomposition): Suppose is a pure state of a composite system, , then there exist orthonormal states for system , and orthonormal states for system such that

 |ψ⟩=∑iλi|iA⟩|iB⟩,

where are non-negative real numbers satisfying , known as Schmidt coefficients.

and are called Schmidt bases with respect to and . The number of non-zero 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 [63] it has been verified that any pure three-qubit state can be uniquely written as

 |Ψ⟩ = λ0|000⟩+λ1eiψ|100⟩+λ2|101⟩+λ3|110⟩+λ4|111⟩ (2.3)

with normalization condition , where , . Eq. (2.3) is called generalized Schmidt decomposition.

For mixed states it is generally very hard to verify if a decomposition like exists. For a given generic separable density matrix, it is also not easy to find the decomposition 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.

#### Partial positive transpose criterion

The positive partial transpose (PPT) criterion provided by Peres [18] says that if a bipartite state is separable, then the new matrix with matrix elements defined in some fixed product basis as:

 ⟨m|⟨μ|ρTBAB|n⟩|ν⟩≡⟨m|⟨ν|ρAB|n⟩|μ⟩

is also a density matrix (i.e. 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 [64].

Afterwards the Horodeckis showed that the Peres’ criterion is also sufficient for and 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 and 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 (), which gives rise to a simple necessary condition for separability in arbitrary dimensions, called the reduction criterion: If is separable, then

 ρA⊗I−ρAB≥0,  I⊗ρB−ρAB≥0,

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.

#### Realignment criterion

There is yet another class of criteria based on linear contractions on product states. They stem from the new criterion discovered in [72, 22] called computable cross norm (CCN) criterion or matrix realignment criterion which is operational and independent on PPT test [18]. If a state is separable then the realigned matrix with elements has trace norm not greater than one,

 ||R(ρ)||KF≤1. (2.4)

Quite remarkably, the realignment criterion can detect some PPT entangled (bound entangled) states [72, 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 -dimensional Hilbert space can be expressed according to the generators of the special unitary group [68]. The generators of can be introduced according to the transition-projection operators , where , , are the orthonormal eigenstates of a linear Hermitian operator on . Set

 ωl=−√2l(l+1)(P11+P22+⋯+Pll−lPl+1,l+1), ujk=Pjk+Pkj,   vjk=i(Pjk−Pkj),

where and . We get a set of operators

 Γ≡{ωl,ω2,⋯,ωN−1,u12,u13,⋯,v12,v13,⋯},

which satisfy the relations

 Tr[λi]=0,Tr[λiλj]=2δij,   ∀ λi∈Γ

and thus generate the [69].

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

 ρ=1NIN+12N2−1∑j=1rjλj, (2.5)

where is a unit matrix 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 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 [70], that is,

 Dr(RN2−1)⊆B(RN2−1)⊆DR(RN2−1).

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

 ρABC = IN1⊗IN2⊗M0+N21−1∑i=1λi(1)⊗IN2⊗Mi+N22−1∑j=1IN1⊗λj(2)⊗˜Mj (2.6) +N21−1∑i=1N22−1∑j=1λi(1)⊗λj(2)⊗Mij,

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

###### Theorem 2.2

Let , and , . For a tripartite quantum state with representation (2.6), we have

 M0−N21−1∑i=1riMi−N22−1∑j=1sj˜Mj+N21−1∑i=1N22−1∑j=1risjMij≥0. (2.7)

[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

 0 ≤ TrAB[AρA] = TrAB[A1⊗A2⊗M0+∑i√A1λi(1)√A1⊗A2⊗Mi + ∑jA1⊗√A2λj(2)√A2⊗˜Mj+∑ij√A1λi(1)√A1⊗√A2λj(2)√A2⊗Mij] = M0−N21−1∑i=1riMi−N22−1∑j=1sj˜Mj+N21−1∑i=1N22−1∑j=1risjMij.

Formula (2.7) is valid for any tripartite states. By setting in (2.7), one can get a result for bipartite systems:

###### Corollary 2.2

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

A separable tripartite state can be written as

 ρABC=∑ipi|ψAi⟩⟨ψAi|⊗|ϕBi⟩⟨ϕBi|⊗|ωCi⟩⟨ωCi|.

From (2.5) it can also be represented as:

 ρABC = ∑ipi12(2N1IN1+N21−1∑k=1a(k)iλk(1))⊗12(2N2IN2+N22−1∑l=1b(l)iλl(2))⊗|ωCi⟩⟨ωCi| (2.8) = IN1⊗IN2⊗1N1N2∑ipi|ωCi⟩⟨ωCi| +N21−1∑k=1λk(1)⊗IN2⊗12N2∑ia(k)ipi|ωCi⟩⟨ωCi| +N22−1∑l=1IN1⊗λl(2)⊗12N1∑ib(l)ipi|ωCi⟩⟨ωCi| +N21−1∑kN22−1∑lλk(1)⊗λl(2)⊗14∑ia(k)ib(l)ipi|ωCi⟩⟨ωCi|,

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

Comparing with , we have

 M0=1N1N2∑ipi|ωCi⟩⟨ωCi|,Mk=12N2∑ia(k)ipi|ωCi⟩⟨ωCi|, ˜Ml=12N1∑ib(l)ipi|ωCi⟩⟨ωCi|,Mkl=14∑ia(k)ib(l)ipi|ωCi⟩⟨ωCi|. (2.9)

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

 R=⎛⎜⎝R(1)000R(2)000T⎞⎟⎠, (2.10)

where is a transformation acting on an matrix by

 T(M)=R(1)MRT(2).

Using we define a new operator ,

 γR(ρABC) = IN1⊗IN2⊗M′0+N21−1∑i=1λi(1)⊗IN2⊗M′i+N22−1∑j=1IN1⊗λj(2)⊗˜M′j (2.11) +N21−1∑i=1N22−1∑j=1λi(1)⊗λj(2)⊗M′ij,

where and .

###### Theorem 2.3

If is separable, then .

[Proof] From and we get

 M′0 = M0=1N1N2∑ipi|ωCi⟩⟨ωCi|, M′k=12N2∑miRkm(1)a(m)ipi|ωCi⟩⟨ωCi|, ˜M′l = 12N1∑niRln(2)b(n)ipi|ωCi⟩⟨ωCi|, M′kl=14∑mniRkm(1)a(m)iRln(2)b(n)ipi|ωCi⟩⟨ωCi|.

A straightforward calculation gives rise to

 γR(ρABC) = ∑ipi12⎛⎜⎝2N1IN1+N21−1∑k=1N21−1∑m=1Rkm(1)a(m)iλk(1)⎞⎟⎠ ⊗12⎛⎜⎝2N2IN2+N22−1∑l=1N22−1∑n=1Rln(2)b(n)iλl(2)⎞⎟⎠⊗|ωCi⟩⟨ωCi|.

As and , we get

 |→a′i|2=|R(1)→ai|2≤1(N1−1)2|→ai|2=2N1(N1−1),
 |→b′i|2=|R(2)→bi|2≤1(N2−1)2|→bi|2=2N2(N2−1).

Therefore is still a density operator, i.e. .

Theorem 2.3 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

 γR(ρAB)=IN1⊗M0+N21−1∑j=1λj⊗M′j,

where .

###### Corollary 2.3

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

For systems, the above corollary is reduced to the results in [71]. As an example we consider the istropic states,

 ρI=1−p9I3⊗I3+p33∑i,j=1|ii⟩⟨jj|=I3⊗(19I3)+5∑i=1λi⊗(p6λi)−8∑i=6λi⊗(p6λi).

If we choose to be , we get that is entangled for .

For tripartite case, we take the following mixed state as an example:

 ρ=1−p27I27+p|ψ⟩⟨ψ|,

where . Taking , we have that is entangled for .

In fact the criterion for systems [71] is equivalent to the PPT criterion [72]. Similarly theorem 2.3 is also equivalent to the PPT criterion for 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 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 [30],

 N2∑k=1(Mk)2=dI,N2∑k=1⟨Mk⟩2=Tr[ρ2AB]. (2.12)

The covariance matrix is defined with entries

 γij(ρAB,{Mk})=⟨MiMj⟩+⟨MjMi⟩2−⟨Mi⟩⟨Mj⟩, (2.13)

which has a block structure [35]:

 γ=(ACCTB), (2.14)

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