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

Authors: Ming Li; Shao-Ming Fei; Xianqing Li-Jost
Journal: Advances in Mathematical Physics
 (2010)

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## Abstract

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


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## Body

## 1. Introduction

Entanglement is the characteristic trait of quantum mechanics, and it reflects the property that a quantum system can simultaneously appear in two or more different states [1]. This feature implies the existence of global states of composite system which cannot be written as a product of the states of individual subsystems. This phenomenon [2], now known as “quantum entanglement,” plays crucial roles in quantum information processing [3]. Quantum entangled states have become the key ingredient in the rapidly expanding field of quantum information science, with remarkable prospective applications such as quantum computation [3, 4], quantum teleportation [5–9], dense coding [10], quantum cryptographic schemes [11–13], entanglement swapping [14–18], and remote states preparation (RSP) [19–24]. All such effects are based on entanglement and have been demonstrated in pioneering experiments.It has become clear that entanglement is not only the subject of philosophical debates, but also a new quantum resource for tasks which cannot be performed by means of classical resources. Although considerable efforts have been taken to understand and characterize the properties of quantum entanglement recently, the physical character and mathematical structure of entangled states have not been satisfactorily understood yet [25, 26]. In this review we mainly introduce some recent results related to our researches on several basic questions in this subject.(1) Separability of Quantum States
We first discuss the separability of a quantum states; namely, for a given quantum state, how we can know whether or not it is entangled.For pure quantum states, there are many ways to verify the separability. For instance, for a bipartite pure quantum state the separability is easily determined in terms of its Schmidt numbers. For multipartite pure states, the generalized concurrence given in [27] can be used to judge if the state is separable or not. In addition separable states must satisfy all possible Bell inequalities [28].For mixed states we still have no general criterion. The well-known PPT (partial positive transposition) criterion was proposed by Peres in 1996 [29]. It says that for any bipartite separable quantum state the density matrix must be positive under partial transposition. By using the method of positive maps Horodecki et al. [30] showed that the Peres' criterion is also sufficient for 2×2 and 2×3 bipartite systems. And for higher dimensional states, the PPT criterion is only necessary. Horodecki [31] has constructed some classes entangled states with positive partial transposes for 3×3 and 2×4 systems. States of this kind are said to be bound entangled (BE). Another powerful operational criterion is the realignment criterion [32, 33]. It demonstrates a remarkable ability to detect many bound entangled states and even genuinely tripartite entanglement [34]. Considerable efforts have been made in finding stronger variants and multipartite generalizations for this criterion [35–39]. It was shown that PPT criterion and realignment criterion are equivalent to the permutations of the density matrix's indices [34]. Another important criterion for separability is the reduction criterion [40, 41]. This criterion is equivalent to the PPT criterion for 2×N composite systems. Although it is generally weaker than the PPT, the reduction criteria have tight relation to the distillation of quantum states.There are also some other necessary criteria for separability. Nielsen and Kempe [42] presented a necessary criterion called majorization: the decreasing ordered vector of the eigenvalues for ρ is majorized by that of ρA1 or ρA2 alone for a separable state. That is, if a state ρ is separable, then λρ↓≺λρA1↓, λρ↓≺λρA2↓. Here λρ↓ denotes the decreasing ordered vector of the eigenvalues of ρ. A d-dimensional vector x↓ is majorized by y↓, x↓≺y↓, if ∑j=1kxj↓≤∑j=1kyj↓ for k=1,…,d-1 and the equality holds for k=d. Zeros are appended to the vectors λρA1,A2↓ such that their dimensions are equal to the one of λρ↓.In [31], another necessary criterion called range criterion was given. If a bipartite state ρ acting on the space ℋA⊗ℋB is separable, then there exists a family of product vectors ψi⊗ϕi such that (i) they span the range of ρ; (ii) the vector {ψi⊗ϕi*}i=1k spans the range of ρTB, where * denotes complex conjugation in the basis in which partial transposition was performed and ρTB is the partially transposed matrix of ρ with respect to the subspace B. In particular, any of the vectors ψi⊗ϕi* belongs to the range of ρ.Recently, some elegant results for the separability problem have been derived. In [43–45], a separability criteria based on the local uncertainty relations (LURs) was obtained. The authors show that, for any separable state ρ∈ℋA⊗ℋB,
(1.1)1-∑k〈GkA⊗GkB〉-12〈GkA⊗I-I⊗GkB〉2≥0,
where GkA or GkB are arbitrary local orthogonal and normalized operators (LOOs) in ℋA⊗ℋB. This criterion is strictly stronger than the realignment criterion. Thus more bound entangled quantum states can be recognized by the LUR criterion. The criterion is optimized in [46] by choosing the optimal LOOs. In [47] a criterion based on the correlation matrix of a state has been presented. The correlation matrix criterion is shown to be independent of PPT and realignment criterion [48], that is, there exist quantum states that can be recognized by correlation criterion while the PPT and realignment criterion fail. The covariance matrix of a quantum state is also used to study separability in [49]. It has been shown that the LUR criterion, including the optimized one, can be derived from the covariance matrix criterion [50].(2) Measure of Quantum Entanglement
One of the most difficult and fundamental problems in entanglement theory is to quantify entanglement. The initial idea to quantify entanglement was connected with its usefulness in terms of communication [51]. A good entanglement measure has to fulfill some conditions [52]. For bipartite quantum systems, we have several good entanglement measures such as Entanglement of Formation (EOF), Concurrence, and Tangle ctc. For two-qubit systems it has been proved that EOF is a monotonically increasing function of the concurrence and an elegant formula for the concurrence was derived analytically by Wootters [53]. However with the increasing dimensions of the subsystems the computation of EOF and concurrence become formidably difficult. A few explicit analytic formulae for EOF and concurrence have been found only for some special symmetric states [54–58].The first analytic lower bound of concurrence for arbitrary dimensional bipartite quantum states was derived by Mintert et al. in [59]. By using the positive partial transposition (PPT) and realignment separability criterion, analytic lower bounds on EOF and concurrence for any dimensional mixed bipartite quantum states have been derived in [60, 61]. These bounds are exact for some special classes of states and can be used to detect many bound entangled states. In [62] another lower bound on EOF for bipartite states has been presented from a new separability criterion [63]. A lower bound of concurrence based on local uncertainty relations (LURs) criterion is derived in [64]. This bound is further optimized in [46]. The lower bound of concurrence for tripartite systems has been studied in [65]. In [66, 67] the authors presented lower bounds of concurrence for bipartite systems by considering the “two-qubit" entanglement of bipartite quantum states with arbitrary dimensions. It has been shown that this lower bound has a tight relationship with the distillability of bipartite quantum states. Tangle is also a good entanglement measure that has a close relation with concurrence, as it is defined by the square of the concurrence for a pure state. It is also meaningful to derive tight lower and upper bounds for tangle [68].In [69] Mintert et al. proposed an experimental method to measure the concurrence directly by using joint measurements on two copies of a pure state. Then Walborn et al. presented an experimental determination of concurrence for two-qubit states [70, 71], where only one-setting measurement is needed, but two copies of the state have to be prepared in every measurement. In [72] another way of experimental determination of concurrence for two-qubit and multiqubit states has been presented, in which only one copy of the state is needed in every measurement. To determine the concurrence of the two-qubit state used in [70, 71], also one-setting measurement is needed, which avoids the preparation of the twin states or the imperfect copy of the unknown state, and the experimental difficulty is dramatically reduced.(3) Fidelity of Quantum Teleportation and Distillation
Quantum teleportation, or entanglement-assisted teleportation, is a technique used to transfer information on a quantum level, usually from one particle (or series of particles) to another particle (or series of particles) in another location via quantum entanglement. It does not transport energy or matter, nor does it allow communication of information at super luminal (faster than light) speed.In [5–7], Bennett et al. first presented a protocol to teleport an unknown qubit state by using a pair of maximally entangled pure qubit state. The protocol is generalized to transmit high-dimensional quantum states [8, 9]. The optimal fidelity of teleportation is shown to be determined by the fully entangled fraction of the entangled resource which is generally a mixed state. Nevertheless similar to the estimation of concurrence, the computation of the fully entangled fraction for a given mixed state is also very difficult.The distillation protocol has been presented to get maximally entangled pure states from many entangled mixed states by means of local quantum operations and classical communication (LQCC) between the parties sharing the pairs of particles in this mixed state [73–76]. Bennett et al. first derived a protocol to distill one maximally entangled pure Bell state from many copies of not maximally entangled quantum mixed states in [73] in 1996. The protocol is then generalized to distill any bipartite quantum state with higher dimension by M. Horodecki and P. Horodecki in 1999 [77]. It is proven that a quantum state can be always distilled if it violates the reduced matrix separability criterion [77].This review mainly contains three parts. In Section2 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 stateρ12⋯N∈ℋ1⊗ℋ2⊗⋯⊗ℋN is said to be fully separable if it can be written as(2.1)ρ12⋯N=ρ1⊗ρ2⊗⋯⊗ρN,
where ρ1 and ρ2,…,ρN are reduced density matrices defined as ρ1=Tr23⋯N[ρ12⋯N], ρ2=Tr13⋯N[ρ12⋯N],…,ρN=Tr12⋯N-1[ρ12⋯N]. This is equivalent to the condition
(2.2)ρ12⋯N=|ψ1〉〈ψ1|⊗|ϕ2〉〈ϕ2|⊗⋯⊗|μN〉〈μN|,
where |ψ1〉∈ℋ1,|ϕ2〉∈ℋ2,…,|μN〉∈ℋN.A multipartite quantum mixed stateρ12⋯N∈ℋ1⊗ℋ2⊗⋯⊗ℋN is said to be fully separable if it can be written as(2.3)ρ12⋯N=∑iqiρi1⊗ρi2⊗⋯⊗ρiN,
where ρi1,ρi2,…,ρiN are the reduced density matrices with respect to the systems 1,2,…,N, respectively, qi>0, and ∑iqi=1. This is equivalent to the condition
(2.4)ρ12⋯N=∑ipi|ψi1〉〈ψi1|⊗|ϕi2〉〈ϕi2|⊗⋯⊗|μiN〉〈μiN|,
where |ψi1〉,|ϕi2〉,…,|μiN〉 are normalized pure states of systems 1,2,…,N, respectively, pi>0, and ∑ipi=1.For pure states, the definition (2.1) itself is an operational separability criterion. In particular, for bipartite case, there are Schmidt decompositions.Theorem 2.1 (see Schmidt decomposition in [78]).
Suppose that|ψ〉∈ℋA⊗ℋB is a pure state of a composite system, AB, then there exist orthonormal states |iA〉 for system A and orthonormal states |iB〉 for system B such that
(2.5)|ψ〉=∑iλi|iA〉|iB〉,
where λi are nonnegative real numbers satisfying ∑iλi2=1, known as Schmidt coefficients.|iA〉 and |iB〉 are called Schmidt bases with respect to ℋA and ℋB. The number of nonzero values λi is called Schmidt number, also known as Schmidt rank, which is invariant under unitary transformations on system A or system B. For a bipartite pure state |ψ〉, |ψ〉 is separable if and only if the Schmidt number of |ψ〉 is one.For multipartite pure states, one has no such Schmidt decomposition. In [79] it has been verified that any pure three-qubit state |Ψ〉 can be uniquely written as(2.6)|Ψ〉=λ0|000〉+λ1eiψ|100〉+λ2|101〉+λ3|110〉+λ4|111〉
with normalization condition λi≥0,0≤ψ≤π, where ∑iμi=1,  μi≡λi2. Equation  (2.6) is called generalized Schmidt decomposition.For mixed states it is generally very hard to verify whether a decomposition like (2.3) exists. For a given generic separable density matrix, it is also not easy to find the decomposition (2.3) in detail.


### 2.1. Separability Criteria for Mixed States

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


#### 2.1.1. Partial Positive Transpose Criterion

The positive partial transpose (PPT) criterion provided by Peres [29] says that if a bipartite state ρAB∈ℋA⊗ℋB is separable, then the new matrix ρABTB with matrix elements defined in some fixed product basis as
(2.7)〈m|〈μ|ρABTB|n〉|ν〉≡〈m|〈ν|ρAB|n〉|μ〉
is also a density matrix (i.e., it has nonnegative spectrum). The operation TB, called partial transpose, just corresponds to the transposition of the indices with respect to the second subsystem B. It has an interpretation as a partial time reversal [80].Afterwards Horodecki et al. showed that Peres' criterion is also sufficient for2×2 and 2×3 bipartite systems [30]. This criterion is now called PPT or Peres-Horodecki (P-H) criterion. For high-dimensional states, the P-H criterion is only necessary. Horodecki has constructed some classes of families of entangled states with positive partial transposes for 3×3 and 2×4 systems [31]. States of this kind are said to be bound entangled (BE).


#### 2.1.2. Reduced Density Matrix Criterion

Cerf et al. [81] and M. Horodecki and P. Horodecki [82], independently, introduced a map Γ:ρ→TrB[ρAB]⊗I-ρAB(I⊗TrA[ρAB]-ρAB), which gives rise to a simple necessary condition for separability in arbitrary dimensions, called the reduction criterion. If ρAB is separable, then
(2.8)ρA⊗I-ρAB≥0,I⊗ρB-ρAB≥0,
where ρA=TrB[ρAB],  ρB=TrA[ρAB]. This criterion is simply equivalent to the P-H criterion for 2×n composite systems. It is also sufficient for 2×2 and 2×3 systems. In higher dimensions the reduction criterion is weaker than the P-H criterion.


#### 2.1.3. Realignment Criterion

There is yet another class of criteria based on linear contractions on product states. They stem from the new criterion discovered in [33, 83] called computable cross-norm (CCN) criterion or matrix realignment criterion which is operational and independent on PPT test [29]. If a state ρAB is separable, then the realigned matrix ℛ(ρ) with elements ℛ(ρ)ij,kl=ρik,jl has trace norm not greater than one:
(2.9)∥ℛ(ρ)∥KF≤1.
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 anN-dimensional Hilbert space ℋ can be expressed according to the generators of the special unitary group SU(N) [84]. The generators of SU(N) can be introduced according to the transition-projection operators Pjk=|j〉〈k|, where |i〉, i=1,…,N, are the orthonormal eigenstates of a linear Hermitian operator on ℋ. Set
(2.10)ωl=-2l(l+1)(P11+P22+⋯+Pll-lPl+1,l+1),ujk=Pjk+Pkj,vjk=i(Pjk-Pkj),
where 1≤l≤N-1 and 1≤j<k≤N. We get a set of N2-1 operators
(2.11)Γ≡{ωl,ω2,…,ωN-1,u12,u13,…,v12,v13,…},
which satisfies the relations
(2.12)Tr[λi]=0,Tr[λiλj]=2δij,∀λi∈Γ,
and thus generate the SU(N) [85].Any Hermitian operatorρ in ℋ can be represented in terms of these generators of SU(N) as(2.13)ρ=1NIN+12∑j=1N2-1rjλj,
where IN is a unit matrix and r=(r1,r2,…,rN2-1)∈ℝN2-1 and r is called Bloch vector. The set of all the Bloch vectors that constitute a density operator is known as the Bloch vector space B(ℝN2-1).A matrix of the form (2.13) is of unit trace and Hermitian, but it might not be positive. To guarantee the positivity restrictions must be imposed on the Bloch vector. It is shown that B(ℝN2-1) is a subset of the ball DR(ℝN2-1) of radius R=2(1-1/N), which is the minimum ball containing it, and that the ball Dr(ℝN2-1) of radius r=2/N(N-1) is included in B(ℝN2-1) [86], that is,(2.14)Dr(ℝN2-1)⊆B(ℝN2-1)⊆DR(ℝN2-1).Let the dimensions of systemsA,B, and C be dA=N1,dB=N2, and dC=N3, respectively. Any tripartite quantum states ρABC∈ℋA⊗ℋB⊗ℋC can be written as(2.15)ρABC=IN1⊗IN2⊗M0+∑i=1N12-1λi(1)⊗IN2⊗Mi+∑j=1N22-1IN1⊗λj(2)⊗M̃j+∑i=1N12-1‍∑j=1N22-1λi(1)⊗λj(2)⊗Mij,
where λi(1), λj(2) are the generators of SU(N1) and SU(N2); Mi,M̃j, and Mij are operators of ℋC.Theorem 2.2.
Letr∈ℝN12-1, s∈ℝN22-1 and |r|≤2/N1(N1-1), |s|≤2/N2(N2-1). For a tripartite quantum state ρ∈ℋA⊗ℋB⊗ℋC with representation (2.15), one has [87]
(2.16)M0-∑i=1N12-1riMi-∑j=1N22-1sjM̃j+∑i=1N12-1‍∑j=1N22-1risjMij≥0.Proof.
Sincer∈ℝN12-1, s∈ℝN22-1 and |r|≤2/N1(N1-1), |s|≤2/N2(N2-1), we have that A1≡(1/2)((2/N1)I-∑i=1N12-1riλi(1)) and A2≡(1/2)((2/N2)I-∑j=1N22-1sjλj(2)) are positive Hermitian operators. Let A=A1⊗A2⊗IN3. Then AρA≥0 and (AρA)†=AρA. The partial trace of AρA over ℋA (and ℋB) should be also positive. Hence
(2.17)0≤TrAB[AρA]=TrAB[A1⊗A2⊗M0+∑iA1λi(1)A1⊗A2⊗Mi+∑jA1⊗A2λj(2)A2⊗M̃j+∑ijA1λi(1)A1⊗A2λj(2)A2⊗Mij]=M0-∑i=1N12-1riMi-∑j=1N22-1sjM̃j+∑i=1N12-1‍∑j=1N22-1risjMij.Formula (2.16) is valid for any tripartite state. By setting s=0 in (2.16), one can get a result for bipartite systems.Corollary 2.3.
LetρAB∈ℋA⊗ℋB, which can be generally written as ρAB=IN1⊗M0+∑j=1N12-1λj⊗Mj, then, for any r∈ℝN12-1 with |r|≤2/N1(N1-1), M0-∑j=1N12-1rjMj≥0.A separable tripartite stateρABC can be written as
(2.18)ρABC=∑ipi|ψiA〉〈ψiA|⊗|ϕiB〉〈ϕiB|⊗|ωiC〉〈ωiC|.
From (2.13) it can also be represented as(2.19)ρABC=∑ipi12(2N1IN1+∑k=1N12-1ai(k)λk(1))⊗12(2N2IN2+∑l=1N22-1bi(l)λl(2))⊗|ωiC〉〈ωiC|=IN1⊗IN2⊗1N1N2∑ipi|ωiC〉〈ωiC|+∑k=1N12-1λk(1)⊗IN2⊗12N2∑iai(k)pi|ωiC〉〈ωiC|+∑l=1N22-1IN1⊗λl(2)⊗12N1∑ibi(l)pi|ωiC〉〈ωiC|+∑kN12-1∑lN22-1λk(1)⊗λl(2)⊗14∑iai(k)bi(l)pi|ωiC〉〈ωiC|,
where (ai(1),ai(2),…,ai(N12-1)) and (bi(1),bi(2),…,bi(N22-1)) are real vectors on the Bloch sphere satisfying |ai⃗|2=∑j=1N12-1(ai(j))2=2(1-1/N1) and |bi⃗|2=∑j=1N22-1(bi(j))2=2(1-1/N2).Comparing (2.15) with (2.19), we have(2.20)M0=1N1N2∑ipi|ωiC〉〈ωiC|,Mk=12N2∑iai(k)pi|ωiC〉〈ωiC|,M̃l=12N1∑ibi(l)pi|ωiC〉〈ωiC|,Mkl=14∑iai(k)bi(l)pi|ωiC〉〈ωiC|.For any(N12-1)×(N12-1) real matrix R(1) and (N22-1)×(N22-1) real matrix R(2) satisfying (1/(N1-1)2)I-R(1)TR(1)≥0 and (1/(N2-1)2)I-R(2)TR(2)≥0, we define a new matrix(2.21)ℛ=(R(1)000R(2)000T),
where T is a transformation acting on an (N12-1)×(N22-1) matrix M by(2.22)T(M)=R(1)MRT(2).Usingℛ, we define a new operator γℛ:(2.23)γℛ(ρABC)=IN1⊗IN2⊗M0′+∑i=1N12-1λi(1)⊗IN2⊗Mi′+∑j=1N22-1IN1⊗λj(2)⊗M̃j′+∑i=1N12-1‍∑j=1N22-1λi(1)⊗λj(2)⊗Mij′,
where M0′=M0,Mk′=∑m=1N12-1Rkm(1)Mm,M̃l′=∑n=1N22-1Rln(2)M̃n, and Mij′=(T(M))ij=(R(1)MRT(2))ij.Theorem 2.4.
IfρABC is separable, then [87] γℛ(ρABC)≥0.Proof.
From (2.20) and (2.23) we get
(2.24)M0′=M0=1N1N2∑ipi|ωiC〉〈ωiC|,Mk′=12N2∑miRkm(1)ai(m)pi|ωiC〉〈ωiC|,M̃l′=12N1∑niRln(2)bi(n)pi|ωiC〉〈ωiC|,Mkl′=14∑mniRkm(1)ai(m)Rln(2)bi(n)pi|ωiC〉〈ωiC|.
A straightforward calculation gives rise to
(2.25)γℛ(ρABC)=∑ipi12(2N1IN1+∑k=1N12-1∑m=1N12-1Rkm(1)ai(m)λk(1))⊗12(2N2IN2+∑l=1N22-1∑n=1N22-1Rln(2)bi(n)λl(2))⊗|ωiC〉〈ωiC|.
As (1/(N1-1)2)I-R(1)TR(1)≥0 and (1/(N2-1)2)I-R(2)TR(2)≥0, we get
(2.26)|ai′⃗|2=|R(1)ai⃗|2≤1(N1-1)2|ai⃗|2=2N1(N1-1),|bi′⃗|2=|R(2)bi⃗|2≤1(N2-1)2|bi⃗|2=2N2(N2-1).
Therefore γℛ(ρABC) is still a density operator, that is, γℛ(ρABC)≥0.Theorem2.4 gives a necessary separability criterion for general tripartite systems. The result can be also applied to bipartite systems. Let ρAB∈ℋA⊗ℋB, ρAB=IN1⊗M0+∑j=1N12-1λj⊗Mj. For any real (N12-1)×(N12-1) matrix ℛ satisfying (1/(N1-1)2)I-ℛTℛ≥0 and any state ρAB, we define
(2.27)γℛ(ρAB)=IN1⊗M0+∑j=1N12-1λj⊗Mj′,
where Mj′=∑kℛjkMk.Corollary 2.5.
ForρAB∈ℋA⊗ℋB, if there exists an ℛ with (1/(N1-1)2)I-ℛTℛ≥0 such that γℛ(ρAB)<0, then ρAB must be entangled.For2×N systems, the above corollary is reduced to the results in [88]. As an example we consider the 3×3 istropic states
(2.28)ρI=1-p9I3⊗I3+p3∑i,j=13|ii〉〈jj|=I3⊗(19I3)+∑i=15λi⊗(p6λi)-∑i=68λi⊗(p6λi).
If we choose ℛ to be Diag{1/2,1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2}, we get that ρI is entangled for 0.5<p≤1.For tripartite case, we take the following3×3×3 mixed state as an example:
(2.29)ρ=1-p27I27+p|ψ〉〈ψ|,
where |ψ〉=(1/3)(|000〉+|111〉+|222〉)(〈000|+〈111|+〈222|). Taking R(1)=R(2)=Diag{1/2,1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2}, we have that ρ is entangled for 0.6248<p≤1.In fact the criterion for2×N systems [88] is equivalent to the PPT criterion [89]. Similarly Theorem 2.4 is also equivalent to the PPT criterion for 2×2×N systems.


#### 2.1.5. Covariance Matrix Criterion

In this subsection we study the separability problem by using the covariance matrix approach. We first give a brief review of covariance matrix criterion proposed in [49]. Let ℋdA and ℋdB be d-dimensional complex vector spaces and ρAB a bipartite quantum state in ℋdA⊗ℋdB. Let Ak (resp., Bk) be d2 observables on ℋdA (resp., ℋdB) such that they form an orthonormal normalized basis of the observable space, satisfying Tr[AkAl]=δk,l (resp., Tr[BkBl]=δk,l). Consider the total set {Mk}={Ak⊗I,I⊗Bk}. It can be proven that [44](2.30)∑k=1N2(Mk)2=dI,∑k=1N2〈Mk〉2=Tr[ρAB2].The covariance matrixγ is defined with entries(2.31)γij(ρAB,{Mk})=〈MiMj〉+〈MjMi〉2-〈Mi〉〈Mj〉,
which has a block structure [49](2.32)γ=(ACCTB),
where A=γ(ρA,{Ak}),B=γ(ρB,{Bk}),Cij=〈Ai⊗Bj〉ρAB-〈Ai〉ρA〈Bj〉ρB,ρA=TrB[ρAB], and ρB=TrA[ρAB]. Such covariance matrix has a concavity property: for a mixed density matrix ρ=∑kpkρk with pk≥0 and ∑kpk=1, one has γ(ρ)≥∑kpkγ(ρk).For a bipartite product stateρAB=ρA⊗ρB, C in (2.32) is zero. Generally if ρAB is separable, then there exist states |ak〉〈ak| on ℋdA,|bk〉〈bk| on ℋdB and pk such that(2.33)γ(ρ)≥κA⊕κB,
where κA=∑pkγ(|ak〉〈ak|,{Ak}), κB=∑pkγ(|bk〉〈bk|,{Bk}).For a separable bipartite state, it has been shown that [49](2.34)∑i=1d2|Cii|≤(1-Tr[ρA2])+(1-Tr[ρB2])2.Criterion (2.34) depends on the choice of the orthonormal normalized basis of the observables. In fact the term ∑i=1d2|Cii| has an upper bound ∥C∥KF which is invariant under unitary transformation and can be attained by choosing proper local orthonormal observable basis, where ∥C∥KF stands for the Ky Fan norm of C, ∥C∥KF=Tr[CC†], with † denoting the transpose and conjugation. It has been shown in [46] that if ρAB is separable, then(2.35)∥C∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2.From the covariance matrix approach, we can also get an alternative criterion. From (2.32) and (2.33) we have that if ρAB is separable, then
(2.36)X≡(A-κACCTB-κB)≥0.
Hence all the 2×2 minor submatrices of X must be positive. Namely, one has
(2.37)|(A-κA)iiCijCji(B-κB)jj|≥0,
that is,  (A-κA)ii(B-κB)jj≥Cij2. Summing over all i, j and using (2.30), we get(2.38)∑i,j=1d2Ci,j2≤(Tr[A]-Tr[κA])(Tr[B]-Tr[κB])=(d-Tr[ρA2]-d+1)(d-Tr[ρB2]-d+1)=(1-Tr[ρA2])(1-Tr[ρB2]).That is,(2.39)∥C∥HS2≤(1-Tr[ρA2])(1-Tr[ρB2]),
where ∥C∥HS stands for the Euclid norm of C, that is, ∥C∥HS=Tr[CC†].Formulae (2.35) and (2.39) are independent and could be complement. When
(2.40)(1-Tr[ρA2])(1-Tr[ρB2])<∥C∥HS≤∥C∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2,(2.39) can recognize the entanglement but (2.35) cannot. When
(2.41)∥C∥HS≤(1-Tr[ρA2])(1-Tr[ρB2])≤(1-Tr[ρA2])+(1-Tr[ρB2])2<∥C∥KF,(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 caseρABC∈ℋdA⊗ℋdB⊗ℋdC. Take d2 observables Ak on ℋA, respectively, Bk on ℋB, respectively, Ck on ℋC. Set {Mk}={Ak⊗I⊗I,I⊗Bk⊗I,I⊗I⊗Ck}. The covariance matrix defined by (2.31) has then the following block structure:(2.42)γ=(ADEDTBFETFTC),
where A=γ(ρA,{Ak}), B=γ(ρB,{Bk}), C=γ(ρC,{Ck}), Dij=〈Ai⊗Bj〉ρAB-〈Ai〉ρA〈Bj〉ρB, Eij=〈Ai⊗Cj〉ρAC-〈Ai〉ρA〈Cj〉ρC, and Fij=〈Bi⊗Cj〉ρBC-〈Bi〉ρB〈Cj〉ρC.Theorem 2.6.
IfρABC is fully separable, then [90]
(2.43)∥D∥HS2≤(1-Tr[ρA2])(1-Tr[ρB2]),(2.44)∥E∥HS2≤(1-Tr[ρA2])(1-Tr[ρC2]),(2.45)∥F∥HS2≤(1-Tr[ρB2])(1-Tr[ρC2]),(2.46)2∥D∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2]),(2.47)2∥E∥KF≤(1-Tr[ρA2])+(1-Tr[ρC2]),(2.48)2∥F∥KF≤(1-Tr[ρB2])+(1-Tr[ρC2]).Proof.
For a tripartite product stateρABC=ρA⊗ρB⊗ρC, D,E, and F in (2.42) are zero. If ρABC is fully separable, then there exist states |ak〉〈ak| in ℋdA, |bk〉〈bk| in ℋdB, and |ck〉〈ck| in ℋdC, and pk such that γ(ρ)≥κA⊕κB⊕κC, where κA=∑pkγ(|ak〉〈ak|,{Ak}), κB=∑pkγ(|bk〉〈bk|,{Bk}), and κC=∑pkγ(|ck〉〈ck|,{Ck}), that is,
(2.49)Y≡(A-κADEDTB-κBFETFTC-κC)≥0.
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
(2.50)|(A-κA)iiDijDji(B-κB)jj|≥0,
that is, (A-κA)ii(B-κB)jj≥|Dij|2. Summing over all i, j and using (2.30), we get
(2.51)∥D∥HS2=∑i,j=1d2Di,j2≤(Tr[A]-Tr[κA])(Tr[B]-Tr[κB]=(d-Tr[ρA2]-d+1)(d-Tr[ρB2]-d+1)=(1-Tr[ρA2])(1-Tr[ρB2]),
which proves (2.43). Equations (2.44) and (2.45) can be similarly proved.
From (2.50) we also have (A-κA)ii+(B-κB)ii≥2|Dii|. Therefore
(2.52)∑i|Dii|≤(Tr[A]-Tr[κA])+(Tr[B]-Tr[κB]2=(d-Tr[ρA2]-d+1)+(d-Tr[ρB2]-d+1)2=(1-Tr[ρA2])+(1-Tr[ρB2])2.
Note that ∑i=1d2Dii≤∑i=1d2|Dii|. By using that Tr[MU]≤∥M∥KF=Tr[MM†] for any matrix M and any unitary U [91], we have ∑i=1d2Dii≤∥D∥KF.
LetD=U†ΛV be the singular value decomposition of D. Make a transformation of the orthonormal normalized basis of the local orthonormal observable space Ãi=∑lUilAl and B̃j=∑mVjm*Bm. In the new basis we have
(2.53)D̃ij=∑lmUilDlmVjm=(UDV†)ij=Λij.
Then (2.52) becomes
(2.54)∑i=1d2D̃ii=∥D∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2,
which proves (2.46). Equations (2.47) and (2.48) can be similarly treated.We consider now the case thatρABC is bipartite separable.Theorem 2.7.
IfρABC is a bipartite separable state with respect to the bipartite partition of the sub-systems A and BC (resp., AB and C; resp., AC and B), then (2.43), (2.44) and (2.46), (2.47) (resp., (2.44), (2.45) and (2.47), (2.48); resp., (2.43), (2.45) and (2.46), (2.48)) must hold [90].Proof.
We prove the case thatρABC is bipartite separable with respect to the A system and BC systems partition. The other cases can be similarly treated. In this case the matrices D and E in the covariance matrix (2.42) are zero. ρABC takes the form ρABC=∑mpmρAm⊗ρBCm. Define κA=∑pmγ(ρAm,{Ak}), κBC=∑pmγ(ρBCm,{Bk⊗I,I⊗Ck}). κBC has a form
(2.55)κBC=(κBF′(F′)TκC),
where κB=∑pkγ(|bk〉〈bk|,{Bk}) and κC=∑pkγ(|ck〉〈ck|,{Ck}), (F′)ij=∑mpm(〈Bi⊗Cj〉ρBCm-〈Bi〉ρBm〈Cj〉ρCm). By using the concavity of covariance matrix we have
(2.56)γ(ρABC)≥∑mpmγ(ρAm⊗ρBCm)=(κA000κBF′0(F′)TκC).
Accounting to the method used in proving Theorem 2, we get (2.43), (2.44), and (2.46), (2.47).From Theorems2.6 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 Theorem2.6 can be generalized to general multipartite case ρ∈ℋd(1)⊗ℋd(2)⊗⋯⊗ℋd(N). Define Âαi=I⊗I⊗⋯λα⊗I⊗⋯⊗I, where λ0=I/d, λα (α=1,2,…,d2-1) are the normalized generators of SU(d) satisfying Tr[λαλβ]=δαβ and acting on the ith system ℋd(i), i=1,2,…,N. Denote {Mk} as the set of all Âαi. Then the covariance matrix of ρ can be written as(2.57)γ(ρ)=(𝒜11𝒜12⋯𝒜1N𝒜12T𝒜22⋯𝒜2N⋮⋮⋮𝒜1NT𝒜2NT⋯𝒜NN),
where 𝒜ii=γ(ρ,{Âki}) and (𝒜ij)mn=〈Âmi⊗Ânj〉-〈Âmi〉〈Ânj〉 for i≠j.For a product stateρ12⋯N, 𝒜ij, i≠j, in (2.57) are zero matrices. Define
(2.58)κ𝒜ii=∑kpkγ(|ψki〉〈ψki|,{Âli}).
Then for a fully separable multipartite state ρ=∑kpk|ψk1〉〈ψk1|⊗|ψk2〉〈ψk2|⊗⋯⊗|ψkN〉〈ψkN|, one has
(2.59)Z=(𝒜11-κ𝒜11𝒜12⋯𝒜1N𝒜12T𝒜22-κ𝒜22⋯𝒜2N⋮⋮⋮𝒜1NT𝒜2NT⋯𝒜NN-κ𝒜NN)≥0,
from which we have the following separability criterion for multipartite systems.Theorem 2.9.
If a stateρ∈ℋd(1)⊗ℋd(2)⊗⋯⊗ℋd(N) is fully separable, then the following inequalities
(2.60)∥𝒜ij∥HS2≤(1-Tr[ρi2])(1-Tr[ρj2]),∥𝒜ij∥KF≤(1-Tr[ρi2])+(1-Tr[ρj2])2
must be fulfilled for any i≠j [90].


### 2.2. Normal Form of Quantum States

In this subsection we show that the correlation matrix (CM) criterion can be improved from the normal form obtained under filtering transformations. Based on CM criterion entanglement witness in terms of local orthogonal observables (LOOs) [92] for both bipartite and multipartite systems can be also constructed.For bipartite case,ρ∈ℋ=ℋA⊗ℋB with dimℋA=M, dimℋB=N, and M≤N is mapped to the following form under local filtering transformations [93]:(2.61)ρ→ρ̃=(FA⊗FB)ρ(FA⊗FB)†Tr[(FA⊗FB)ρ(FA⊗FB)†],
where FA/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 positiveρ into a normal form [94]:(2.62)ρ̃=(FA⊗FB)ρ(FA⊗FB)†Tr[(FA⊗FB)ρ(FA⊗FB)†]=1MN(I+∑i=1M2-1ξiGiA⊗GiB),
where ξi≥0 and GiA and GiB are some traceless orthogonal observables. The matrices FA and FB can be obtained by minimizing the function
(2.63)f(A,B)=Tr[ρ(A⊗B)](detA)1/M(detB)1/N,
where A=FA†FA and B=FB†FB. In fact, one can choose FA0≡|det(ρA)|1/2M(ρA)-1, and FB0≡|det(ρB′)|1/2N(ρB′)-1, where ρB′=TrA[I⊗(ρA)-1ρI⊗(ρA)-1]. Then by iterations one can get the optimal A and B. In particular, there is a matlab code available in [95].For bipartite separable statesρ, the CM separability criterion [96] says that(2.64)∥T∥KF≤MN(M-1)(N-1),
where T is an (M2-1)×(N2-1) matrix with Tij=MN·Tr[ρλiA⊗λjB], ∥T∥KF stands for the trace norm of T, λkA/Bs are the generators of SU(M/N) and have been chosen to be normalized, and Tr[λk(A/B)λl(A/B)]=δkl.As the filtering transformation does not change the separability of a state, one can study the separability ofρ̃ instead of ρ. Under the normal form (2.62) the criterion (2.64) becomes(2.65)∑iξi≤MN(M-1)(N-1).In [44] a separability criterion based on local uncertainty relation (LUR) has been obtained. It says that, for any separable state ρ,(2.66)1-∑k〈GkA⊗GkB〉-12〈GkA⊗I-I⊗GkB〉2≥0,
where GkA/Bs are LOOs such as the normalized generators of SU(M/N) and GkA=0 for k=M2+1,…,N2. The criterion is shown to be strictly stronger than the realignment criterion [61]. Under the normal form (2.62) criterion (2.66) becomes
(2.67)1-∑k〈GkA⊗GkB〉-12〈GkA⊗I-I⊗GkB〉2=1-1MN∑kξk-12(1M+1N)≥0,
that is,(2.68)∑k‍ξk≤MN-M+N2.
As MN(M-1)(N-1)≤MN-(M+N)/2 holds for any M and N, from (2.65) and (2.68) it is obvious that the CM criterion recognizes entanglement better when the normal form is taken into account.We now consider multipartite systems. Letρ be a strictly positive density matrix in ℋ=ℋ1⊗ℋ2⊗⋯⊗ℋN and dimℋi=di. ρ can be generally expressed in terms of the SU(n) generators λαk [97] as(2.69)ρ=1ΠiNdi(⊗jNIdj+∑{μ1}∑α1𝒯α1{μ1}λα1{μ1}+∑{μ1μ2}∑α1α2𝒯α1α2{μ1μ2}λα1{μ1}λα2{μ2}+∑{μ1μ2μ3}∑α1α2α3𝒯α1α2α3{μ1μ2μ3}λα1{μ1}λα2{μ2}λα3{μ3}+⋯+∑{μ1μ2⋯μM}∑α1α2⋯αM𝒯α1α2⋯αM{μ1μ2⋯μM}λα1{μ1}λα2{μ2}⋯λαM{μM}+⋯+∑α1α2⋯αN𝒯α1α2⋯αM{1,2,…,N}λα1{1}λα2{2}⋯λαN{N}),
where λαk{μk}=Id1⊗Id2⊗⋯⊗λαk⊗Idμk+1⊗⋯⊗IdN with λαk appears at the μkth position and(2.70)𝒯α1α2⋯αM{μ1μ2⋯μM}=∏i=1Mdμi2MTr[ρλα1{μ1}λα2{μ2}⋯λαM{μM}].The generalized CM criterion says that ifρ in (2.69) is fully separable, then(2.71)∥𝒯{μ1,μ2,…,μM}∥KF≤12M∏k=1Mdμk(dμk-1),
for 2≤M≤N,{μ1,μ2,…,μM}⊂{1,2,…,N}. The KF norm is defined by
(2.72)∥𝒯{μ1,μ2,…,μM}∥KF=maxm=1,2,…,M∥𝒯(m)∥KF,
where 𝒯(m) is a kind of matrix unfolding of 𝒯{μ1,μ2,…,μM}.The criterion (2.71) can be improved by investigating the normal form of (2.69).Theorem 2.10.
By filtering transformations of the form(2.73)ρ̃=F1⊗F2⊗⋯⊗FNρF1†⊗F2†⊗FN†,
where Fi∈GL(di,ℂ),i=1,2,…N, followed by normalization, any strictly positive state ρ can be transformed into a normal form [98]:
(2.74)ρ=1ΠiNdi(⊗jNIdj+∑{μ1μ2}∑α1α2𝒯α1α2{μ1μ2}λα1{μ1}λα2{μ2}+∑{μ1μ2μ3}∑α1α2α3𝒯α1α2α3{μ1μ2μ3}λα1{μ1}λα2{μ2}λα3{μ3}+⋯+∑{μ1μ2⋯μM}∑α1α2⋯αM𝒯α1α2⋯αM{μ1μ2⋯μM}λα1{μ1}λα2{μ2}⋯λαM{μM}+⋯+∑α1α2⋯αN𝒯α1α2⋯αM{1,2,…,N}λα1{1}λα2{2}⋯λαN{N}).Proof.
LetD1,D2,…,DN be the sets of density matrices of the N subsystems. The cartesian product D1×D2×⋯×DN consisting of all product density matrices ρ1⊗ρ2⊗⋯⊗ρN with normalization Tr[ρi]=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 ρi:
(2.75)f(ρ1,ρ2,…,ρN)=Tr[ρ(ρ1⊗ρ2⊗⋯⊗ρN)]∏i=1Ndet(ρi)1/di.
The function is well defined on the interior of D1×D2×⋯×DN where detρi>0. As ρ is assumed to be strictly positive, we have Tr[ρ(ρ1⊗ρ2⊗⋯⊗ρN)]>0. Since D1×D2×⋯×DN is compact, we have Tr[ρ(ρ1⊗ρ2⊗⋯⊗ρN)]≥C>0 with a lower bound C depending on ρ.
It follows thatf→∞ on the boundary of D1×D2×⋯×DN where at least one of the ρis satisfies detρi=0. It follows further that f has a positive minimum on the interior of D1×D2×⋯×DN with the minimum value attained for at least one product density matrix τ1⊗τ2⊗⋯⊗τN with detτi>0, i=1,2,…,N. Any positive density matrix τi with detτi>0 can be factorized in terms of Hermitian matrices Fi as
(2.76)τi=Fi†Fi,
where Fi∈GL(di,ℂ). Denote that F=F1⊗F2⊗⋯⊗FN, so that τ1⊗τ2⊗⋯⊗τN=F†F. Set ρ̃=FρF† and define
(2.77)f̃(ρ1,ρ2,…,ρN)=Tr[ρ̃(ρ1⊗ρ2⊗⋯⊗ρN)]∏i=1Ndet(ρi)1/di=∏i=1Ndet(τi)1/di·Tr[ρ(F1†ρ1F1⊗F2†ρ2F2⊗⋯⊗FN†ρNFN)]∏i=1Ndet(τi)1/didet(ρi)1/di=∏i=1Ndet(τi)1/di·f(F1†ρ1F1,F2†ρ2F2,…,FN†ρNFN).
We see that, whenFi†ρiFi=τi, f̃ has a minimum and
(2.78)ρ1⊗ρ2⊗⋯⊗ρN=(F†)-1τ1⊗τ2⊗⋯⊗τNF-1=I.
Sincef̃ is stationary under infinitesimal variations about the minimum, it follows that
(2.79)Tr[ρ̃δ(ρ1⊗ρ2⊗⋯⊗ρN)]=0
for all infinitesimal variations
(2.80)δ(ρ1⊗ρ2⊗⋯⊗ρN)=δρ1⊗Id2⊗⋯⊗IdN+Id1⊗δρ2⊗Id3⊗⋯⊗IdN+⋯+Id1⊗Id2⊗⋯⊗IdN-1⊗δρN,
subjected to the constraint det(Idi+δρi)=1, which is equivalent to Tr[δρi]=0, i=1,2,…,N, using det(eA)=eTr[A] for a given matrix A. Thus, δρi can be represented by the SU generators, δρi=∑kδckiλki. It follows that Tr[ρ̃λαk{μk}]=0 for any αk and μk. Hence the terms proportional to λαk{μk} in (2.69) disappear.Corollary 2.11.
The normal form of a product state inℋ must be proportional to the identity.Proof.
Letρ be such a state. From (2.74), we get that
(2.81)ρ̃i=Tr1,2,…,i-1,i+1,…,N[ρ]=1diIdi.
Therefore for a product state ρ we have
(2.82)ρ̃=ρ1⊗ρ2⊗⋯⊗ρN=1∏i=1Ndi⊗i=1NIdi.As an example for separability of multipartite states in terms of their normal forms (2.74), we consider the PPT entangled edge state [79]
(2.83)ρ=(100000010a00000000b00000000c000000001c000000001b000000001a010000001)
mixed with noises
(2.84)ρp=pρ+(1-p)8I8.
Select a=2,b=3, and c=0.6. Using the criterion in [97] we get that ρp is entangled for 0.92744<p≤1. But after transforming ρp to its normal form (2.74), the criterion can detect entanglement for 0.90285<p≤1.Here we indicate that the filtering transformation does not change the PPT property. Letρ∈ℋA⊗ℋB be PPT, that is, ρTA≥0, and ρTB≥0. Let ρ̃ be the normal form of ρ. From (2.61) we have
(2.85)ρ̃TA=(FA*⊗FB)ρTA(FAT⊗FB†)Tr[(FA⊗FB)ρ(FA⊗FB)†].
For any vector |ψ〉, we have
(2.86)〈ψ|ρ̃TA|ψ〉=〈ψ|(FA*⊗FB)ρTA(FAT⊗FB†)|ψ〉Tr[(FA⊗FB)ρ(FA⊗FB)†]≡〈ψ′|ρTA|ψ′〉≥0,
where |ψ′〉=(FAT⊗FB†)|ψ〉/Tr[(FA⊗FB)ρ(FA⊗FB)†].  ρ̃TB≥0 can be proved similarly. This property is also valid for multipartite case. Hence a bound entangled state will be bound entangled under filtering transformations.


### 2.3. Entanglement Witness Based on Correlation Matrix Criterion

Entanglement witness (EW) is another way to describe separability. Based on CM criterion we can further construct entanglement witness in terms of LOOs. EW [92] is an observable of the composite system such that (i) nonnegative expectation values in all separable states and (ii) at least one negative eigenvalue (can recognize at least one entangled state). Consider bipartite systems in ℋAM⊗ℋBN with M≤N.Theorem 2.12.
For any properly selected LOOsGkA and GkB,
(2.87)W=I-α∑k=0N2-1GkA⊗GkB
is an EW [98], where α=MN/((M-1)(N-1)+1) and
(2.88)G0A=1MIM,G0B=1NIN.Proof.
Letρ=∑l,m=0N2-1TlmλlA⊗λmB be a separable state, where λkA/B are normalized generators of SU(M/N) with λ0A=(1/M)IM, λ0B=(1/N)IN. Any other LOOs GkA/B fulfilling (2.88) can be obtained from these λs through orthogonal transformations 𝒪A/B,GkA/B=∑l=0N2-1𝒪klA/Bλl, where 𝒪A/B=(100ℛA/B) and ℛA/B are (N2-1)×(N2-1) orthogonal matrices. We have
(2.89)Tr[ρW]=1-α1MN-α∑k=1N2-1∑l,m=1N2-1ℛklAℛkmBTr[ρ(λlA⊗λmB)]=(M-1)(N-1)(M-1)(N-1)+1-1MN((M-1)(N-1)+1)∑k=1N2-1∑l,m=1N2-1ℛklATlmℛkmB≥MN(M-1)(N-1)-∥T∥KFMN((M-1)(N-1)+1)≥0,
where we have used Tr[ℛT]≤∥T∥KF for any unitary ℛ in the first inequality and the CM criterion in the second inequality.
Now letρ=(1/MN)(IMN+∑i=1M2-1siλiA⊗IN+∑j=1N2-1rjIM⊗λjB+∑i=1M2-1∑j=1N2-1TijλiA⊗λjB) be a state in ℋAM⊗ℋBN which violates the CM criterion. Denote σk(T) as the singular values of T. By singular value decomposition, one has T=U†ΛV*, where Λ is a diagonal matrix with Λkk=σk(T). Now choose LOOs to be GkA=∑lUklλlA, GkB=∑mVkmλmB for k=1,2,…,N2-1 and G0A=(1/M)IM,G0B=(1/N)IN. We obtain
(2.90)Tr[ρW]=1-α1MN-α∑k=1N2-1∑l,m=1N2-1UklVkmTr[ρ(λlA⊗λmB)]=(M-1)(N-1)(M-1)(N-1)+1-1MN((M-1)(N-1)+1)Tr[UTVT]=MN(M-1)(N-1)-∥T∥KFMN((M-1)(N-1)+1)<0,
where the CM criterion has been used in the last step.As the CM criterion can be generalized to multipartite form [97], we can also define entanglement witness for multipartite system in ℋ1d1⊗ℋ2d2⊗⋯⊗ℋNdN. Set d(M)=max{dμi,i=1,2,…,M}. Choose LOOs Gk{μi} for 0≤k≤d(M)2-1 with G0{μi}=(1/dμi)Idμi and define(2.91)W(M)=I-β(M)∑k=0d(M)2-1Gk{μ1}⊗Gk{μ2}⊗⋯⊗Gk{μM},
where β(M)=∏i=1Mdμi/(1+∏i=1M(dμi-1)),2≤M≤N. One can prove that (2.91) is an EW candidate for multipartite states. First we assume that ∥𝒯(M)∥KF=∥𝒯(m0)∥KF. Note that, for any 𝒯(m0), there must exist an elementary transformation P such that ∑k=1d(M)2-1𝒯kk⋯k{μ1μ2⋯μM}=Tr[𝒯(m0)P]. Then for an N-partite separable state we have
(2.92)Tr[ρW(M)]=1-β(M)1∏i=1Mdμi-β(M)1∏i=1MdμiTr[𝒯(m0)P]≥1-β(M)1∏i=1Mdμi-β(M)1∏i=1Mdμi∥𝒯(m0)∥KF≥1-β(M)1∏i=1Mdμi-β(M)1∏i=1Mdμi∏k=1Mdμk(dμk-1)=0
for any 2≤M≤N, where we have taken into account that P is orthogonal and Tr[MU]≤∥M∥KF for any unitary U at the first inequality. The second inequality is due to the generalized CM criterion.By choosing proper LOOs, it is also easy to show thatW(M) has negative eigenvalues. For example, for three-qubit case, taking the normalized Pauli matrices as LOOs, one finds a negative eigenvalue of W(M), (1-3)/2.


## 2.1. Separability Criteria for Mixed States

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


### 2.1.1. Partial Positive Transpose Criterion

The positive partial transpose (PPT) criterion provided by Peres [29] says that if a bipartite state ρAB∈ℋA⊗ℋB is separable, then the new matrix ρABTB with matrix elements defined in some fixed product basis as
(2.7)〈m|〈μ|ρABTB|n〉|ν〉≡〈m|〈ν|ρAB|n〉|μ〉
is also a density matrix (i.e., it has nonnegative spectrum). The operation TB, called partial transpose, just corresponds to the transposition of the indices with respect to the second subsystem B. It has an interpretation as a partial time reversal [80].Afterwards Horodecki et al. showed that Peres' criterion is also sufficient for2×2 and 2×3 bipartite systems [30]. This criterion is now called PPT or Peres-Horodecki (P-H) criterion. For high-dimensional states, the P-H criterion is only necessary. Horodecki has constructed some classes of families of entangled states with positive partial transposes for 3×3 and 2×4 systems [31]. States of this kind are said to be bound entangled (BE).


### 2.1.2. Reduced Density Matrix Criterion

Cerf et al. [81] and M. Horodecki and P. Horodecki [82], independently, introduced a map Γ:ρ→TrB[ρAB]⊗I-ρAB(I⊗TrA[ρAB]-ρAB), which gives rise to a simple necessary condition for separability in arbitrary dimensions, called the reduction criterion. If ρAB is separable, then
(2.8)ρA⊗I-ρAB≥0,I⊗ρB-ρAB≥0,
where ρA=TrB[ρAB],  ρB=TrA[ρAB]. This criterion is simply equivalent to the P-H criterion for 2×n composite systems. It is also sufficient for 2×2 and 2×3 systems. In higher dimensions the reduction criterion is weaker than the P-H criterion.


### 2.1.3. Realignment Criterion

There is yet another class of criteria based on linear contractions on product states. They stem from the new criterion discovered in [33, 83] called computable cross-norm (CCN) criterion or matrix realignment criterion which is operational and independent on PPT test [29]. If a state ρAB is separable, then the realigned matrix ℛ(ρ) with elements ℛ(ρ)ij,kl=ρik,jl has trace norm not greater than one:
(2.9)∥ℛ(ρ)∥KF≤1.
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 anN-dimensional Hilbert space ℋ can be expressed according to the generators of the special unitary group SU(N) [84]. The generators of SU(N) can be introduced according to the transition-projection operators Pjk=|j〉〈k|, where |i〉, i=1,…,N, are the orthonormal eigenstates of a linear Hermitian operator on ℋ. Set
(2.10)ωl=-2l(l+1)(P11+P22+⋯+Pll-lPl+1,l+1),ujk=Pjk+Pkj,vjk=i(Pjk-Pkj),
where 1≤l≤N-1 and 1≤j<k≤N. We get a set of N2-1 operators
(2.11)Γ≡{ωl,ω2,…,ωN-1,u12,u13,…,v12,v13,…},
which satisfies the relations
(2.12)Tr[λi]=0,Tr[λiλj]=2δij,∀λi∈Γ,
and thus generate the SU(N) [85].Any Hermitian operatorρ in ℋ can be represented in terms of these generators of SU(N) as(2.13)ρ=1NIN+12∑j=1N2-1rjλj,
where IN is a unit matrix and r=(r1,r2,…,rN2-1)∈ℝN2-1 and r is called Bloch vector. The set of all the Bloch vectors that constitute a density operator is known as the Bloch vector space B(ℝN2-1).A matrix of the form (2.13) is of unit trace and Hermitian, but it might not be positive. To guarantee the positivity restrictions must be imposed on the Bloch vector. It is shown that B(ℝN2-1) is a subset of the ball DR(ℝN2-1) of radius R=2(1-1/N), which is the minimum ball containing it, and that the ball Dr(ℝN2-1) of radius r=2/N(N-1) is included in B(ℝN2-1) [86], that is,(2.14)Dr(ℝN2-1)⊆B(ℝN2-1)⊆DR(ℝN2-1).Let the dimensions of systemsA,B, and C be dA=N1,dB=N2, and dC=N3, respectively. Any tripartite quantum states ρABC∈ℋA⊗ℋB⊗ℋC can be written as(2.15)ρABC=IN1⊗IN2⊗M0+∑i=1N12-1λi(1)⊗IN2⊗Mi+∑j=1N22-1IN1⊗λj(2)⊗M̃j+∑i=1N12-1‍∑j=1N22-1λi(1)⊗λj(2)⊗Mij,
where λi(1), λj(2) are the generators of SU(N1) and SU(N2); Mi,M̃j, and Mij are operators of ℋC.Theorem 2.2.
Letr∈ℝN12-1, s∈ℝN22-1 and |r|≤2/N1(N1-1), |s|≤2/N2(N2-1). For a tripartite quantum state ρ∈ℋA⊗ℋB⊗ℋC with representation (2.15), one has [87]
(2.16)M0-∑i=1N12-1riMi-∑j=1N22-1sjM̃j+∑i=1N12-1‍∑j=1N22-1risjMij≥0.Proof.
Sincer∈ℝN12-1, s∈ℝN22-1 and |r|≤2/N1(N1-1), |s|≤2/N2(N2-1), we have that A1≡(1/2)((2/N1)I-∑i=1N12-1riλi(1)) and A2≡(1/2)((2/N2)I-∑j=1N22-1sjλj(2)) are positive Hermitian operators. Let A=A1⊗A2⊗IN3. Then AρA≥0 and (AρA)†=AρA. The partial trace of AρA over ℋA (and ℋB) should be also positive. Hence
(2.17)0≤TrAB[AρA]=TrAB[A1⊗A2⊗M0+∑iA1λi(1)A1⊗A2⊗Mi+∑jA1⊗A2λj(2)A2⊗M̃j+∑ijA1λi(1)A1⊗A2λj(2)A2⊗Mij]=M0-∑i=1N12-1riMi-∑j=1N22-1sjM̃j+∑i=1N12-1‍∑j=1N22-1risjMij.Formula (2.16) is valid for any tripartite state. By setting s=0 in (2.16), one can get a result for bipartite systems.Corollary 2.3.
LetρAB∈ℋA⊗ℋB, which can be generally written as ρAB=IN1⊗M0+∑j=1N12-1λj⊗Mj, then, for any r∈ℝN12-1 with |r|≤2/N1(N1-1), M0-∑j=1N12-1rjMj≥0.A separable tripartite stateρABC can be written as
(2.18)ρABC=∑ipi|ψiA〉〈ψiA|⊗|ϕiB〉〈ϕiB|⊗|ωiC〉〈ωiC|.
From (2.13) it can also be represented as(2.19)ρABC=∑ipi12(2N1IN1+∑k=1N12-1ai(k)λk(1))⊗12(2N2IN2+∑l=1N22-1bi(l)λl(2))⊗|ωiC〉〈ωiC|=IN1⊗IN2⊗1N1N2∑ipi|ωiC〉〈ωiC|+∑k=1N12-1λk(1)⊗IN2⊗12N2∑iai(k)pi|ωiC〉〈ωiC|+∑l=1N22-1IN1⊗λl(2)⊗12N1∑ibi(l)pi|ωiC〉〈ωiC|+∑kN12-1∑lN22-1λk(1)⊗λl(2)⊗14∑iai(k)bi(l)pi|ωiC〉〈ωiC|,
where (ai(1),ai(2),…,ai(N12-1)) and (bi(1),bi(2),…,bi(N22-1)) are real vectors on the Bloch sphere satisfying |ai⃗|2=∑j=1N12-1(ai(j))2=2(1-1/N1) and |bi⃗|2=∑j=1N22-1(bi(j))2=2(1-1/N2).Comparing (2.15) with (2.19), we have(2.20)M0=1N1N2∑ipi|ωiC〉〈ωiC|,Mk=12N2∑iai(k)pi|ωiC〉〈ωiC|,M̃l=12N1∑ibi(l)pi|ωiC〉〈ωiC|,Mkl=14∑iai(k)bi(l)pi|ωiC〉〈ωiC|.For any(N12-1)×(N12-1) real matrix R(1) and (N22-1)×(N22-1) real matrix R(2) satisfying (1/(N1-1)2)I-R(1)TR(1)≥0 and (1/(N2-1)2)I-R(2)TR(2)≥0, we define a new matrix(2.21)ℛ=(R(1)000R(2)000T),
where T is a transformation acting on an (N12-1)×(N22-1) matrix M by(2.22)T(M)=R(1)MRT(2).Usingℛ, we define a new operator γℛ:(2.23)γℛ(ρABC)=IN1⊗IN2⊗M0′+∑i=1N12-1λi(1)⊗IN2⊗Mi′+∑j=1N22-1IN1⊗λj(2)⊗M̃j′+∑i=1N12-1‍∑j=1N22-1λi(1)⊗λj(2)⊗Mij′,
where M0′=M0,Mk′=∑m=1N12-1Rkm(1)Mm,M̃l′=∑n=1N22-1Rln(2)M̃n, and Mij′=(T(M))ij=(R(1)MRT(2))ij.Theorem 2.4.
IfρABC is separable, then [87] γℛ(ρABC)≥0.Proof.
From (2.20) and (2.23) we get
(2.24)M0′=M0=1N1N2∑ipi|ωiC〉〈ωiC|,Mk′=12N2∑miRkm(1)ai(m)pi|ωiC〉〈ωiC|,M̃l′=12N1∑niRln(2)bi(n)pi|ωiC〉〈ωiC|,Mkl′=14∑mniRkm(1)ai(m)Rln(2)bi(n)pi|ωiC〉〈ωiC|.
A straightforward calculation gives rise to
(2.25)γℛ(ρABC)=∑ipi12(2N1IN1+∑k=1N12-1∑m=1N12-1Rkm(1)ai(m)λk(1))⊗12(2N2IN2+∑l=1N22-1∑n=1N22-1Rln(2)bi(n)λl(2))⊗|ωiC〉〈ωiC|.
As (1/(N1-1)2)I-R(1)TR(1)≥0 and (1/(N2-1)2)I-R(2)TR(2)≥0, we get
(2.26)|ai′⃗|2=|R(1)ai⃗|2≤1(N1-1)2|ai⃗|2=2N1(N1-1),|bi′⃗|2=|R(2)bi⃗|2≤1(N2-1)2|bi⃗|2=2N2(N2-1).
Therefore γℛ(ρABC) is still a density operator, that is, γℛ(ρABC)≥0.Theorem2.4 gives a necessary separability criterion for general tripartite systems. The result can be also applied to bipartite systems. Let ρAB∈ℋA⊗ℋB, ρAB=IN1⊗M0+∑j=1N12-1λj⊗Mj. For any real (N12-1)×(N12-1) matrix ℛ satisfying (1/(N1-1)2)I-ℛTℛ≥0 and any state ρAB, we define
(2.27)γℛ(ρAB)=IN1⊗M0+∑j=1N12-1λj⊗Mj′,
where Mj′=∑kℛjkMk.Corollary 2.5.
ForρAB∈ℋA⊗ℋB, if there exists an ℛ with (1/(N1-1)2)I-ℛTℛ≥0 such that γℛ(ρAB)<0, then ρAB must be entangled.For2×N systems, the above corollary is reduced to the results in [88]. As an example we consider the 3×3 istropic states
(2.28)ρI=1-p9I3⊗I3+p3∑i,j=13|ii〉〈jj|=I3⊗(19I3)+∑i=15λi⊗(p6λi)-∑i=68λi⊗(p6λi).
If we choose ℛ to be Diag{1/2,1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2}, we get that ρI is entangled for 0.5<p≤1.For tripartite case, we take the following3×3×3 mixed state as an example:
(2.29)ρ=1-p27I27+p|ψ〉〈ψ|,
where |ψ〉=(1/3)(|000〉+|111〉+|222〉)(〈000|+〈111|+〈222|). Taking R(1)=R(2)=Diag{1/2,1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2}, we have that ρ is entangled for 0.6248<p≤1.In fact the criterion for2×N systems [88] is equivalent to the PPT criterion [89]. Similarly Theorem 2.4 is also equivalent to the PPT criterion for 2×2×N systems.


### 2.1.5. Covariance Matrix Criterion

In this subsection we study the separability problem by using the covariance matrix approach. We first give a brief review of covariance matrix criterion proposed in [49]. Let ℋdA and ℋdB be d-dimensional complex vector spaces and ρAB a bipartite quantum state in ℋdA⊗ℋdB. Let Ak (resp., Bk) be d2 observables on ℋdA (resp., ℋdB) such that they form an orthonormal normalized basis of the observable space, satisfying Tr[AkAl]=δk,l (resp., Tr[BkBl]=δk,l). Consider the total set {Mk}={Ak⊗I,I⊗Bk}. It can be proven that [44](2.30)∑k=1N2(Mk)2=dI,∑k=1N2〈Mk〉2=Tr[ρAB2].The covariance matrixγ is defined with entries(2.31)γij(ρAB,{Mk})=〈MiMj〉+〈MjMi〉2-〈Mi〉〈Mj〉,
which has a block structure [49](2.32)γ=(ACCTB),
where A=γ(ρA,{Ak}),B=γ(ρB,{Bk}),Cij=〈Ai⊗Bj〉ρAB-〈Ai〉ρA〈Bj〉ρB,ρA=TrB[ρAB], and ρB=TrA[ρAB]. Such covariance matrix has a concavity property: for a mixed density matrix ρ=∑kpkρk with pk≥0 and ∑kpk=1, one has γ(ρ)≥∑kpkγ(ρk).For a bipartite product stateρAB=ρA⊗ρB, C in (2.32) is zero. Generally if ρAB is separable, then there exist states |ak〉〈ak| on ℋdA,|bk〉〈bk| on ℋdB and pk such that(2.33)γ(ρ)≥κA⊕κB,
where κA=∑pkγ(|ak〉〈ak|,{Ak}), κB=∑pkγ(|bk〉〈bk|,{Bk}).For a separable bipartite state, it has been shown that [49](2.34)∑i=1d2|Cii|≤(1-Tr[ρA2])+(1-Tr[ρB2])2.Criterion (2.34) depends on the choice of the orthonormal normalized basis of the observables. In fact the term ∑i=1d2|Cii| has an upper bound ∥C∥KF which is invariant under unitary transformation and can be attained by choosing proper local orthonormal observable basis, where ∥C∥KF stands for the Ky Fan norm of C, ∥C∥KF=Tr[CC†], with † denoting the transpose and conjugation. It has been shown in [46] that if ρAB is separable, then(2.35)∥C∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2.From the covariance matrix approach, we can also get an alternative criterion. From (2.32) and (2.33) we have that if ρAB is separable, then
(2.36)X≡(A-κACCTB-κB)≥0.
Hence all the 2×2 minor submatrices of X must be positive. Namely, one has
(2.37)|(A-κA)iiCijCji(B-κB)jj|≥0,
that is,  (A-κA)ii(B-κB)jj≥Cij2. Summing over all i, j and using (2.30), we get(2.38)∑i,j=1d2Ci,j2≤(Tr[A]-Tr[κA])(Tr[B]-Tr[κB])=(d-Tr[ρA2]-d+1)(d-Tr[ρB2]-d+1)=(1-Tr[ρA2])(1-Tr[ρB2]).That is,(2.39)∥C∥HS2≤(1-Tr[ρA2])(1-Tr[ρB2]),
where ∥C∥HS stands for the Euclid norm of C, that is, ∥C∥HS=Tr[CC†].Formulae (2.35) and (2.39) are independent and could be complement. When
(2.40)(1-Tr[ρA2])(1-Tr[ρB2])<∥C∥HS≤∥C∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2,(2.39) can recognize the entanglement but (2.35) cannot. When
(2.41)∥C∥HS≤(1-Tr[ρA2])(1-Tr[ρB2])≤(1-Tr[ρA2])+(1-Tr[ρB2])2<∥C∥KF,(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 caseρABC∈ℋdA⊗ℋdB⊗ℋdC. Take d2 observables Ak on ℋA, respectively, Bk on ℋB, respectively, Ck on ℋC. Set {Mk}={Ak⊗I⊗I,I⊗Bk⊗I,I⊗I⊗Ck}. The covariance matrix defined by (2.31) has then the following block structure:(2.42)γ=(ADEDTBFETFTC),
where A=γ(ρA,{Ak}), B=γ(ρB,{Bk}), C=γ(ρC,{Ck}), Dij=〈Ai⊗Bj〉ρAB-〈Ai〉ρA〈Bj〉ρB, Eij=〈Ai⊗Cj〉ρAC-〈Ai〉ρA〈Cj〉ρC, and Fij=〈Bi⊗Cj〉ρBC-〈Bi〉ρB〈Cj〉ρC.Theorem 2.6.
IfρABC is fully separable, then [90]
(2.43)∥D∥HS2≤(1-Tr[ρA2])(1-Tr[ρB2]),(2.44)∥E∥HS2≤(1-Tr[ρA2])(1-Tr[ρC2]),(2.45)∥F∥HS2≤(1-Tr[ρB2])(1-Tr[ρC2]),(2.46)2∥D∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2]),(2.47)2∥E∥KF≤(1-Tr[ρA2])+(1-Tr[ρC2]),(2.48)2∥F∥KF≤(1-Tr[ρB2])+(1-Tr[ρC2]).Proof.
For a tripartite product stateρABC=ρA⊗ρB⊗ρC, D,E, and F in (2.42) are zero. If ρABC is fully separable, then there exist states |ak〉〈ak| in ℋdA, |bk〉〈bk| in ℋdB, and |ck〉〈ck| in ℋdC, and pk such that γ(ρ)≥κA⊕κB⊕κC, where κA=∑pkγ(|ak〉〈ak|,{Ak}), κB=∑pkγ(|bk〉〈bk|,{Bk}), and κC=∑pkγ(|ck〉〈ck|,{Ck}), that is,
(2.49)Y≡(A-κADEDTB-κBFETFTC-κC)≥0.
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
(2.50)|(A-κA)iiDijDji(B-κB)jj|≥0,
that is, (A-κA)ii(B-κB)jj≥|Dij|2. Summing over all i, j and using (2.30), we get
(2.51)∥D∥HS2=∑i,j=1d2Di,j2≤(Tr[A]-Tr[κA])(Tr[B]-Tr[κB]=(d-Tr[ρA2]-d+1)(d-Tr[ρB2]-d+1)=(1-Tr[ρA2])(1-Tr[ρB2]),
which proves (2.43). Equations (2.44) and (2.45) can be similarly proved.
From (2.50) we also have (A-κA)ii+(B-κB)ii≥2|Dii|. Therefore
(2.52)∑i|Dii|≤(Tr[A]-Tr[κA])+(Tr[B]-Tr[κB]2=(d-Tr[ρA2]-d+1)+(d-Tr[ρB2]-d+1)2=(1-Tr[ρA2])+(1-Tr[ρB2])2.
Note that ∑i=1d2Dii≤∑i=1d2|Dii|. By using that Tr[MU]≤∥M∥KF=Tr[MM†] for any matrix M and any unitary U [91], we have ∑i=1d2Dii≤∥D∥KF.
LetD=U†ΛV be the singular value decomposition of D. Make a transformation of the orthonormal normalized basis of the local orthonormal observable space Ãi=∑lUilAl and B̃j=∑mVjm*Bm. In the new basis we have
(2.53)D̃ij=∑lmUilDlmVjm=(UDV†)ij=Λij.
Then (2.52) becomes
(2.54)∑i=1d2D̃ii=∥D∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2,
which proves (2.46). Equations (2.47) and (2.48) can be similarly treated.We consider now the case thatρABC is bipartite separable.Theorem 2.7.
IfρABC is a bipartite separable state with respect to the bipartite partition of the sub-systems A and BC (resp., AB and C; resp., AC and B), then (2.43), (2.44) and (2.46), (2.47) (resp., (2.44), (2.45) and (2.47), (2.48); resp., (2.43), (2.45) and (2.46), (2.48)) must hold [90].Proof.
We prove the case thatρABC is bipartite separable with respect to the A system and BC systems partition. The other cases can be similarly treated. In this case the matrices D and E in the covariance matrix (2.42) are zero. ρABC takes the form ρABC=∑mpmρAm⊗ρBCm. Define κA=∑pmγ(ρAm,{Ak}), κBC=∑pmγ(ρBCm,{Bk⊗I,I⊗Ck}). κBC has a form
(2.55)κBC=(κBF′(F′)TκC),
where κB=∑pkγ(|bk〉〈bk|,{Bk}) and κC=∑pkγ(|ck〉〈ck|,{Ck}), (F′)ij=∑mpm(〈Bi⊗Cj〉ρBCm-〈Bi〉ρBm〈Cj〉ρCm). By using the concavity of covariance matrix we have
(2.56)γ(ρABC)≥∑mpmγ(ρAm⊗ρBCm)=(κA000κBF′0(F′)TκC).
Accounting to the method used in proving Theorem 2, we get (2.43), (2.44), and (2.46), (2.47).From Theorems2.6 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 Theorem2.6 can be generalized to general multipartite case ρ∈ℋd(1)⊗ℋd(2)⊗⋯⊗ℋd(N). Define Âαi=I⊗I⊗⋯λα⊗I⊗⋯⊗I, where λ0=I/d, λα (α=1,2,…,d2-1) are the normalized generators of SU(d) satisfying Tr[λαλβ]=δαβ and acting on the ith system ℋd(i), i=1,2,…,N. Denote {Mk} as the set of all Âαi. Then the covariance matrix of ρ can be written as(2.57)γ(ρ)=(𝒜11𝒜12⋯𝒜1N𝒜12T𝒜22⋯𝒜2N⋮⋮⋮𝒜1NT𝒜2NT⋯𝒜NN),
where 𝒜ii=γ(ρ,{Âki}) and (𝒜ij)mn=〈Âmi⊗Ânj〉-〈Âmi〉〈Ânj〉 for i≠j.For a product stateρ12⋯N, 𝒜ij, i≠j, in (2.57) are zero matrices. Define
(2.58)κ𝒜ii=∑kpkγ(|ψki〉〈ψki|,{Âli}).
Then for a fully separable multipartite state ρ=∑kpk|ψk1〉〈ψk1|⊗|ψk2〉〈ψk2|⊗⋯⊗|ψkN〉〈ψkN|, one has
(2.59)Z=(𝒜11-κ𝒜11𝒜12⋯𝒜1N𝒜12T𝒜22-κ𝒜22⋯𝒜2N⋮⋮⋮𝒜1NT𝒜2NT⋯𝒜NN-κ𝒜NN)≥0,
from which we have the following separability criterion for multipartite systems.Theorem 2.9.
If a stateρ∈ℋd(1)⊗ℋd(2)⊗⋯⊗ℋd(N) is fully separable, then the following inequalities
(2.60)∥𝒜ij∥HS2≤(1-Tr[ρi2])(1-Tr[ρj2]),∥𝒜ij∥KF≤(1-Tr[ρi2])+(1-Tr[ρj2])2
must be fulfilled for any i≠j [90].


## 2.1.1. Partial Positive Transpose Criterion

The positive partial transpose (PPT) criterion provided by Peres [29] says that if a bipartite state ρAB∈ℋA⊗ℋB is separable, then the new matrix ρABTB with matrix elements defined in some fixed product basis as
(2.7)〈m|〈μ|ρABTB|n〉|ν〉≡〈m|〈ν|ρAB|n〉|μ〉
is also a density matrix (i.e., it has nonnegative spectrum). The operation TB, called partial transpose, just corresponds to the transposition of the indices with respect to the second subsystem B. It has an interpretation as a partial time reversal [80].Afterwards Horodecki et al. showed that Peres' criterion is also sufficient for2×2 and 2×3 bipartite systems [30]. This criterion is now called PPT or Peres-Horodecki (P-H) criterion. For high-dimensional states, the P-H criterion is only necessary. Horodecki has constructed some classes of families of entangled states with positive partial transposes for 3×3 and 2×4 systems [31]. States of this kind are said to be bound entangled (BE).


## 2.1.2. Reduced Density Matrix Criterion

Cerf et al. [81] and M. Horodecki and P. Horodecki [82], independently, introduced a map Γ:ρ→TrB[ρAB]⊗I-ρAB(I⊗TrA[ρAB]-ρAB), which gives rise to a simple necessary condition for separability in arbitrary dimensions, called the reduction criterion. If ρAB is separable, then
(2.8)ρA⊗I-ρAB≥0,I⊗ρB-ρAB≥0,
where ρA=TrB[ρAB],  ρB=TrA[ρAB]. This criterion is simply equivalent to the P-H criterion for 2×n composite systems. It is also sufficient for 2×2 and 2×3 systems. In higher dimensions the reduction criterion is weaker than the P-H criterion.


## 2.1.3. Realignment Criterion

There is yet another class of criteria based on linear contractions on product states. They stem from the new criterion discovered in [33, 83] called computable cross-norm (CCN) criterion or matrix realignment criterion which is operational and independent on PPT test [29]. If a state ρAB is separable, then the realigned matrix ℛ(ρ) with elements ℛ(ρ)ij,kl=ρik,jl has trace norm not greater than one:
(2.9)∥ℛ(ρ)∥KF≤1.
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 anN-dimensional Hilbert space ℋ can be expressed according to the generators of the special unitary group SU(N) [84]. The generators of SU(N) can be introduced according to the transition-projection operators Pjk=|j〉〈k|, where |i〉, i=1,…,N, are the orthonormal eigenstates of a linear Hermitian operator on ℋ. Set
(2.10)ωl=-2l(l+1)(P11+P22+⋯+Pll-lPl+1,l+1),ujk=Pjk+Pkj,vjk=i(Pjk-Pkj),
where 1≤l≤N-1 and 1≤j<k≤N. We get a set of N2-1 operators
(2.11)Γ≡{ωl,ω2,…,ωN-1,u12,u13,…,v12,v13,…},
which satisfies the relations
(2.12)Tr[λi]=0,Tr[λiλj]=2δij,∀λi∈Γ,
and thus generate the SU(N) [85].Any Hermitian operatorρ in ℋ can be represented in terms of these generators of SU(N) as(2.13)ρ=1NIN+12∑j=1N2-1rjλj,
where IN is a unit matrix and r=(r1,r2,…,rN2-1)∈ℝN2-1 and r is called Bloch vector. The set of all the Bloch vectors that constitute a density operator is known as the Bloch vector space B(ℝN2-1).A matrix of the form (2.13) is of unit trace and Hermitian, but it might not be positive. To guarantee the positivity restrictions must be imposed on the Bloch vector. It is shown that B(ℝN2-1) is a subset of the ball DR(ℝN2-1) of radius R=2(1-1/N), which is the minimum ball containing it, and that the ball Dr(ℝN2-1) of radius r=2/N(N-1) is included in B(ℝN2-1) [86], that is,(2.14)Dr(ℝN2-1)⊆B(ℝN2-1)⊆DR(ℝN2-1).Let the dimensions of systemsA,B, and C be dA=N1,dB=N2, and dC=N3, respectively. Any tripartite quantum states ρABC∈ℋA⊗ℋB⊗ℋC can be written as(2.15)ρABC=IN1⊗IN2⊗M0+∑i=1N12-1λi(1)⊗IN2⊗Mi+∑j=1N22-1IN1⊗λj(2)⊗M̃j+∑i=1N12-1‍∑j=1N22-1λi(1)⊗λj(2)⊗Mij,
where λi(1), λj(2) are the generators of SU(N1) and SU(N2); Mi,M̃j, and Mij are operators of ℋC.Theorem 2.2.
Letr∈ℝN12-1, s∈ℝN22-1 and |r|≤2/N1(N1-1), |s|≤2/N2(N2-1). For a tripartite quantum state ρ∈ℋA⊗ℋB⊗ℋC with representation (2.15), one has [87]
(2.16)M0-∑i=1N12-1riMi-∑j=1N22-1sjM̃j+∑i=1N12-1‍∑j=1N22-1risjMij≥0.Proof.
Sincer∈ℝN12-1, s∈ℝN22-1 and |r|≤2/N1(N1-1), |s|≤2/N2(N2-1), we have that A1≡(1/2)((2/N1)I-∑i=1N12-1riλi(1)) and A2≡(1/2)((2/N2)I-∑j=1N22-1sjλj(2)) are positive Hermitian operators. Let A=A1⊗A2⊗IN3. Then AρA≥0 and (AρA)†=AρA. The partial trace of AρA over ℋA (and ℋB) should be also positive. Hence
(2.17)0≤TrAB[AρA]=TrAB[A1⊗A2⊗M0+∑iA1λi(1)A1⊗A2⊗Mi+∑jA1⊗A2λj(2)A2⊗M̃j+∑ijA1λi(1)A1⊗A2λj(2)A2⊗Mij]=M0-∑i=1N12-1riMi-∑j=1N22-1sjM̃j+∑i=1N12-1‍∑j=1N22-1risjMij.Formula (2.16) is valid for any tripartite state. By setting s=0 in (2.16), one can get a result for bipartite systems.Corollary 2.3.
LetρAB∈ℋA⊗ℋB, which can be generally written as ρAB=IN1⊗M0+∑j=1N12-1λj⊗Mj, then, for any r∈ℝN12-1 with |r|≤2/N1(N1-1), M0-∑j=1N12-1rjMj≥0.A separable tripartite stateρABC can be written as
(2.18)ρABC=∑ipi|ψiA〉〈ψiA|⊗|ϕiB〉〈ϕiB|⊗|ωiC〉〈ωiC|.
From (2.13) it can also be represented as(2.19)ρABC=∑ipi12(2N1IN1+∑k=1N12-1ai(k)λk(1))⊗12(2N2IN2+∑l=1N22-1bi(l)λl(2))⊗|ωiC〉〈ωiC|=IN1⊗IN2⊗1N1N2∑ipi|ωiC〉〈ωiC|+∑k=1N12-1λk(1)⊗IN2⊗12N2∑iai(k)pi|ωiC〉〈ωiC|+∑l=1N22-1IN1⊗λl(2)⊗12N1∑ibi(l)pi|ωiC〉〈ωiC|+∑kN12-1∑lN22-1λk(1)⊗λl(2)⊗14∑iai(k)bi(l)pi|ωiC〉〈ωiC|,
where (ai(1),ai(2),…,ai(N12-1)) and (bi(1),bi(2),…,bi(N22-1)) are real vectors on the Bloch sphere satisfying |ai⃗|2=∑j=1N12-1(ai(j))2=2(1-1/N1) and |bi⃗|2=∑j=1N22-1(bi(j))2=2(1-1/N2).Comparing (2.15) with (2.19), we have(2.20)M0=1N1N2∑ipi|ωiC〉〈ωiC|,Mk=12N2∑iai(k)pi|ωiC〉〈ωiC|,M̃l=12N1∑ibi(l)pi|ωiC〉〈ωiC|,Mkl=14∑iai(k)bi(l)pi|ωiC〉〈ωiC|.For any(N12-1)×(N12-1) real matrix R(1) and (N22-1)×(N22-1) real matrix R(2) satisfying (1/(N1-1)2)I-R(1)TR(1)≥0 and (1/(N2-1)2)I-R(2)TR(2)≥0, we define a new matrix(2.21)ℛ=(R(1)000R(2)000T),
where T is a transformation acting on an (N12-1)×(N22-1) matrix M by(2.22)T(M)=R(1)MRT(2).Usingℛ, we define a new operator γℛ:(2.23)γℛ(ρABC)=IN1⊗IN2⊗M0′+∑i=1N12-1λi(1)⊗IN2⊗Mi′+∑j=1N22-1IN1⊗λj(2)⊗M̃j′+∑i=1N12-1‍∑j=1N22-1λi(1)⊗λj(2)⊗Mij′,
where M0′=M0,Mk′=∑m=1N12-1Rkm(1)Mm,M̃l′=∑n=1N22-1Rln(2)M̃n, and Mij′=(T(M))ij=(R(1)MRT(2))ij.Theorem 2.4.
IfρABC is separable, then [87] γℛ(ρABC)≥0.Proof.
From (2.20) and (2.23) we get
(2.24)M0′=M0=1N1N2∑ipi|ωiC〉〈ωiC|,Mk′=12N2∑miRkm(1)ai(m)pi|ωiC〉〈ωiC|,M̃l′=12N1∑niRln(2)bi(n)pi|ωiC〉〈ωiC|,Mkl′=14∑mniRkm(1)ai(m)Rln(2)bi(n)pi|ωiC〉〈ωiC|.
A straightforward calculation gives rise to
(2.25)γℛ(ρABC)=∑ipi12(2N1IN1+∑k=1N12-1∑m=1N12-1Rkm(1)ai(m)λk(1))⊗12(2N2IN2+∑l=1N22-1∑n=1N22-1Rln(2)bi(n)λl(2))⊗|ωiC〉〈ωiC|.
As (1/(N1-1)2)I-R(1)TR(1)≥0 and (1/(N2-1)2)I-R(2)TR(2)≥0, we get
(2.26)|ai′⃗|2=|R(1)ai⃗|2≤1(N1-1)2|ai⃗|2=2N1(N1-1),|bi′⃗|2=|R(2)bi⃗|2≤1(N2-1)2|bi⃗|2=2N2(N2-1).
Therefore γℛ(ρABC) is still a density operator, that is, γℛ(ρABC)≥0.Theorem2.4 gives a necessary separability criterion for general tripartite systems. The result can be also applied to bipartite systems. Let ρAB∈ℋA⊗ℋB, ρAB=IN1⊗M0+∑j=1N12-1λj⊗Mj. For any real (N12-1)×(N12-1) matrix ℛ satisfying (1/(N1-1)2)I-ℛTℛ≥0 and any state ρAB, we define
(2.27)γℛ(ρAB)=IN1⊗M0+∑j=1N12-1λj⊗Mj′,
where Mj′=∑kℛjkMk.Corollary 2.5.
ForρAB∈ℋA⊗ℋB, if there exists an ℛ with (1/(N1-1)2)I-ℛTℛ≥0 such that γℛ(ρAB)<0, then ρAB must be entangled.For2×N systems, the above corollary is reduced to the results in [88]. As an example we consider the 3×3 istropic states
(2.28)ρI=1-p9I3⊗I3+p3∑i,j=13|ii〉〈jj|=I3⊗(19I3)+∑i=15λi⊗(p6λi)-∑i=68λi⊗(p6λi).
If we choose ℛ to be Diag{1/2,1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2}, we get that ρI is entangled for 0.5<p≤1.For tripartite case, we take the following3×3×3 mixed state as an example:
(2.29)ρ=1-p27I27+p|ψ〉〈ψ|,
where |ψ〉=(1/3)(|000〉+|111〉+|222〉)(〈000|+〈111|+〈222|). Taking R(1)=R(2)=Diag{1/2,1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2}, we have that ρ is entangled for 0.6248<p≤1.In fact the criterion for2×N systems [88] is equivalent to the PPT criterion [89]. Similarly Theorem 2.4 is also equivalent to the PPT criterion for 2×2×N systems.


## 2.1.5. Covariance Matrix Criterion

In this subsection we study the separability problem by using the covariance matrix approach. We first give a brief review of covariance matrix criterion proposed in [49]. Let ℋdA and ℋdB be d-dimensional complex vector spaces and ρAB a bipartite quantum state in ℋdA⊗ℋdB. Let Ak (resp., Bk) be d2 observables on ℋdA (resp., ℋdB) such that they form an orthonormal normalized basis of the observable space, satisfying Tr[AkAl]=δk,l (resp., Tr[BkBl]=δk,l). Consider the total set {Mk}={Ak⊗I,I⊗Bk}. It can be proven that [44](2.30)∑k=1N2(Mk)2=dI,∑k=1N2〈Mk〉2=Tr[ρAB2].The covariance matrixγ is defined with entries(2.31)γij(ρAB,{Mk})=〈MiMj〉+〈MjMi〉2-〈Mi〉〈Mj〉,
which has a block structure [49](2.32)γ=(ACCTB),
where A=γ(ρA,{Ak}),B=γ(ρB,{Bk}),Cij=〈Ai⊗Bj〉ρAB-〈Ai〉ρA〈Bj〉ρB,ρA=TrB[ρAB], and ρB=TrA[ρAB]. Such covariance matrix has a concavity property: for a mixed density matrix ρ=∑kpkρk with pk≥0 and ∑kpk=1, one has γ(ρ)≥∑kpkγ(ρk).For a bipartite product stateρAB=ρA⊗ρB, C in (2.32) is zero. Generally if ρAB is separable, then there exist states |ak〉〈ak| on ℋdA,|bk〉〈bk| on ℋdB and pk such that(2.33)γ(ρ)≥κA⊕κB,
where κA=∑pkγ(|ak〉〈ak|,{Ak}), κB=∑pkγ(|bk〉〈bk|,{Bk}).For a separable bipartite state, it has been shown that [49](2.34)∑i=1d2|Cii|≤(1-Tr[ρA2])+(1-Tr[ρB2])2.Criterion (2.34) depends on the choice of the orthonormal normalized basis of the observables. In fact the term ∑i=1d2|Cii| has an upper bound ∥C∥KF which is invariant under unitary transformation and can be attained by choosing proper local orthonormal observable basis, where ∥C∥KF stands for the Ky Fan norm of C, ∥C∥KF=Tr[CC†], with † denoting the transpose and conjugation. It has been shown in [46] that if ρAB is separable, then(2.35)∥C∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2.From the covariance matrix approach, we can also get an alternative criterion. From (2.32) and (2.33) we have that if ρAB is separable, then
(2.36)X≡(A-κACCTB-κB)≥0.
Hence all the 2×2 minor submatrices of X must be positive. Namely, one has
(2.37)|(A-κA)iiCijCji(B-κB)jj|≥0,
that is,  (A-κA)ii(B-κB)jj≥Cij2. Summing over all i, j and using (2.30), we get(2.38)∑i,j=1d2Ci,j2≤(Tr[A]-Tr[κA])(Tr[B]-Tr[κB])=(d-Tr[ρA2]-d+1)(d-Tr[ρB2]-d+1)=(1-Tr[ρA2])(1-Tr[ρB2]).That is,(2.39)∥C∥HS2≤(1-Tr[ρA2])(1-Tr[ρB2]),
where ∥C∥HS stands for the Euclid norm of C, that is, ∥C∥HS=Tr[CC†].Formulae (2.35) and (2.39) are independent and could be complement. When
(2.40)(1-Tr[ρA2])(1-Tr[ρB2])<∥C∥HS≤∥C∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2,(2.39) can recognize the entanglement but (2.35) cannot. When
(2.41)∥C∥HS≤(1-Tr[ρA2])(1-Tr[ρB2])≤(1-Tr[ρA2])+(1-Tr[ρB2])2<∥C∥KF,(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 caseρABC∈ℋdA⊗ℋdB⊗ℋdC. Take d2 observables Ak on ℋA, respectively, Bk on ℋB, respectively, Ck on ℋC. Set {Mk}={Ak⊗I⊗I,I⊗Bk⊗I,I⊗I⊗Ck}. The covariance matrix defined by (2.31) has then the following block structure:(2.42)γ=(ADEDTBFETFTC),
where A=γ(ρA,{Ak}), B=γ(ρB,{Bk}), C=γ(ρC,{Ck}), Dij=〈Ai⊗Bj〉ρAB-〈Ai〉ρA〈Bj〉ρB, Eij=〈Ai⊗Cj〉ρAC-〈Ai〉ρA〈Cj〉ρC, and Fij=〈Bi⊗Cj〉ρBC-〈Bi〉ρB〈Cj〉ρC.Theorem 2.6.
IfρABC is fully separable, then [90]
(2.43)∥D∥HS2≤(1-Tr[ρA2])(1-Tr[ρB2]),(2.44)∥E∥HS2≤(1-Tr[ρA2])(1-Tr[ρC2]),(2.45)∥F∥HS2≤(1-Tr[ρB2])(1-Tr[ρC2]),(2.46)2∥D∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2]),(2.47)2∥E∥KF≤(1-Tr[ρA2])+(1-Tr[ρC2]),(2.48)2∥F∥KF≤(1-Tr[ρB2])+(1-Tr[ρC2]).Proof.
For a tripartite product stateρABC=ρA⊗ρB⊗ρC, D,E, and F in (2.42) are zero. If ρABC is fully separable, then there exist states |ak〉〈ak| in ℋdA, |bk〉〈bk| in ℋdB, and |ck〉〈ck| in ℋdC, and pk such that γ(ρ)≥κA⊕κB⊕κC, where κA=∑pkγ(|ak〉〈ak|,{Ak}), κB=∑pkγ(|bk〉〈bk|,{Bk}), and κC=∑pkγ(|ck〉〈ck|,{Ck}), that is,
(2.49)Y≡(A-κADEDTB-κBFETFTC-κC)≥0.
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
(2.50)|(A-κA)iiDijDji(B-κB)jj|≥0,
that is, (A-κA)ii(B-κB)jj≥|Dij|2. Summing over all i, j and using (2.30), we get
(2.51)∥D∥HS2=∑i,j=1d2Di,j2≤(Tr[A]-Tr[κA])(Tr[B]-Tr[κB]=(d-Tr[ρA2]-d+1)(d-Tr[ρB2]-d+1)=(1-Tr[ρA2])(1-Tr[ρB2]),
which proves (2.43). Equations (2.44) and (2.45) can be similarly proved.
From (2.50) we also have (A-κA)ii+(B-κB)ii≥2|Dii|. Therefore
(2.52)∑i|Dii|≤(Tr[A]-Tr[κA])+(Tr[B]-Tr[κB]2=(d-Tr[ρA2]-d+1)+(d-Tr[ρB2]-d+1)2=(1-Tr[ρA2])+(1-Tr[ρB2])2.
Note that ∑i=1d2Dii≤∑i=1d2|Dii|. By using that Tr[MU]≤∥M∥KF=Tr[MM†] for any matrix M and any unitary U [91], we have ∑i=1d2Dii≤∥D∥KF.
LetD=U†ΛV be the singular value decomposition of D. Make a transformation of the orthonormal normalized basis of the local orthonormal observable space Ãi=∑lUilAl and B̃j=∑mVjm*Bm. In the new basis we have
(2.53)D̃ij=∑lmUilDlmVjm=(UDV†)ij=Λij.
Then (2.52) becomes
(2.54)∑i=1d2D̃ii=∥D∥KF≤(1-Tr[ρA2])+(1-Tr[ρB2])2,
which proves (2.46). Equations (2.47) and (2.48) can be similarly treated.We consider now the case thatρABC is bipartite separable.Theorem 2.7.
IfρABC is a bipartite separable state with respect to the bipartite partition of the sub-systems A and BC (resp., AB and C; resp., AC and B), then (2.43), (2.44) and (2.46), (2.47) (resp., (2.44), (2.45) and (2.47), (2.48); resp., (2.43), (2.45) and (2.46), (2.48)) must hold [90].Proof.
We prove the case thatρABC is bipartite separable with respect to the A system and BC systems partition. The other cases can be similarly treated. In this case the matrices D and E in the covariance matrix (2.42) are zero. ρABC takes the form ρABC=∑mpmρAm⊗ρBCm. Define κA=∑pmγ(ρAm,{Ak}), κBC=∑pmγ(ρBCm,{Bk⊗I,I⊗Ck}). κBC has a form
(2.55)κBC=(κBF′(F′)TκC),
where κB=∑pkγ(|bk〉〈bk|,{Bk}) and κC=∑pkγ(|ck〉〈ck|,{Ck}), (F′)ij=∑mpm(〈Bi⊗Cj〉ρBCm-〈Bi〉ρBm〈Cj〉ρCm). By using the concavity of covariance matrix we have
(2.56)γ(ρABC)≥∑mpmγ(ρAm⊗ρBCm)=(κA000κBF′0(F′)TκC).
Accounting to the method used in proving Theorem 2, we get (2.43), (2.44), and (2.46), (2.47).From Theorems2.6 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 Theorem2.6 can be generalized to general multipartite case ρ∈ℋd(1)⊗ℋd(2)⊗⋯⊗ℋd(N). Define Âαi=I⊗I⊗⋯λα⊗I⊗⋯⊗I, where λ0=I/d, λα (α=1,2,…,d2-1) are the normalized generators of SU(d) satisfying Tr[λαλβ]=δαβ and acting on the ith system ℋd(i), i=1,2,…,N. Denote {Mk} as the set of all Âαi. Then the covariance matrix of ρ can be written as(2.57)γ(ρ)=(𝒜11𝒜12⋯𝒜1N𝒜12T𝒜22⋯𝒜2N⋮⋮⋮𝒜1NT𝒜2NT⋯𝒜NN),
where 𝒜ii=γ(ρ,{Âki}) and (𝒜ij)mn=〈Âmi⊗Ânj〉-〈Âmi〉〈Ânj〉 for i≠j.For a product stateρ12⋯N, 𝒜ij, i≠j, in (2.57) are zero matrices. Define
(2.58)κ𝒜ii=∑kpkγ(|ψki〉〈ψki|,{Âli}).
Then for a fully separable multipartite state ρ=∑kpk|ψk1〉〈ψk1|⊗|ψk2〉〈ψk2|⊗⋯⊗|ψkN〉〈ψkN|, one has
(2.59)Z=(𝒜11-κ𝒜11𝒜12⋯𝒜1N𝒜12T𝒜22-κ𝒜22⋯𝒜2N⋮⋮⋮𝒜1NT𝒜2NT⋯𝒜NN-κ𝒜NN)≥0,
from which we have the following separability criterion for multipartite systems.Theorem 2.9.
If a stateρ∈ℋd(1)⊗ℋd(2)⊗⋯⊗ℋd(N) is fully separable, then the following inequalities
(2.60)∥𝒜ij∥HS2≤(1-Tr[ρi2])(1-Tr[ρj2]),∥𝒜ij∥KF≤(1-Tr[ρi2])+(1-Tr[ρj2])2
must be fulfilled for any i≠j [90].


## 2.2. Normal Form of Quantum States

In this subsection we show that the correlation matrix (CM) criterion can be improved from the normal form obtained under filtering transformations. Based on CM criterion entanglement witness in terms of local orthogonal observables (LOOs) [92] for both bipartite and multipartite systems can be also constructed.For bipartite case,ρ∈ℋ=ℋA⊗ℋB with dimℋA=M, dimℋB=N, and M≤N is mapped to the following form under local filtering transformations [93]:(2.61)ρ→ρ̃=(FA⊗FB)ρ(FA⊗FB)†Tr[(FA⊗FB)ρ(FA⊗FB)†],
where FA/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 positiveρ into a normal form [94]:(2.62)ρ̃=(FA⊗FB)ρ(FA⊗FB)†Tr[(FA⊗FB)ρ(FA⊗FB)†]=1MN(I+∑i=1M2-1ξiGiA⊗GiB),
where ξi≥0 and GiA and GiB are some traceless orthogonal observables. The matrices FA and FB can be obtained by minimizing the function
(2.63)f(A,B)=Tr[ρ(A⊗B)](detA)1/M(detB)1/N,
where A=FA†FA and B=FB†FB. In fact, one can choose FA0≡|det(ρA)|1/2M(ρA)-1, and FB0≡|det(ρB′)|1/2N(ρB′)-1, where ρB′=TrA[I⊗(ρA)-1ρI⊗(ρA)-1]. Then by iterations one can get the optimal A and B. In particular, there is a matlab code available in [95].For bipartite separable statesρ, the CM separability criterion [96] says that(2.64)∥T∥KF≤MN(M-1)(N-1),
where T is an (M2-1)×(N2-1) matrix with Tij=MN·Tr[ρλiA⊗λjB], ∥T∥KF stands for the trace norm of T, λkA/Bs are the generators of SU(M/N) and have been chosen to be normalized, and Tr[λk(A/B)λl(A/B)]=δkl.As the filtering transformation does not change the separability of a state, one can study the separability ofρ̃ instead of ρ. Under the normal form (2.62) the criterion (2.64) becomes(2.65)∑iξi≤MN(M-1)(N-1).In [44] a separability criterion based on local uncertainty relation (LUR) has been obtained. It says that, for any separable state ρ,(2.66)1-∑k〈GkA⊗GkB〉-12〈GkA⊗I-I⊗GkB〉2≥0,
where GkA/Bs are LOOs such as the normalized generators of SU(M/N) and GkA=0 for k=M2+1,…,N2. The criterion is shown to be strictly stronger than the realignment criterion [61]. Under the normal form (2.62) criterion (2.66) becomes
(2.67)1-∑k〈GkA⊗GkB〉-12〈GkA⊗I-I⊗GkB〉2=1-1MN∑kξk-12(1M+1N)≥0,
that is,(2.68)∑k‍ξk≤MN-M+N2.
As MN(M-1)(N-1)≤MN-(M+N)/2 holds for any M and N, from (2.65) and (2.68) it is obvious that the CM criterion recognizes entanglement better when the normal form is taken into account.We now consider multipartite systems. Letρ be a strictly positive density matrix in ℋ=ℋ1⊗ℋ2⊗⋯⊗ℋN and dimℋi=di. ρ can be generally expressed in terms of the SU(n) generators λαk [97] as(2.69)ρ=1ΠiNdi(⊗jNIdj+∑{μ1}∑α1𝒯α1{μ1}λα1{μ1}+∑{μ1μ2}∑α1α2𝒯α1α2{μ1μ2}λα1{μ1}λα2{μ2}+∑{μ1μ2μ3}∑α1α2α3𝒯α1α2α3{μ1μ2μ3}λα1{μ1}λα2{μ2}λα3{μ3}+⋯+∑{μ1μ2⋯μM}∑α1α2⋯αM𝒯α1α2⋯αM{μ1μ2⋯μM}λα1{μ1}λα2{μ2}⋯λαM{μM}+⋯+∑α1α2⋯αN𝒯α1α2⋯αM{1,2,…,N}λα1{1}λα2{2}⋯λαN{N}),
where λαk{μk}=Id1⊗Id2⊗⋯⊗λαk⊗Idμk+1⊗⋯⊗IdN with λαk appears at the μkth position and(2.70)𝒯α1α2⋯αM{μ1μ2⋯μM}=∏i=1Mdμi2MTr[ρλα1{μ1}λα2{μ2}⋯λαM{μM}].The generalized CM criterion says that ifρ in (2.69) is fully separable, then(2.71)∥𝒯{μ1,μ2,…,μM}∥KF≤12M∏k=1Mdμk(dμk-1),
for 2≤M≤N,{μ1,μ2,…,μM}⊂{1,2,…,N}. The KF norm is defined by
(2.72)∥𝒯{μ1,μ2,…,μM}∥KF=maxm=1,2,…,M∥𝒯(m)∥KF,
where 𝒯(m) is a kind of matrix unfolding of 𝒯{μ1,μ2,…,μM}.The criterion (2.71) can be improved by investigating the normal form of (2.69).Theorem 2.10.
By filtering transformations of the form(2.73)ρ̃=F1⊗F2⊗⋯⊗FNρF1†⊗F2†⊗FN†,
where Fi∈GL(di,ℂ),i=1,2,…N, followed by normalization, any strictly positive state ρ can be transformed into a normal form [98]:
(2.74)ρ=1ΠiNdi(⊗jNIdj+∑{μ1μ2}∑α1α2𝒯α1α2{μ1μ2}λα1{μ1}λα2{μ2}+∑{μ1μ2μ3}∑α1α2α3𝒯α1α2α3{μ1μ2μ3}λα1{μ1}λα2{μ2}λα3{μ3}+⋯+∑{μ1μ2⋯μM}∑α1α2⋯αM𝒯α1α2⋯αM{μ1μ2⋯μM}λα1{μ1}λα2{μ2}⋯λαM{μM}+⋯+∑α1α2⋯αN𝒯α1α2⋯αM{1,2,…,N}λα1{1}λα2{2}⋯λαN{N}).Proof.
LetD1,D2,…,DN be the sets of density matrices of the N subsystems. The cartesian product D1×D2×⋯×DN consisting of all product density matrices ρ1⊗ρ2⊗⋯⊗ρN with normalization Tr[ρi]=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 ρi:
(2.75)f(ρ1,ρ2,…,ρN)=Tr[ρ(ρ1⊗ρ2⊗⋯⊗ρN)]∏i=1Ndet(ρi)1/di.
The function is well defined on the interior of D1×D2×⋯×DN where detρi>0. As ρ is assumed to be strictly positive, we have Tr[ρ(ρ1⊗ρ2⊗⋯⊗ρN)]>0. Since D1×D2×⋯×DN is compact, we have Tr[ρ(ρ1⊗ρ2⊗⋯⊗ρN)]≥C>0 with a lower bound C depending on ρ.
It follows thatf→∞ on the boundary of D1×D2×⋯×DN where at least one of the ρis satisfies detρi=0. It follows further that f has a positive minimum on the interior of D1×D2×⋯×DN with the minimum value attained for at least one product density matrix τ1⊗τ2⊗⋯⊗τN with detτi>0, i=1,2,…,N. Any positive density matrix τi with detτi>0 can be factorized in terms of Hermitian matrices Fi as
(2.76)τi=Fi†Fi,
where Fi∈GL(di,ℂ). Denote that F=F1⊗F2⊗⋯⊗FN, so that τ1⊗τ2⊗⋯⊗τN=F†F. Set ρ̃=FρF† and define
(2.77)f̃(ρ1,ρ2,…,ρN)=Tr[ρ̃(ρ1⊗ρ2⊗⋯⊗ρN)]∏i=1Ndet(ρi)1/di=∏i=1Ndet(τi)1/di·Tr[ρ(F1†ρ1F1⊗F2†ρ2F2⊗⋯⊗FN†ρNFN)]∏i=1Ndet(τi)1/didet(ρi)1/di=∏i=1Ndet(τi)1/di·f(F1†ρ1F1,F2†ρ2F2,…,FN†ρNFN).
We see that, whenFi†ρiFi=τi, f̃ has a minimum and
(2.78)ρ1⊗ρ2⊗⋯⊗ρN=(F†)-1τ1⊗τ2⊗⋯⊗τNF-1=I.
Sincef̃ is stationary under infinitesimal variations about the minimum, it follows that
(2.79)Tr[ρ̃δ(ρ1⊗ρ2⊗⋯⊗ρN)]=0
for all infinitesimal variations
(2.80)δ(ρ1⊗ρ2⊗⋯⊗ρN)=δρ1⊗Id2⊗⋯⊗IdN+Id1⊗δρ2⊗Id3⊗⋯⊗IdN+⋯+Id1⊗Id2⊗⋯⊗IdN-1⊗δρN,
subjected to the constraint det(Idi+δρi)=1, which is equivalent to Tr[δρi]=0, i=1,2,…,N, using det(eA)=eTr[A] for a given matrix A. Thus, δρi can be represented by the SU generators, δρi=∑kδckiλki. It follows that Tr[ρ̃λαk{μk}]=0 for any αk and μk. Hence the terms proportional to λαk{μk} in (2.69) disappear.Corollary 2.11.
The normal form of a product state inℋ must be proportional to the identity.Proof.
Letρ be such a state. From (2.74), we get that
(2.81)ρ̃i=Tr1,2,…,i-1,i+1,…,N[ρ]=1diIdi.
Therefore for a product state ρ we have
(2.82)ρ̃=ρ1⊗ρ2⊗⋯⊗ρN=1∏i=1Ndi⊗i=1NIdi.As an example for separability of multipartite states in terms of their normal forms (2.74), we consider the PPT entangled edge state [79]
(2.83)ρ=(100000010a00000000b00000000c000000001c000000001b000000001a010000001)
mixed with noises
(2.84)ρp=pρ+(1-p)8I8.
Select a=2,b=3, and c=0.6. Using the criterion in [97] we get that ρp is entangled for 0.92744<p≤1. But after transforming ρp to its normal form (2.74), the criterion can detect entanglement for 0.90285<p≤1.Here we indicate that the filtering transformation does not change the PPT property. Letρ∈ℋA⊗ℋB be PPT, that is, ρTA≥0, and ρTB≥0. Let ρ̃ be the normal form of ρ. From (2.61) we have
(2.85)ρ̃TA=(FA*⊗FB)ρTA(FAT⊗FB†)Tr[(FA⊗FB)ρ(FA⊗FB)†].
For any vector |ψ〉, we have
(2.86)〈ψ|ρ̃TA|ψ〉=〈ψ|(FA*⊗FB)ρTA(FAT⊗FB†)|ψ〉Tr[(FA⊗FB)ρ(FA⊗FB)†]≡〈ψ′|ρTA|ψ′〉≥0,
where |ψ′〉=(FAT⊗FB†)|ψ〉/Tr[(FA⊗FB)ρ(FA⊗FB)†].  ρ̃TB≥0 can be proved similarly. This property is also valid for multipartite case. Hence a bound entangled state will be bound entangled under filtering transformations.


## 2.3. Entanglement Witness Based on Correlation Matrix Criterion

Entanglement witness (EW) is another way to describe separability. Based on CM criterion we can further construct entanglement witness in terms of LOOs. EW [92] is an observable of the composite system such that (i) nonnegative expectation values in all separable states and (ii) at least one negative eigenvalue (can recognize at least one entangled state). Consider bipartite systems in ℋAM⊗ℋBN with M≤N.Theorem 2.12.
For any properly selected LOOsGkA and GkB,
(2.87)W=I-α∑k=0N2-1GkA⊗GkB
is an EW [98], where α=MN/((M-1)(N-1)+1) and
(2.88)G0A=1MIM,G0B=1NIN.Proof.
Letρ=∑l,m=0N2-1TlmλlA⊗λmB be a separable state, where λkA/B are normalized generators of SU(M/N) with λ0A=(1/M)IM, λ0B=(1/N)IN. Any other LOOs GkA/B fulfilling (2.88) can be obtained from these λs through orthogonal transformations 𝒪A/B,GkA/B=∑l=0N2-1𝒪klA/Bλl, where 𝒪A/B=(100ℛA/B) and ℛA/B are (N2-1)×(N2-1) orthogonal matrices. We have
(2.89)Tr[ρW]=1-α1MN-α∑k=1N2-1∑l,m=1N2-1ℛklAℛkmBTr[ρ(λlA⊗λmB)]=(M-1)(N-1)(M-1)(N-1)+1-1MN((M-1)(N-1)+1)∑k=1N2-1∑l,m=1N2-1ℛklATlmℛkmB≥MN(M-1)(N-1)-∥T∥KFMN((M-1)(N-1)+1)≥0,
where we have used Tr[ℛT]≤∥T∥KF for any unitary ℛ in the first inequality and the CM criterion in the second inequality.
Now letρ=(1/MN)(IMN+∑i=1M2-1siλiA⊗IN+∑j=1N2-1rjIM⊗λjB+∑i=1M2-1∑j=1N2-1TijλiA⊗λjB) be a state in ℋAM⊗ℋBN which violates the CM criterion. Denote σk(T) as the singular values of T. By singular value decomposition, one has T=U†ΛV*, where Λ is a diagonal matrix with Λkk=σk(T). Now choose LOOs to be GkA=∑lUklλlA, GkB=∑mVkmλmB for k=1,2,…,N2-1 and G0A=(1/M)IM,G0B=(1/N)IN. We obtain
(2.90)Tr[ρW]=1-α1MN-α∑k=1N2-1∑l,m=1N2-1UklVkmTr[ρ(λlA⊗λmB)]=(M-1)(N-1)(M-1)(N-1)+1-1MN((M-1)(N-1)+1)Tr[UTVT]=MN(M-1)(N-1)-∥T∥KFMN((M-1)(N-1)+1)<0,
where the CM criterion has been used in the last step.As the CM criterion can be generalized to multipartite form [97], we can also define entanglement witness for multipartite system in ℋ1d1⊗ℋ2d2⊗⋯⊗ℋNdN. Set d(M)=max{dμi,i=1,2,…,M}. Choose LOOs Gk{μi} for 0≤k≤d(M)2-1 with G0{μi}=(1/dμi)Idμi and define(2.91)W(M)=I-β(M)∑k=0d(M)2-1Gk{μ1}⊗Gk{μ2}⊗⋯⊗Gk{μM},
where β(M)=∏i=1Mdμi/(1+∏i=1M(dμi-1)),2≤M≤N. One can prove that (2.91) is an EW candidate for multipartite states. First we assume that ∥𝒯(M)∥KF=∥𝒯(m0)∥KF. Note that, for any 𝒯(m0), there must exist an elementary transformation P such that ∑k=1d(M)2-1𝒯kk⋯k{μ1μ2⋯μM}=Tr[𝒯(m0)P]. Then for an N-partite separable state we have
(2.92)Tr[ρW(M)]=1-β(M)1∏i=1Mdμi-β(M)1∏i=1MdμiTr[𝒯(m0)P]≥1-β(M)1∏i=1Mdμi-β(M)1∏i=1Mdμi∥𝒯(m0)∥KF≥1-β(M)1∏i=1Mdμi-β(M)1∏i=1Mdμi∏k=1Mdμk(dμk-1)=0
for any 2≤M≤N, where we have taken into account that P is orthogonal and Tr[MU]≤∥M∥KF for any unitary U at the first inequality. The second inequality is due to the generalized CM criterion.By choosing proper LOOs, it is also easy to show thatW(M) has negative eigenvalues. For example, for three-qubit case, taking the normalized Pauli matrices as LOOs, one finds a negative eigenvalue of W(M), (1-3)/2.


## 3. Concurrence and Tangle

In this section, we focus on two important measures: concurrence and tangle (see [99, 100]). An elegant formula for concurrence of two-qubit states is derived analytically by Wootters [53, 101]. This quantity has recently been shown to play an essential role in describing quantum phase transition in various interacting quantum many-body systems [102, 103] and may affect macroscopic properties of solids significantly [104, 105]. Furthermore, concurrence also provides an estimation [106, 107] for the entanglement of formation (EOF) [76], which quantifies the required minimally physical resources to prepare a quantum state.LetℋA (resp., ℋB) 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 ℋA⊗ℋB is of the form(3.1)|Ψ〉=∑i=1M∑j=1Naij|i〉⊗|j〉,
where aij∈ℂ satisfy the normalization ∑i=1M∑j=1Naijaij*=1.The concurrence of (3.1) is defined by [27, 108, 109]
(3.2)C(|ψ〉)=2(1-Tr[ρA2]),
where ρA=TrB[|ψ〉〈ψ|]. The definition is extended to general mixed states ρ=∑ipi|ψi〉〈ψi| by the convex roof(3.3)C(ρ)=min{pi,|ψi〉}∑ipiC(|ψi〉).For two-qubits systems, the concurrence of|Ψ〉 is given by(3.4)C(|Ψ〉)=|〈Ψ∣Ψ̃〉|=2|a11a22-a12a21|,
where |Ψ̃〉=σy⊗σy|Ψ*〉, |Ψ*〉 is the complex conjugate of |Ψ〉, and σy is the Pauli matrix, σy=(0-ii0).For a mixed two-qubit quantum stateρ, the entanglement of formation E(ρ) has a simple relation with the concurrence [53, 101]
(3.5)E(ρ)=h(1+1-C(ρ)22),
where h(x)=-xlog2x-(1-x)log2(1-x),
(3.6)C(ρ)=max{λ1-λ2-λ3-λ4,0},
where the λis are the eigenvalues, in decreasing order, of the Hermitian matrix ρρ̃ρ and ρ̃=(σy⊗σy)ρ*(σy⊗σy).Another entanglement measure called tangle is defined by(3.7)τ(|ψ〉)=C2(|ψ〉)=2(1-Tr[ρA2])
for a pure state |ψ〉. For mixed state ρ=∑ipi|ψi〉〈ψi|, the definition is given by(3.8)τ(ρ)=min{pi,|ψi〉}∑ipiτ(|ψi〉).For multipartite state|ψ〉∈ℋ1⊗ℋ2⊗⋯⊗ℋN, dimℋi=di,i=1,…,N, the concurrence of |ψ〉 is defined by [110, 111](3.9)CN(|ψ〉〈ψ|)=21-N/2(2N-2)-∑αTr[ρα2],
where α labels all different reduced density matrices.Up to constant factor (3.9) can be also expressed in another way. Let H denote a d-dimensional vector space with basis |i〉, i=1,2,…,d. An N-partite pure state in H⊗⋯⊗H is generally of the form(3.10)|Ψ〉=∑i1,i2,…iN=1dai1,i2,…iN|i1,i2,…iN〉,ai1,i2,…iN∈ℂ.Letα 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 subindix of a. The generalized concurrence of |Ψ〉 is then given by [27](3.11)CdN(|Ψ〉)=d2m(d-1)∑p∑{α,α′,β,β′}d|aαβaα′β′-aαβ′aα′β|2,
where m=2N-1-1 and ∑p stands for the summation over all possible combinations of the indices of α and β.For a mixed multipartite quantum state,ρ=∑ipi|ψi〉〈ψi| in ℋ1⊗ℋ2⊗⋯⊗ℋN, the corresponding concurrence is given by the convex roof:(3.12)CN(ρ)=min{pi,|ψi}〉∑ipiCN(|ψi〉).


### 3.1. Lower and Upper Bounds of Concurrence

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


#### 3.1.1. Lower Bound of Concurrence from Covariance Matrix Criterion

In [61] a lower bound of C(ρ) has been obtained as(3.13)C(ρ)≥2M(M-1)[Max(∥TA(ρ)∥,∥R(ρ)∥)-1],
where TA and R stand for partial transpose with respect to subsystem A and the realignment, respectively. This bound is further improved based on local uncertainty relations [64](3.14)C(ρ)≥M+N-2-∑iΔρ2(GiA⊗I+I⊗GiB)2M(M-1),
where GiA and GiB are any set of local orthonormal observables, Δρ2(X)=Tr[X2ρ]-(Tr[Xρ])2.Bound (3.14) again depends on the choice of the local orthonormal observables. This bound can be optimized, in the sense that a local orthonormal observable-independent up bound of the right-hand side of (3.14) can be obtained.Theorem 3.1.
Letρ be a bipartite state in ℋMA⊗ℋNB. Then C(ρ) satisfies [90]
(3.15)C(ρ)≥2∥C∥KF-(1-Tr[ρA2])-(1-Tr[ρB2])2M(M-1).Proof.
The other orthonormal normalized basis of the local orthonormal observable space can be obtained fromAi and Bi by unitary transformations U and V: Ãi=∑lUilAl and B̃j=∑mVjm*Bm. Select U and V so that C=U†ΛV is the singular value decomposition of C. Then the new observables can be written as Ãi=∑lUilAl, B̃j=-∑mVjm*Bm. We have
(3.16)∑iΔρ2(Ãi⊗I+I⊗B̃i)=∑i[ΔρA2(Ãi)+ΔρA2(B̃i)+2(〈Ãi⊗B̃i〉-〈Ãi〉〈B̃i〉)]=M-Tr[ρA2]+N-Tr[ρB2]-2∑i(UCV†)ii=M-Tr[ρA2]+N-Tr[ρB2]-2∥C∥KF.
Substituting above relation to (3.14), one gets (3.15).Bound (3.15) does not depend on the choice of local orthonormal observables. It can be easily applied and realized by direct measurements in experiments. It is in accord with the result in [46] where optimization of entanglement witness based on local uncertainty relation has been taken into account. As an example, let us consider the 3×3 bound entangled state [76](3.17)ρ=14(I9-∑i=04|ξi〉〈ξi|),
where I9 is the 9×9 identity matrix, |ξ0〉=(1/2)|0〉(|0〉-|1〉), |ξ1〉=(1/2)(|0〉-|1〉)|2〉, |ξ2〉=(1/2)|2〉(|1〉-|2〉), |ξ3〉=(1/2)(|1〉-|2〉)|0〉, and |ξ4〉=(1/3)(|0〉+|1〉+|2〉)(|0〉+|1〉+|2〉). We simply choose the local orthonormal observables to be the normalized generators of SU(3). Formula (3.13) gives C(ρ)≥0.050. Formula (3.14) gives C(ρ)≥0.052 [64], while formula (3.15) yields a better lower bound C(ρ)≥0.0555.If we mix the bound entangled state (3.17) with |ψ〉=(1/3)∑i=02|ii〉, that is,(3.18)ρ′=(1-x)ρ+x|ψ〉〈ψ|,
then it is easily seen that (3.15) gives a better lower bound of concurrence than formula (3.13) (Figure 1).Figure 1
Lower bounds from (3.15) (dashed line) and (3.13) (solid line) for mixed state (3.18).


#### 3.1.2. Lower Bound of Concurrence from “Two-Qubit” Decomposition

In [67] the authors derived an analytical lower bound of concurrence for arbitrary bipartite quantum states by decomposing the joint Hilbert space into many 2⊗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 boundτ2 of concurrence for bipartite states has been obtained in [67]. For a bipartite quantum state ρ in H⊗H, the concurrence C(ρ) satisfies
(3.19)τ2(ρ)≡d2(d-1)∑m,n=1d(d-1)/2Cmn2(ρ)≤C2(ρ),
where Cmn(ρ)=max{0,λmn(1)-λmn(2)-λmn(3)-λmn(4)} with λmn(1),…,λmn(4) being the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃mn with ρ̃mn=(Lm⊗Ln)ρ*(Lm⊗Ln), while Lm and Ln are the generators of SO(d).The lower boundτ2 in (3.19) in fact characterizes all two qubits’ entanglement in a high dimensional bipartite state. One can directly verify that there are at most 4×4=16 nonzero elements in each matrix ρρ̃mn. These elements constitute a 4×4 matrix ϱ(σy⊗σy)ϱ*(σy⊗σy), where σy is the Pauli matrix, the matrix ϱ is a submatrix of the original ρ:(3.20)ϱ=(ρik,ikρik,ilρik,jkρik,jlρil,ikρil,ilρil,jkρil,jlρjk,ikρjk,ilρjk,jkρjk,jlρjl,ikρjl,ilρjl,jkρjl,jl),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 ϱ is not normalized but positive semidefinite. 𝒞mn are just the concurrences of these states (3.20).The boundτ2 provides a much clearer structure of entanglement, which not only yields an effective separability criterion and an easy evaluation of entanglement, but also helps one to classify mixed-state entanglement.(2) Lower Bound of Concurrence for Multipartite States
We first consider tripartite case. A general pure state onH⊗H⊗H is of the form
(3.21)|Ψ〉=∑i,j,k=1daijk|ijk〉,aijk∈ℂ,∑i,j,k=1daijkaijk*=1
with
(3.22)Cd3(|Ψ〉)=d6(d-1)∑(|aijkapqm-aijmapqk|2+|aijkapqm-aiqkapjm|2+|aijkapqm-apjkaiqm|2),
or equivalently
(3.23)Cd3(|Ψ〉)=d6(d-1)(3-(Tr[ρ12]+Tr[ρ22]+Tr[ρ32])),
where ρ1=Tr23[ρ],ρ2=Tr13[ρ],andρ3=Tr12[ρ] are the reduced density matrices of ρ=|Ψ〉〈Ψ|.DefineCαβ12|3(|Ψ〉)=|aijkapqm-aijmapqk|, Cαβ13|2(|Ψ〉)=|aijkapqm-aiqkapjm|, and Cαβ23|1(|Ψ〉)=|aijkapqm-apjkaiqm|, where α and β of Cαβ12|3 (resp., Cαβ13|2, resp., Cαβ23|1) stand for the subindices of a associated with the subspaces 1,2, and 3 (resp., 1,3, and 2; resp., 2,3, and 1). Let Li1i2⋯iN denote the generators of group SO(di1di2⋯diN) associated to the subsystems i1,i2,…,iN. Then for a tripartite pure state (3.21), one has
(3.24)Cd3(|Ψ〉)=d6(d-1)∑αd2(d2-1)/2∑βd(d-1)/2[(Cαβ12|3(|Ψ〉))2+(Cαβ13|2(|Ψ〉))2+(Cαβ23|1(|Ψ〉))2]=d6(d-1)∑αβ[(|〈Ψ|Sαβ12|3|Ψ*〉|)2+(|〈Ψ|Sαβ13|2|Ψ*〉|)2+(|〈Ψ|Sαβ23|2|Ψ*〉|)2],
where Sαβ12|3=(Lα12⊗Lβ3), Sαβ13|2=(Lα13⊗Lβ2), and Sαβ23|1=(Lβ1⊗Lα23).Theorem 3.2.
For an arbitrary mixed stateρ in H⊗H⊗H, the concurrence C(ρ) satisfies [112]
(3.25)τ3(ρ)≡d6(d-1)∑αd2(d2-1)/2∑βd(d-1)/2[(Cαβ12|3(ρ))2+(Cαβ13|2(ρ))2+(Cαβ23|1(ρ))2]≤C2(ρ),
where τ3(ρ) is a lower bound of C(ρ)(3.26)Cαβ12|3(ρ)=max{0,λ(1)αβ12∣3-λ(2)αβ12∣3-λ(3)αβ12∣3-λ(4)αβ12∣3},
where λ(1)αβ12|3,λ(2)αβ12|3,λ(3)αβ12|3,andλ(4)αβ12|3 are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃αβ12|3 with ρ̃αβ12|3=Sαβ12|3ρ*Sαβ12|3. Cαβ13|2(ρ) and Cαβ23|1(ρ) are defined in a similar way to Cαβ12|3(ρ).Proof.
Set|ξi〉=pi|ψi〉, xαβi=|〈ξi|Sαβ12|3|ξi*〉|, yαβi=|〈ξi|Sαβ13|2|ξi*〉|, and zαβi=|〈ξi|Sαβ1|23|ξi*〉|. We have, from Minkowski inequality,
(3.27)C(ρ)=min∑i‍d6(d-1)∑αβ[(xαβi)2+(yαβi)2+(zαβi)2]≥mind6(d-1)∑αβ(∑i[(xαβi)2+(yαβi)2+(zαβi)2]1/2)2.
Noting that for nonnegative real variablesxα, yα, zα and given that X=∑α=1Nxα, Y=∑α=1NYα, and Z=∑α=1Nzα, by using Lagrange multipliers, one obtains that the following inequality holds:
(3.28)∑α=1N(xα2+yα2+zα2)1/2≥(X2+Y2+Z2)1/2.
Therefore we have
(3.29)C(ρ)≥mind6(d-1)∑αβ‍[(∑ixαβi)2+(∑iyαβi)2+(∑izαβi)2]≥d6(d-1)∑αβ[(min∑ixαβi)2+(min∑iyαβi)2+(min∑izαβi)2].
The values ofCαβ12|3(ρ)≡min∑ixαβi, Cαβ13|2(ρ)≡min∑iyαβi, and Cαβ23|1(ρ)≡min∑izαβi can be calculated by using the similar procedure in [53]. Here we compute the value of Cαβ12|3(ρ) in detail. The values of Cαβ13|2(ρ) and Cαβ23|1(ρ) can be obtained analogously.
Letλi and |χi〉 be eigenvalues and eigenvectors of ρ, respectively. Any decomposition of ρ can be obtained from a unitary d3×d3 matrix Vij, |ξj〉=∑i=1d3Vij*(λi|χi〉). Therefore one has 〈ξi|Sαβ12|3|ξj*〉=(VYαβVT)ij, where the matrix Yαβ is defined by (Yαβ)ij=〈χi|Sαβ12|3|χj*〉. Namely, Cαβ12|3(ρ)=min∑i|[VYαβVT]ii|, which has an analytical expression [53], that Cαβ12|3(ρ)=max{0,λ(1)αβ12|3-∑j>1λ(j)αβ12|3}, where λαβ12|3(k) are the square roots of the eigenvalues of the positive Hermitian matrix YαβYαβ†, or equivalently the non-Hermitian matrix ρρ̃αβ, in decreasing order. Here as the matrix Sαβ12|3 has d2-4 rows and d2-4 columns that are identically zero, the matrix ρρ̃αβ has a rank not greater than 4, that is, λαβ12|3(j)=0 for j≥5. From (3.29) we have (3.25).Theorem3.2 can be directly generalized to arbitrary multipartite case.Theorem 3.3.
For an arbitraryN-partite state ρ∈H⊗H⊗⋯⊗H, the concurrence defined in (4.1) satisfies [112],
(3.30)τN(ρ)≡d2m(d-1)∑p∑αβ(Cαβp(ρ))2≤C2(ρ),
where τN(ρ) is the lower bound of C(ρ), ∑p stands for the summation over all possible combinations of the indices of α,β, Cαβp(ρ)=max{0,λ(1)αβp-λ(2)αβp-λ(3)αβp-λ(4)αβp}, and λ(i)αβp, i=1,2,3,4, are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃αβp where ρ̃αβp=Sαβpρ*Sαβp.Lower Bound and Separability
An N-partite quantum stateρ is fully separable if and only if there exist pi with pi≥0,∑ipi=1, and pure states ρij=|ψij〉〈ψij| such that
(3.31)ρ=∑ipiρi1⊗ρi2⊗⋯⊗ρiN.It is easily verified that, for a fully separable multipartite stateρ, τN(ρ)=0. Thus τN(ρ)>0 indicates that there must be some kinds of entanglement inside the quantum state, which shows that the lower bound τN(ρ) can be used to recognize entanglement.As an example, we consider a tripartite quantum state [79] ρ=((1-p)/8)I8+p|W〉〈W|, where I8 is the 8×8 identity matrix, and |W〉=(1/3)(|100〉+|010〉+|001〉) is the tripartite W state. Select an entanglement witness operator to be 𝒲=(1/2)I8-|GHZ〉〈GHZ|, where |GHZ〉=(1/2)(|000〉+|111〉) is to be the tripartite GHZ-state. From the condition Tr[𝒲ρ]<0, the entanglement of ρ is detected for (3/5)<p≤1 in [79]. In [97] the authors have obtained the generalized correlation matrix criterion which says that if an N-qubit quantum state is fully separable, then the inequality ∥𝒯N∥KF≤1 must hold, where ∥𝒯N∥KF=max{∥𝒯nN∥KF}, 𝒯nN is a kind of matrix unfold of tα1α2⋯αN defined by tα1α2⋯αN=Tr[ρσα1(1)σα2(2)⋯σαN(N)], and σαi(i) stands for the Pauli matrix. Now using the generalized correlation, matrix criterion the entanglement of ρ is detected for 0.3068<p≤1. From Theorem 3.2, we have the lower bound τ3(ρ)>0 for 0.2727<p≤1. Therefore the bound (3.71) detects entanglement better than these two criteria in this case. If we replace W with GHZ state in ρ, then the criterion in [97] detects the entanglement of ρ for 0.35355<p≤1, while τ3(ρ) detects, again better, the entanglement for 0.2<p≤1.Nevertheless for PPT statesρ, we have τ3(ρ)=0, which can be seen in the following way. A density matrix ρ is called PPT if the partial transposition of ρ over any subsystem(s) is still positive. Let ρTi denote the partial transposition with respect to the ith subsystem. Assume that there is a PPT state ρ with τ(ρ)>0. Then at least one term in (3.25), say Cα0β012|3(ρ), is not zero. Define ρα0β0=Lα012⊗Lβ03ρ(Lα012⊗Lβ03)†. By using the PPT property of ρ, we have(3.32)ρα0β0T3=Lα012⊗(Lβ03)*ρT3(Lα012)†⊗(Lβ03)T≥0.
Noting that both Lα012 and Lβ03 are projectors to two-dimensional subsystems, ρα0β0 can be considered as a 4×4 density matrix, while a PPT 4×4 density matrix ρα0β0 must be a separable state, which contradicts with Cα0β012|3(ρ)≠0.Relation between Lower Bounds of Bi- and Tripartite Concurrence
τ3 is basically different from τ2 as τ3 characterizes also genuine tripartite entanglement that can not be described by bipartite decompositions. Nevertheless, there are interesting relations between them.Theorem 3.4.
For any pure tripartite state (3.21), the following inequality holds [112]:
(3.33)τ2(ρ12)+τ2(ρ13)+τ2(ρ23)≤3τ3(ρ),
where τ2 is the lower bound of bipartite concurrence (3.19), τ3 is the lower bound of tripartite concurrence (3.25), and ρ12=Tr3[ρ], ρ13=Tr2[ρ], ρ23=Tr1[ρ], and ρ=|Ψ〉123〈Ψ|.Proof.
SinceCαβ2≤(λαβ(1))2≤∑i=14(λαβ(i))2=Tr[ρρ̃αβ] for ρ=ρ12, ρ=ρ13, and ρ=ρ23, we have
(3.34)τ2(ρ12)+τ2(ρ13)+τ2(ρ23)≤d2(d-1)(∑α,β=1d(d-1)/2Tr[ρ12(ρ̃12)αβ]+∑α,β=1d(d-1)/2Tr[ρ13(ρ̃13)αβ]+∑α,β=1d(d-1)/2Tr[ρ23(ρ̃23)αβ])=d2(d-1)(3-Tr[ρ12]-Tr[ρ22]-Tr[ρ32])=3C2(ρ)=3τ3(ρ),
where we have used the similar analysis in [67, 113] to obtain the equalities ∑α,βTr[ρ12(ρ̃12)αβ]=1-Tr[ρ12]-Tr[ρ22]+Tr[ρ32], ∑α,βTr[ρ13(ρ̃13)αβ]=1-Tr[ρ12]+Tr[ρ22]-Tr[ρ32], and ∑α,βTr[ρ23(ρ̃23)αβ]=1+Tr[ρ12]-Tr[ρ22]-Tr[ρ32]. The last equality is due to the fact that ρ is a pure state.In fact, the bipartite entanglement inside a tripartite state is useful for distilling maximally entangled states. Assume that there are two of the qualities{τ(ρ12),τ(ρ13),τ(ρ23)} larger than zero; say τ(ρ12)>0 and τ(ρ13)>0. According to [67], one can distill two maximal entangled states |ψ12〉 and |ψ13〉 which belong to ℋ1⊗ℋ2 and ℋ1⊗ℋ3, respectively. In terms of the result in [114], one can use them to produce a GHZ state.


#### 3.1.3. Estimation of Multipartite Entanglement

For a pureN-partite quantum state |ψ〉∈ℋ1⊗ℋ2⊗⋯⊗ℋN, dimℋi=di, i=1,…,N, the concurrence of bipartite decomposition between subsystems 12⋯M and M+1⋯N is defined by(3.35)C2(|ψ〉)=2(1-Tr[ρ12⋯M2])
where ρ12⋯M2=TrM+1⋯N[|ψ〉〈ψ|] is the reduced density matrix of ρ=|ψ〉〈ψ| by tracing over subsystems M+1⋯N. On the other hand, the concurrence of |ψ〉 is defined by (3.9).For a mixed multipartite quantum stateρ=∑ipi|ψi〉〈ψi|∈ℋ1⊗ℋ2⊗⋯⊗ℋN, the corresponding concurrences of (3.35) and (3.9) are then given by the convex roof(3.36)C2(ρ)=min{pi,|ψi}〉∑i‍piC2(|ψi〉〈ψi|),
and (3.12). We now investigate the relation between these two kinds of concurrences.Lemma 3.5.
For a bipartite density matrixρ∈ℋA⊗ℋB, one has
(3.37)1-Tr[ρ2]≤1-Tr[ρA2]+1-Tr[ρB2],
where ρA/B=TrB/A[ρ] are the reduced density matrices of ρ.Proof.
Letρ=∑ijλij|ij〉〈ij| be the spectral decomposition, where λij≥0,∑ijλij=1. Then ρ1=∑ijλij|i〉〈i|, ρ2=∑ijλij|j〉〈j|. Therefore
(3.38)1-Tr[ρA2]+1-Tr[ρB2]-1+Tr[ρ2]=1-Tr[ρA2]-Tr[ρB2]+Tr[ρ2]=(∑ijλij)2-∑i,j,j′λijλij′-∑i,i′,jλijλi′j+∑ijλij2=(∑i=i′,j=j′λij2+∑i=i′,j≠j′λijλij′+∑i≠i′,j=j′λijλi′j+∑i≠i′,j≠j′λijλi′j′)=∑i≠i′,j≠j′λijλi′j′≥0.This lemma can be also derived in another way [46, 115].Theorem 3.6.
For a multipartite quantum stateρ∈ℋ1⊗ℋ2⊗⋯⊗ℋN with N≥3, the following inequality holds [116]:
(3.39)CN(ρ)≥max2(3-N)/2C2(ρ),
where the maximum is taken over all kinds of bipartite concurrence.Proof.
Without loss of generality, we suppose that the maximal bipartite concurrence is attained between subsystems12⋯M and (M+1)⋯N.
For a pure multipartite state|ψ〉∈ℋ1⊗ℋ2⊗⋯⊗ℋN, Tr[ρ12⋯M2]=Tr[ρ(M+1)⋯N2]. From (3.37) we have
(3.40)CN2(|ψ〉〈ψ|)=22-N((2N-2)-∑αTr[ρα2])≥23-N(N-∑k=1NTr[ρk2])≥23-N(1-Tr[ρ12⋯M2]+1-Tr[ρ(M+1)⋯N2])=23-N*2(1-Tr[ρ12⋯M2])=23-NC22(|ψ〉〈ψ|),
that is, CN(|ψ〉〈ψ|)≥2(3-N)/2C2(|ψ〉〈ψ|).
Letρ=∑ipi|ψi〉〈ψi| attain the minimal decomposition of the multipartite concurrence. One has
(3.41)CN(ρ)=∑ipiCN(|ψi〉〈ψi|)≥2(3-N)/2∑ipiC2(|ψi〉〈ψi|)≥2(3-N)/2min{pi,|ψi}∑ipiC2(|ψi〉〈ψi|)=2(3-N)/2C2(ρ).Corollary 3.7.
For a tripartite quantum stateρ∈ℋ1⊗ℋ2⊗ℋ3, the following inequality holds:
(3.42)C3(ρ)≥maxC2(ρ),
where the maximum is taken over all kinds of bipartite concurrence.In [46, 64], from the separability criteria related to local uncertainty relation, covariance matrix, and correlation matrix, the following lower bounds for bipartite concurrence are obtained:(3.43)C2(ρ)≥2∥C(ρ)∥-(1-Tr[ρA2])-(1-Tr[ρB2])2dA(dA-1),(3.44)C2(ρ)≥8dA3dB2(dA-1)(∥T(ρ)∥-dAdB(dA-1)(dB-1)2),
where the entries of the matrix C, Cij=〈λiA⊗λjB〉-〈λiA⊗IdB〉〈IdA⊗λjB〉, Tij=dAdB/2〈λiA⊗λjB〉, λkA/B stand for the normalized generator of SU(dA/dB), that is, Tr[λkA/BλlA/B]=δkl and 〈X〉=Tr[ρX]. It is shown that the lower bounds (3.43) and (3.44) are independent of (3.13).Now we consider a multipartite quantum stateρ∈ℋ1⊗ℋ2⊗⋯⊗ℋN as a bipartite state belonging to ℋA⊗ℋB with the dimensions of the subsystems A and B being dA=ds1ds2⋯dsm and dB=dsm+1dsm+2⋯dsN, respectively. By using Corollary 3.7, (3.13), (3.43), and (3.44), one has the following lower bound.Theorem 3.8.
For any N-partite quantum stateρ [116],
(3.45)CN(ρ)≥2(3-N)/2max{B1,B2,B3},
where
(3.46)B1=max{i}2Mi(Mi-1)[max(∥𝒯A(ρi)∥,∥R(ρi)∥)-1],B2=max{i}2∥C(ρi)∥-(1-Tr[(ρAi)2])-(1-Tr[(ρBi)2])2Mi(Mi-1),B3=max{i}8Mi3Ni2(Mi-1)(∥T(ρi)∥-MiNi(Mi-1)(Ni-1)2),ρi’s are all possible bipartite decompositions of ρ, and
(3.47)Mi=min{ds1ds2⋯dsm,dsm+1dsm+2⋯dsN},Ni=max{ds1ds2⋯dsm,dsm+1dsm+2⋯dsN}.In [46, 106, 107, 117], it is shown that the upper and lower bounds of multipartite concurrence satisfy(3.48)(4-23-N)Tr[ρ2]-22-N∑αTr[ρα2]≤CN(ρ)≤22-N[(2N-2)-∑αTr[ρα2]].In fact one can obtain a more effective upper bound for multipartite concurrence. Letρ=∑iλi|ψi〉〈ψi|∈ℋ1⊗ℋ2⊗⋯⊗ℋN, where |ψi〉’s are the orthogonal pure states and ∑iλi=1. We have(3.49)CN(ρ)=min{pi,|φi}〉∑ipiCN(|φi〉〈φi|)≤∑iλiCN(|ψi〉〈ψi|).
The right side of (3.49) gives a new upper bound of CN(ρ). Since
(3.50)∑iλiCN(|ψi〉〈ψi|)=21-N/2∑iλi(2N-2)-∑αTr[(ραi)2]≤21-N/2(2N-2)-∑αTr[∑iλi(ραi)2]≤21-N/2(2N-2)-∑αTr[(ρα)2],
the upper bound obtained in (3.49) is better than that in (3.48).


#### 3.1.4. Bounds of Concurrence and Tangle

In [68], a lower bound for tangle defined in (3.8) has been derived as(3.51)τ(ρ)≥8MN(M+N)(∥T(ρ)∥HS2-MN(M-1)(N-1)4),
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 et al. in [106, 107] and Zhang et al. in [46]:(3.52)2(Tr[ρ2]-Tr[ρA2])≤C(ρ)≤2(1-Tr[ρA2]).Since the convexity ofC2(ρ), we have that τ(ρ)≥C2(ρ) always holds. For two-qubit quantum systems, tangle τ is always equal to the square of concurrence C2 [58, 113], as a decomposition {pi,|ψi〉} achieving the minimum in (3.3) has the property that C(|ψi〉)=C(|ψj〉)∀i,j. For higher dimensional systems we do not have similar relations. Thus it is meaningful to derive valid upper bound for tangle and lower bound for concurrence.Theorem 3.9.
For any quantum stateρ∈ℋA⊗ℋB, one has [118]
(3.53)τ(ρ)≤min{1-Tr[ρA2],1-Tr[ρB2]},(3.54)C(ρ)≥8MN(M+N)(∥T(ρ)∥HS-MN(M-1)(N-1)2),
where ρA is the reduced matrix of ρ, and T(ρ) is the correlation matrix of ρ defined in (3.44).Proof.
We assume that1-Tr[ρA2]≤1-Tr[ρB2] for convenience. By the definition of τ, we have that for a pure state |ψ〉,τ(|ψ〉)=2(1-Tr[(ρA|ψ〉)2]). Let ρ=∑ipiρi be the optimal decomposition such that τ(ρ)=∑ipiτ(ρi). We get
(3.55)τ(ρ)=∑ipiτ(ρi)=∑ipi2[1-Tr[(ρA|ψi〉)2]]=2[1-Tr[∑ipi(ρA|ψi〉)2]]≤2[1-Tr[ρA2]].
Note that, for pure state|ψ〉∈ℋA⊗ℋB [68],
(3.56)C(|ψ〉)=8MN(M+N)(∥T(|ψ〉)∥2-MN(M-1)(N-1)4).
Using the inequality a-b≥a-b for any a≥b, we get
(3.57)C(|ψ〉)≥8MN(M+N)(∥T(|ψ〉)∥HS-MN(M-1)(N-1)2).
Now let ρ=∑ipiρi be the optimal decomposition such that C(ρ)=∑ipiC(ρi). We get
(3.58)C(ρ)=∑ipiC(ρi)≥∑ipi8MN(M+N)(∥T(ρi)∥HS-MN(M-1)(N-1)2)=8MN(M+N)(∑ipi∥T(ρi)∥HS-MN(M-1)(N-1)2)≥8MN(M+N)(∥T(ρ)∥HS-MN(M-1)(N-1)2),
which ends the proof.The upper bound (3.53), together with the lower bounds (3.54), (3.43), (3.44), (3.51), and (3.52), can allow for estimations of entanglement for arbitrary quantum states. Moreover, since the upper bound is exactly the value of tangle for pure states, the upper bound can be a good estimation when the state is very weakly mixed.


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

In this subsection we show that although bound entangled states cannot be distilled, the concurrence and tangle of two entangled states will be always strictly larger than those of one, even if the two entangled states are both bound entangled.Letρ=∑ijklρij,kl|ij〉〈kl|∈ℋA⊗ℋB and σ=∑i′j′k′l′σi′j′,k′l′|i′j′〉〈k′l′|∈ℋA′⊗ℋB′ be two quantum states shared by subsystems AA′ and BB′. We use ρ⊗σ=∑ijkl,i′j′k′l′ρij,klσi′j′,k′l′|ii′〉AA′〈kk′|⊗|jj′〉BB′〈ll′| to denote the state of the whole system.Lemma 3.10.
For pure states|ψ〉∈ℋA⊗ℋB and |φ〉∈ℋA′⊗ℋB′, the inequalities
(3.59)C(|ψ〉⊗|φ〉)≥max{C(|ψ〉),C(|φ〉)},(3.60)τ(|ψ〉⊗|φ〉)≥max{τ(|ψ〉),τ(|φ〉)}
always hold, and  “=” in the two inequalities hold if and only if at least one of {|ψ〉,|φ〉} is separable.Proof.
Without loss of generality we assume thatC(|ψ〉)≥C(|φ〉). First note that
(3.61)ρAA′|ψ〉⊗|φ〉=ρA|ψ〉⊗ρA′|φ〉.
Let ρA|ψ〉=∑iλi|i〉〈i| and ρA′|φ〉=∑jπj|j〉〈j| be the spectral decomposition of ρA|ψ〉 and ρA′|φ〉, with ∑iλi=1 and ∑jπj=1, respectively. By using (3.61) one obtains that
(3.62)Tr[(ρAA′|ψ〉⊗|φ〉)2]=∑λiπjλi′πj′|ij〉〈ij∣i′j′〉〈i′j′|=∑λi2πj2,
while
(3.63)Tr[(ρA|ψ〉)2]=∑iλi2.
Now using the definition of concurrence and the normalization conditions ofλi and πj, one immediately gets
(3.64)C(|ψ〉⊗|φ〉)=2(1-Tr[(ρAA′|ψ〉⊗|φ〉)2])≥2(1-Tr[(ρA|ψ〉)2])=C(|ψ〉).
If one of {|ψ〉,|φ〉} is separable, say |φ〉, then the rank of ρA′|φ〉 must be one, which means that there is only one item in the spectral decomposition in ρA′|φ〉. Using the normalization condition of πj, we obtain Tr[(ρAA′|ψ〉⊗|φ〉)2]=Tr[(ρA|ψ〉)2]. Then inequality (3.64) becomes an equality.
On the other hand, if both|ψ〉 and |φ〉 are entangled (not separable), then there must be at least two items in the decomposition of their reduced density matrices ρA|ψ〉 and ρA′|φ〉, which means that Tr[(ρAA′|ψ〉⊗|φ〉)2] is strictly larger than Tr[(ρA|ψ〉)2].
The inequality (3.60) also holds because, for pure quantum state ρ, τ(ρ)=C2(ρ).From the lemma, we have, for mixed states the following.Theorem 3.11.
For any quantum statesρ∈ℋA⊗ℋB and σ∈ℋA′⊗ℋB′, the inequalities
(3.65)C(ρ⊗σ)≥max{C(ρ),C(σ)},τ(ρ⊗σ)≥max{τ(ρ),τ(σ)}
always hold, and the “=’’ in the two inequalities hold if and only if at least one of {ρ,σ} is separable, that is,  if both ρ and σ are entangled (even if bound entangled), then C(ρ⊗σ)>max{C(ρ),C(σ)} and τ(ρ⊗σ)>max{τ(ρ),τ(σ)} always hold [118].Proof.
We still assume thatC(ρ)≥C(σ) for convenience. Let ρ=∑ipiρi and σ=∑jqjσj be the optimal decomposition such that C(ρ⊗σ)=∑ipiqjC(ρi⊗σj). By using the inequality obtained in Lemma 3.10, we have
(3.66)C(ρ⊗σ)=∑ipiqjC(ρi⊗σj)≥∑ipiqjC(ρi)=∑ipiC(ρi)≥C(ρ).Case 1.
Now let one of{ρ,σ} be separable, say σ, with ensemble representation σ=∑jqjσj, where ∑jqj=1, and σj is the density matrix of separable pure state. Suppose that ρ=∑ipiρi is the optimal decomposition of ρ such that C(ρ)=∑ipiC(ρi). Using Lemma 3.10, we have
(3.67)C(ρ⊗σ)≤∑ipiqjC(ρi⊗σj)=∑ipiqjC(ρi)=∑ipiC(ρi)=C(ρ).
The inequalities (3.66) and (3.67) show that if σ is separable, then C(ρ⊗σ)=C(ρ).Case 2.
If bothρ and σ are inseparable, that is,  there is at least one pure state in the ensemble decomposition of ρ (and σ, resp.), using Lemma 3.10, then we have
(3.68)C(ρ⊗σ)=∑ipiqjC(ρi⊗σj)>∑ipiqjC(ρi)=∑ipiC(ρi)≥C(ρ).
The inequality for tangleτ can be proved in a similar way.Remark 3.12.
In [119] it is shown that any entangled state ρ can enhance the teleportation power of another state σ. This holds even if the state ρ is bound entangled. But if ρ is bound entangled, then the corresponding σ must be free entangled (distillable). By Theorem 3.11, we can see that even if two entangled quantum states ρ and σ are bound entangled, their concurrence and tangle are strictly larger than those of one state.


### 3.3. Subadditivity of Concurrence and Tangle

We now give a proof of the subadditivity of concurrence and tangle, which illustrates that concurrence and tangle may be proper entanglement measurements.Theorem 3.13.
Letρ and σ be quantum states in ℋA⊗ℋB, the one has [118]
(3.69)C(ρ⊗σ)≤C(ρ)+C(σ),τ(ρ⊗σ)≤τ(ρ)+τ(σ).Proof.
We first prove that the theorem holds for pure states, that is,  for|ψ〉 and |ϕ〉 in ℋA⊗ℋB:
(3.70)C(|ψ〉⊗|ϕ〉)≤C(|ψ〉)+C(|ϕ〉),τ(|ψ〉⊗|ϕ〉)≤τ(|ψ〉)+τ(|ϕ〉).
Assume that ρA|ψ〉=∑iλi|i〉〈i| and ρA|ϕ〉=∑jπj|j〉〈j| are the spectral decomposition of the reduced matrices ρA|ψ〉 and ρA|ϕ〉. One has
(3.71)12[C(|ψ〉)+C(|ϕ〉)]2≥1-Tr[(ρA|ψ〉)2]+1-Tr[(ρA|ϕ〉)2]=1-∑iλi2+1-∑jπj2≥1-∑ijλi2πj2=12C2(|ψ〉⊗|ϕ〉).
Now we prove that (3.69) holds for any mixed-quantum states ρ and σ. Let ρ=∑ipiρi and σ=∑jqjσj be the optimal decomposition such that C(ρ)=∑ipiC(ρi) and C(σ)=∑jqjC(σj). We have
(3.72)C(ρ)+C(σ)=∑ijpiqj[C(ρi)+C(σj)]≥∑ijpiqjC(ρi⊗σj)≥C(ρ⊗σ).
The inequality forτ can be derived in a similar way.


## 3.1. Lower and Upper Bounds of Concurrence

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


### 3.1.1. Lower Bound of Concurrence from Covariance Matrix Criterion

In [61] a lower bound of C(ρ) has been obtained as(3.13)C(ρ)≥2M(M-1)[Max(∥TA(ρ)∥,∥R(ρ)∥)-1],
where TA and R stand for partial transpose with respect to subsystem A and the realignment, respectively. This bound is further improved based on local uncertainty relations [64](3.14)C(ρ)≥M+N-2-∑iΔρ2(GiA⊗I+I⊗GiB)2M(M-1),
where GiA and GiB are any set of local orthonormal observables, Δρ2(X)=Tr[X2ρ]-(Tr[Xρ])2.Bound (3.14) again depends on the choice of the local orthonormal observables. This bound can be optimized, in the sense that a local orthonormal observable-independent up bound of the right-hand side of (3.14) can be obtained.Theorem 3.1.
Letρ be a bipartite state in ℋMA⊗ℋNB. Then C(ρ) satisfies [90]
(3.15)C(ρ)≥2∥C∥KF-(1-Tr[ρA2])-(1-Tr[ρB2])2M(M-1).Proof.
The other orthonormal normalized basis of the local orthonormal observable space can be obtained fromAi and Bi by unitary transformations U and V: Ãi=∑lUilAl and B̃j=∑mVjm*Bm. Select U and V so that C=U†ΛV is the singular value decomposition of C. Then the new observables can be written as Ãi=∑lUilAl, B̃j=-∑mVjm*Bm. We have
(3.16)∑iΔρ2(Ãi⊗I+I⊗B̃i)=∑i[ΔρA2(Ãi)+ΔρA2(B̃i)+2(〈Ãi⊗B̃i〉-〈Ãi〉〈B̃i〉)]=M-Tr[ρA2]+N-Tr[ρB2]-2∑i(UCV†)ii=M-Tr[ρA2]+N-Tr[ρB2]-2∥C∥KF.
Substituting above relation to (3.14), one gets (3.15).Bound (3.15) does not depend on the choice of local orthonormal observables. It can be easily applied and realized by direct measurements in experiments. It is in accord with the result in [46] where optimization of entanglement witness based on local uncertainty relation has been taken into account. As an example, let us consider the 3×3 bound entangled state [76](3.17)ρ=14(I9-∑i=04|ξi〉〈ξi|),
where I9 is the 9×9 identity matrix, |ξ0〉=(1/2)|0〉(|0〉-|1〉), |ξ1〉=(1/2)(|0〉-|1〉)|2〉, |ξ2〉=(1/2)|2〉(|1〉-|2〉), |ξ3〉=(1/2)(|1〉-|2〉)|0〉, and |ξ4〉=(1/3)(|0〉+|1〉+|2〉)(|0〉+|1〉+|2〉). We simply choose the local orthonormal observables to be the normalized generators of SU(3). Formula (3.13) gives C(ρ)≥0.050. Formula (3.14) gives C(ρ)≥0.052 [64], while formula (3.15) yields a better lower bound C(ρ)≥0.0555.If we mix the bound entangled state (3.17) with |ψ〉=(1/3)∑i=02|ii〉, that is,(3.18)ρ′=(1-x)ρ+x|ψ〉〈ψ|,
then it is easily seen that (3.15) gives a better lower bound of concurrence than formula (3.13) (Figure 1).Figure 1
Lower bounds from (3.15) (dashed line) and (3.13) (solid line) for mixed state (3.18).


### 3.1.2. Lower Bound of Concurrence from “Two-Qubit” Decomposition

In [67] the authors derived an analytical lower bound of concurrence for arbitrary bipartite quantum states by decomposing the joint Hilbert space into many 2⊗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 boundτ2 of concurrence for bipartite states has been obtained in [67]. For a bipartite quantum state ρ in H⊗H, the concurrence C(ρ) satisfies
(3.19)τ2(ρ)≡d2(d-1)∑m,n=1d(d-1)/2Cmn2(ρ)≤C2(ρ),
where Cmn(ρ)=max{0,λmn(1)-λmn(2)-λmn(3)-λmn(4)} with λmn(1),…,λmn(4) being the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃mn with ρ̃mn=(Lm⊗Ln)ρ*(Lm⊗Ln), while Lm and Ln are the generators of SO(d).The lower boundτ2 in (3.19) in fact characterizes all two qubits’ entanglement in a high dimensional bipartite state. One can directly verify that there are at most 4×4=16 nonzero elements in each matrix ρρ̃mn. These elements constitute a 4×4 matrix ϱ(σy⊗σy)ϱ*(σy⊗σy), where σy is the Pauli matrix, the matrix ϱ is a submatrix of the original ρ:(3.20)ϱ=(ρik,ikρik,ilρik,jkρik,jlρil,ikρil,ilρil,jkρil,jlρjk,ikρjk,ilρjk,jkρjk,jlρjl,ikρjl,ilρjl,jkρjl,jl),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 ϱ is not normalized but positive semidefinite. 𝒞mn are just the concurrences of these states (3.20).The boundτ2 provides a much clearer structure of entanglement, which not only yields an effective separability criterion and an easy evaluation of entanglement, but also helps one to classify mixed-state entanglement.(2) Lower Bound of Concurrence for Multipartite States
We first consider tripartite case. A general pure state onH⊗H⊗H is of the form
(3.21)|Ψ〉=∑i,j,k=1daijk|ijk〉,aijk∈ℂ,∑i,j,k=1daijkaijk*=1
with
(3.22)Cd3(|Ψ〉)=d6(d-1)∑(|aijkapqm-aijmapqk|2+|aijkapqm-aiqkapjm|2+|aijkapqm-apjkaiqm|2),
or equivalently
(3.23)Cd3(|Ψ〉)=d6(d-1)(3-(Tr[ρ12]+Tr[ρ22]+Tr[ρ32])),
where ρ1=Tr23[ρ],ρ2=Tr13[ρ],andρ3=Tr12[ρ] are the reduced density matrices of ρ=|Ψ〉〈Ψ|.DefineCαβ12|3(|Ψ〉)=|aijkapqm-aijmapqk|, Cαβ13|2(|Ψ〉)=|aijkapqm-aiqkapjm|, and Cαβ23|1(|Ψ〉)=|aijkapqm-apjkaiqm|, where α and β of Cαβ12|3 (resp., Cαβ13|2, resp., Cαβ23|1) stand for the subindices of a associated with the subspaces 1,2, and 3 (resp., 1,3, and 2; resp., 2,3, and 1). Let Li1i2⋯iN denote the generators of group SO(di1di2⋯diN) associated to the subsystems i1,i2,…,iN. Then for a tripartite pure state (3.21), one has
(3.24)Cd3(|Ψ〉)=d6(d-1)∑αd2(d2-1)/2∑βd(d-1)/2[(Cαβ12|3(|Ψ〉))2+(Cαβ13|2(|Ψ〉))2+(Cαβ23|1(|Ψ〉))2]=d6(d-1)∑αβ[(|〈Ψ|Sαβ12|3|Ψ*〉|)2+(|〈Ψ|Sαβ13|2|Ψ*〉|)2+(|〈Ψ|Sαβ23|2|Ψ*〉|)2],
where Sαβ12|3=(Lα12⊗Lβ3), Sαβ13|2=(Lα13⊗Lβ2), and Sαβ23|1=(Lβ1⊗Lα23).Theorem 3.2.
For an arbitrary mixed stateρ in H⊗H⊗H, the concurrence C(ρ) satisfies [112]
(3.25)τ3(ρ)≡d6(d-1)∑αd2(d2-1)/2∑βd(d-1)/2[(Cαβ12|3(ρ))2+(Cαβ13|2(ρ))2+(Cαβ23|1(ρ))2]≤C2(ρ),
where τ3(ρ) is a lower bound of C(ρ)(3.26)Cαβ12|3(ρ)=max{0,λ(1)αβ12∣3-λ(2)αβ12∣3-λ(3)αβ12∣3-λ(4)αβ12∣3},
where λ(1)αβ12|3,λ(2)αβ12|3,λ(3)αβ12|3,andλ(4)αβ12|3 are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃αβ12|3 with ρ̃αβ12|3=Sαβ12|3ρ*Sαβ12|3. Cαβ13|2(ρ) and Cαβ23|1(ρ) are defined in a similar way to Cαβ12|3(ρ).Proof.
Set|ξi〉=pi|ψi〉, xαβi=|〈ξi|Sαβ12|3|ξi*〉|, yαβi=|〈ξi|Sαβ13|2|ξi*〉|, and zαβi=|〈ξi|Sαβ1|23|ξi*〉|. We have, from Minkowski inequality,
(3.27)C(ρ)=min∑i‍d6(d-1)∑αβ[(xαβi)2+(yαβi)2+(zαβi)2]≥mind6(d-1)∑αβ(∑i[(xαβi)2+(yαβi)2+(zαβi)2]1/2)2.
Noting that for nonnegative real variablesxα, yα, zα and given that X=∑α=1Nxα, Y=∑α=1NYα, and Z=∑α=1Nzα, by using Lagrange multipliers, one obtains that the following inequality holds:
(3.28)∑α=1N(xα2+yα2+zα2)1/2≥(X2+Y2+Z2)1/2.
Therefore we have
(3.29)C(ρ)≥mind6(d-1)∑αβ‍[(∑ixαβi)2+(∑iyαβi)2+(∑izαβi)2]≥d6(d-1)∑αβ[(min∑ixαβi)2+(min∑iyαβi)2+(min∑izαβi)2].
The values ofCαβ12|3(ρ)≡min∑ixαβi, Cαβ13|2(ρ)≡min∑iyαβi, and Cαβ23|1(ρ)≡min∑izαβi can be calculated by using the similar procedure in [53]. Here we compute the value of Cαβ12|3(ρ) in detail. The values of Cαβ13|2(ρ) and Cαβ23|1(ρ) can be obtained analogously.
Letλi and |χi〉 be eigenvalues and eigenvectors of ρ, respectively. Any decomposition of ρ can be obtained from a unitary d3×d3 matrix Vij, |ξj〉=∑i=1d3Vij*(λi|χi〉). Therefore one has 〈ξi|Sαβ12|3|ξj*〉=(VYαβVT)ij, where the matrix Yαβ is defined by (Yαβ)ij=〈χi|Sαβ12|3|χj*〉. Namely, Cαβ12|3(ρ)=min∑i|[VYαβVT]ii|, which has an analytical expression [53], that Cαβ12|3(ρ)=max{0,λ(1)αβ12|3-∑j>1λ(j)αβ12|3}, where λαβ12|3(k) are the square roots of the eigenvalues of the positive Hermitian matrix YαβYαβ†, or equivalently the non-Hermitian matrix ρρ̃αβ, in decreasing order. Here as the matrix Sαβ12|3 has d2-4 rows and d2-4 columns that are identically zero, the matrix ρρ̃αβ has a rank not greater than 4, that is, λαβ12|3(j)=0 for j≥5. From (3.29) we have (3.25).Theorem3.2 can be directly generalized to arbitrary multipartite case.Theorem 3.3.
For an arbitraryN-partite state ρ∈H⊗H⊗⋯⊗H, the concurrence defined in (4.1) satisfies [112],
(3.30)τN(ρ)≡d2m(d-1)∑p∑αβ(Cαβp(ρ))2≤C2(ρ),
where τN(ρ) is the lower bound of C(ρ), ∑p stands for the summation over all possible combinations of the indices of α,β, Cαβp(ρ)=max{0,λ(1)αβp-λ(2)αβp-λ(3)αβp-λ(4)αβp}, and λ(i)αβp, i=1,2,3,4, are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃αβp where ρ̃αβp=Sαβpρ*Sαβp.Lower Bound and Separability
An N-partite quantum stateρ is fully separable if and only if there exist pi with pi≥0,∑ipi=1, and pure states ρij=|ψij〉〈ψij| such that
(3.31)ρ=∑ipiρi1⊗ρi2⊗⋯⊗ρiN.It is easily verified that, for a fully separable multipartite stateρ, τN(ρ)=0. Thus τN(ρ)>0 indicates that there must be some kinds of entanglement inside the quantum state, which shows that the lower bound τN(ρ) can be used to recognize entanglement.As an example, we consider a tripartite quantum state [79] ρ=((1-p)/8)I8+p|W〉〈W|, where I8 is the 8×8 identity matrix, and |W〉=(1/3)(|100〉+|010〉+|001〉) is the tripartite W state. Select an entanglement witness operator to be 𝒲=(1/2)I8-|GHZ〉〈GHZ|, where |GHZ〉=(1/2)(|000〉+|111〉) is to be the tripartite GHZ-state. From the condition Tr[𝒲ρ]<0, the entanglement of ρ is detected for (3/5)<p≤1 in [79]. In [97] the authors have obtained the generalized correlation matrix criterion which says that if an N-qubit quantum state is fully separable, then the inequality ∥𝒯N∥KF≤1 must hold, where ∥𝒯N∥KF=max{∥𝒯nN∥KF}, 𝒯nN is a kind of matrix unfold of tα1α2⋯αN defined by tα1α2⋯αN=Tr[ρσα1(1)σα2(2)⋯σαN(N)], and σαi(i) stands for the Pauli matrix. Now using the generalized correlation, matrix criterion the entanglement of ρ is detected for 0.3068<p≤1. From Theorem 3.2, we have the lower bound τ3(ρ)>0 for 0.2727<p≤1. Therefore the bound (3.71) detects entanglement better than these two criteria in this case. If we replace W with GHZ state in ρ, then the criterion in [97] detects the entanglement of ρ for 0.35355<p≤1, while τ3(ρ) detects, again better, the entanglement for 0.2<p≤1.Nevertheless for PPT statesρ, we have τ3(ρ)=0, which can be seen in the following way. A density matrix ρ is called PPT if the partial transposition of ρ over any subsystem(s) is still positive. Let ρTi denote the partial transposition with respect to the ith subsystem. Assume that there is a PPT state ρ with τ(ρ)>0. Then at least one term in (3.25), say Cα0β012|3(ρ), is not zero. Define ρα0β0=Lα012⊗Lβ03ρ(Lα012⊗Lβ03)†. By using the PPT property of ρ, we have(3.32)ρα0β0T3=Lα012⊗(Lβ03)*ρT3(Lα012)†⊗(Lβ03)T≥0.
Noting that both Lα012 and Lβ03 are projectors to two-dimensional subsystems, ρα0β0 can be considered as a 4×4 density matrix, while a PPT 4×4 density matrix ρα0β0 must be a separable state, which contradicts with Cα0β012|3(ρ)≠0.Relation between Lower Bounds of Bi- and Tripartite Concurrence
τ3 is basically different from τ2 as τ3 characterizes also genuine tripartite entanglement that can not be described by bipartite decompositions. Nevertheless, there are interesting relations between them.Theorem 3.4.
For any pure tripartite state (3.21), the following inequality holds [112]:
(3.33)τ2(ρ12)+τ2(ρ13)+τ2(ρ23)≤3τ3(ρ),
where τ2 is the lower bound of bipartite concurrence (3.19), τ3 is the lower bound of tripartite concurrence (3.25), and ρ12=Tr3[ρ], ρ13=Tr2[ρ], ρ23=Tr1[ρ], and ρ=|Ψ〉123〈Ψ|.Proof.
SinceCαβ2≤(λαβ(1))2≤∑i=14(λαβ(i))2=Tr[ρρ̃αβ] for ρ=ρ12, ρ=ρ13, and ρ=ρ23, we have
(3.34)τ2(ρ12)+τ2(ρ13)+τ2(ρ23)≤d2(d-1)(∑α,β=1d(d-1)/2Tr[ρ12(ρ̃12)αβ]+∑α,β=1d(d-1)/2Tr[ρ13(ρ̃13)αβ]+∑α,β=1d(d-1)/2Tr[ρ23(ρ̃23)αβ])=d2(d-1)(3-Tr[ρ12]-Tr[ρ22]-Tr[ρ32])=3C2(ρ)=3τ3(ρ),
where we have used the similar analysis in [67, 113] to obtain the equalities ∑α,βTr[ρ12(ρ̃12)αβ]=1-Tr[ρ12]-Tr[ρ22]+Tr[ρ32], ∑α,βTr[ρ13(ρ̃13)αβ]=1-Tr[ρ12]+Tr[ρ22]-Tr[ρ32], and ∑α,βTr[ρ23(ρ̃23)αβ]=1+Tr[ρ12]-Tr[ρ22]-Tr[ρ32]. The last equality is due to the fact that ρ is a pure state.In fact, the bipartite entanglement inside a tripartite state is useful for distilling maximally entangled states. Assume that there are two of the qualities{τ(ρ12),τ(ρ13),τ(ρ23)} larger than zero; say τ(ρ12)>0 and τ(ρ13)>0. According to [67], one can distill two maximal entangled states |ψ12〉 and |ψ13〉 which belong to ℋ1⊗ℋ2 and ℋ1⊗ℋ3, respectively. In terms of the result in [114], one can use them to produce a GHZ state.


### 3.1.3. Estimation of Multipartite Entanglement

For a pureN-partite quantum state |ψ〉∈ℋ1⊗ℋ2⊗⋯⊗ℋN, dimℋi=di, i=1,…,N, the concurrence of bipartite decomposition between subsystems 12⋯M and M+1⋯N is defined by(3.35)C2(|ψ〉)=2(1-Tr[ρ12⋯M2])
where ρ12⋯M2=TrM+1⋯N[|ψ〉〈ψ|] is the reduced density matrix of ρ=|ψ〉〈ψ| by tracing over subsystems M+1⋯N. On the other hand, the concurrence of |ψ〉 is defined by (3.9).For a mixed multipartite quantum stateρ=∑ipi|ψi〉〈ψi|∈ℋ1⊗ℋ2⊗⋯⊗ℋN, the corresponding concurrences of (3.35) and (3.9) are then given by the convex roof(3.36)C2(ρ)=min{pi,|ψi}〉∑i‍piC2(|ψi〉〈ψi|),
and (3.12). We now investigate the relation between these two kinds of concurrences.Lemma 3.5.
For a bipartite density matrixρ∈ℋA⊗ℋB, one has
(3.37)1-Tr[ρ2]≤1-Tr[ρA2]+1-Tr[ρB2],
where ρA/B=TrB/A[ρ] are the reduced density matrices of ρ.Proof.
Letρ=∑ijλij|ij〉〈ij| be the spectral decomposition, where λij≥0,∑ijλij=1. Then ρ1=∑ijλij|i〉〈i|, ρ2=∑ijλij|j〉〈j|. Therefore
(3.38)1-Tr[ρA2]+1-Tr[ρB2]-1+Tr[ρ2]=1-Tr[ρA2]-Tr[ρB2]+Tr[ρ2]=(∑ijλij)2-∑i,j,j′λijλij′-∑i,i′,jλijλi′j+∑ijλij2=(∑i=i′,j=j′λij2+∑i=i′,j≠j′λijλij′+∑i≠i′,j=j′λijλi′j+∑i≠i′,j≠j′λijλi′j′)=∑i≠i′,j≠j′λijλi′j′≥0.This lemma can be also derived in another way [46, 115].Theorem 3.6.
For a multipartite quantum stateρ∈ℋ1⊗ℋ2⊗⋯⊗ℋN with N≥3, the following inequality holds [116]:
(3.39)CN(ρ)≥max2(3-N)/2C2(ρ),
where the maximum is taken over all kinds of bipartite concurrence.Proof.
Without loss of generality, we suppose that the maximal bipartite concurrence is attained between subsystems12⋯M and (M+1)⋯N.
For a pure multipartite state|ψ〉∈ℋ1⊗ℋ2⊗⋯⊗ℋN, Tr[ρ12⋯M2]=Tr[ρ(M+1)⋯N2]. From (3.37) we have
(3.40)CN2(|ψ〉〈ψ|)=22-N((2N-2)-∑αTr[ρα2])≥23-N(N-∑k=1NTr[ρk2])≥23-N(1-Tr[ρ12⋯M2]+1-Tr[ρ(M+1)⋯N2])=23-N*2(1-Tr[ρ12⋯M2])=23-NC22(|ψ〉〈ψ|),
that is, CN(|ψ〉〈ψ|)≥2(3-N)/2C2(|ψ〉〈ψ|).
Letρ=∑ipi|ψi〉〈ψi| attain the minimal decomposition of the multipartite concurrence. One has
(3.41)CN(ρ)=∑ipiCN(|ψi〉〈ψi|)≥2(3-N)/2∑ipiC2(|ψi〉〈ψi|)≥2(3-N)/2min{pi,|ψi}∑ipiC2(|ψi〉〈ψi|)=2(3-N)/2C2(ρ).Corollary 3.7.
For a tripartite quantum stateρ∈ℋ1⊗ℋ2⊗ℋ3, the following inequality holds:
(3.42)C3(ρ)≥maxC2(ρ),
where the maximum is taken over all kinds of bipartite concurrence.In [46, 64], from the separability criteria related to local uncertainty relation, covariance matrix, and correlation matrix, the following lower bounds for bipartite concurrence are obtained:(3.43)C2(ρ)≥2∥C(ρ)∥-(1-Tr[ρA2])-(1-Tr[ρB2])2dA(dA-1),(3.44)C2(ρ)≥8dA3dB2(dA-1)(∥T(ρ)∥-dAdB(dA-1)(dB-1)2),
where the entries of the matrix C, Cij=〈λiA⊗λjB〉-〈λiA⊗IdB〉〈IdA⊗λjB〉, Tij=dAdB/2〈λiA⊗λjB〉, λkA/B stand for the normalized generator of SU(dA/dB), that is, Tr[λkA/BλlA/B]=δkl and 〈X〉=Tr[ρX]. It is shown that the lower bounds (3.43) and (3.44) are independent of (3.13).Now we consider a multipartite quantum stateρ∈ℋ1⊗ℋ2⊗⋯⊗ℋN as a bipartite state belonging to ℋA⊗ℋB with the dimensions of the subsystems A and B being dA=ds1ds2⋯dsm and dB=dsm+1dsm+2⋯dsN, respectively. By using Corollary 3.7, (3.13), (3.43), and (3.44), one has the following lower bound.Theorem 3.8.
For any N-partite quantum stateρ [116],
(3.45)CN(ρ)≥2(3-N)/2max{B1,B2,B3},
where
(3.46)B1=max{i}2Mi(Mi-1)[max(∥𝒯A(ρi)∥,∥R(ρi)∥)-1],B2=max{i}2∥C(ρi)∥-(1-Tr[(ρAi)2])-(1-Tr[(ρBi)2])2Mi(Mi-1),B3=max{i}8Mi3Ni2(Mi-1)(∥T(ρi)∥-MiNi(Mi-1)(Ni-1)2),ρi’s are all possible bipartite decompositions of ρ, and
(3.47)Mi=min{ds1ds2⋯dsm,dsm+1dsm+2⋯dsN},Ni=max{ds1ds2⋯dsm,dsm+1dsm+2⋯dsN}.In [46, 106, 107, 117], it is shown that the upper and lower bounds of multipartite concurrence satisfy(3.48)(4-23-N)Tr[ρ2]-22-N∑αTr[ρα2]≤CN(ρ)≤22-N[(2N-2)-∑αTr[ρα2]].In fact one can obtain a more effective upper bound for multipartite concurrence. Letρ=∑iλi|ψi〉〈ψi|∈ℋ1⊗ℋ2⊗⋯⊗ℋN, where |ψi〉’s are the orthogonal pure states and ∑iλi=1. We have(3.49)CN(ρ)=min{pi,|φi}〉∑ipiCN(|φi〉〈φi|)≤∑iλiCN(|ψi〉〈ψi|).
The right side of (3.49) gives a new upper bound of CN(ρ). Since
(3.50)∑iλiCN(|ψi〉〈ψi|)=21-N/2∑iλi(2N-2)-∑αTr[(ραi)2]≤21-N/2(2N-2)-∑αTr[∑iλi(ραi)2]≤21-N/2(2N-2)-∑αTr[(ρα)2],
the upper bound obtained in (3.49) is better than that in (3.48).


### 3.1.4. Bounds of Concurrence and Tangle

In [68], a lower bound for tangle defined in (3.8) has been derived as(3.51)τ(ρ)≥8MN(M+N)(∥T(ρ)∥HS2-MN(M-1)(N-1)4),
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 et al. in [106, 107] and Zhang et al. in [46]:(3.52)2(Tr[ρ2]-Tr[ρA2])≤C(ρ)≤2(1-Tr[ρA2]).Since the convexity ofC2(ρ), we have that τ(ρ)≥C2(ρ) always holds. For two-qubit quantum systems, tangle τ is always equal to the square of concurrence C2 [58, 113], as a decomposition {pi,|ψi〉} achieving the minimum in (3.3) has the property that C(|ψi〉)=C(|ψj〉)∀i,j. For higher dimensional systems we do not have similar relations. Thus it is meaningful to derive valid upper bound for tangle and lower bound for concurrence.Theorem 3.9.
For any quantum stateρ∈ℋA⊗ℋB, one has [118]
(3.53)τ(ρ)≤min{1-Tr[ρA2],1-Tr[ρB2]},(3.54)C(ρ)≥8MN(M+N)(∥T(ρ)∥HS-MN(M-1)(N-1)2),
where ρA is the reduced matrix of ρ, and T(ρ) is the correlation matrix of ρ defined in (3.44).Proof.
We assume that1-Tr[ρA2]≤1-Tr[ρB2] for convenience. By the definition of τ, we have that for a pure state |ψ〉,τ(|ψ〉)=2(1-Tr[(ρA|ψ〉)2]). Let ρ=∑ipiρi be the optimal decomposition such that τ(ρ)=∑ipiτ(ρi). We get
(3.55)τ(ρ)=∑ipiτ(ρi)=∑ipi2[1-Tr[(ρA|ψi〉)2]]=2[1-Tr[∑ipi(ρA|ψi〉)2]]≤2[1-Tr[ρA2]].
Note that, for pure state|ψ〉∈ℋA⊗ℋB [68],
(3.56)C(|ψ〉)=8MN(M+N)(∥T(|ψ〉)∥2-MN(M-1)(N-1)4).
Using the inequality a-b≥a-b for any a≥b, we get
(3.57)C(|ψ〉)≥8MN(M+N)(∥T(|ψ〉)∥HS-MN(M-1)(N-1)2).
Now let ρ=∑ipiρi be the optimal decomposition such that C(ρ)=∑ipiC(ρi). We get
(3.58)C(ρ)=∑ipiC(ρi)≥∑ipi8MN(M+N)(∥T(ρi)∥HS-MN(M-1)(N-1)2)=8MN(M+N)(∑ipi∥T(ρi)∥HS-MN(M-1)(N-1)2)≥8MN(M+N)(∥T(ρ)∥HS-MN(M-1)(N-1)2),
which ends the proof.The upper bound (3.53), together with the lower bounds (3.54), (3.43), (3.44), (3.51), and (3.52), can allow for estimations of entanglement for arbitrary quantum states. Moreover, since the upper bound is exactly the value of tangle for pure states, the upper bound can be a good estimation when the state is very weakly mixed.


## 3.1.1. Lower Bound of Concurrence from Covariance Matrix Criterion

In [61] a lower bound of C(ρ) has been obtained as(3.13)C(ρ)≥2M(M-1)[Max(∥TA(ρ)∥,∥R(ρ)∥)-1],
where TA and R stand for partial transpose with respect to subsystem A and the realignment, respectively. This bound is further improved based on local uncertainty relations [64](3.14)C(ρ)≥M+N-2-∑iΔρ2(GiA⊗I+I⊗GiB)2M(M-1),
where GiA and GiB are any set of local orthonormal observables, Δρ2(X)=Tr[X2ρ]-(Tr[Xρ])2.Bound (3.14) again depends on the choice of the local orthonormal observables. This bound can be optimized, in the sense that a local orthonormal observable-independent up bound of the right-hand side of (3.14) can be obtained.Theorem 3.1.
Letρ be a bipartite state in ℋMA⊗ℋNB. Then C(ρ) satisfies [90]
(3.15)C(ρ)≥2∥C∥KF-(1-Tr[ρA2])-(1-Tr[ρB2])2M(M-1).Proof.
The other orthonormal normalized basis of the local orthonormal observable space can be obtained fromAi and Bi by unitary transformations U and V: Ãi=∑lUilAl and B̃j=∑mVjm*Bm. Select U and V so that C=U†ΛV is the singular value decomposition of C. Then the new observables can be written as Ãi=∑lUilAl, B̃j=-∑mVjm*Bm. We have
(3.16)∑iΔρ2(Ãi⊗I+I⊗B̃i)=∑i[ΔρA2(Ãi)+ΔρA2(B̃i)+2(〈Ãi⊗B̃i〉-〈Ãi〉〈B̃i〉)]=M-Tr[ρA2]+N-Tr[ρB2]-2∑i(UCV†)ii=M-Tr[ρA2]+N-Tr[ρB2]-2∥C∥KF.
Substituting above relation to (3.14), one gets (3.15).Bound (3.15) does not depend on the choice of local orthonormal observables. It can be easily applied and realized by direct measurements in experiments. It is in accord with the result in [46] where optimization of entanglement witness based on local uncertainty relation has been taken into account. As an example, let us consider the 3×3 bound entangled state [76](3.17)ρ=14(I9-∑i=04|ξi〉〈ξi|),
where I9 is the 9×9 identity matrix, |ξ0〉=(1/2)|0〉(|0〉-|1〉), |ξ1〉=(1/2)(|0〉-|1〉)|2〉, |ξ2〉=(1/2)|2〉(|1〉-|2〉), |ξ3〉=(1/2)(|1〉-|2〉)|0〉, and |ξ4〉=(1/3)(|0〉+|1〉+|2〉)(|0〉+|1〉+|2〉). We simply choose the local orthonormal observables to be the normalized generators of SU(3). Formula (3.13) gives C(ρ)≥0.050. Formula (3.14) gives C(ρ)≥0.052 [64], while formula (3.15) yields a better lower bound C(ρ)≥0.0555.If we mix the bound entangled state (3.17) with |ψ〉=(1/3)∑i=02|ii〉, that is,(3.18)ρ′=(1-x)ρ+x|ψ〉〈ψ|,
then it is easily seen that (3.15) gives a better lower bound of concurrence than formula (3.13) (Figure 1).Figure 1
Lower bounds from (3.15) (dashed line) and (3.13) (solid line) for mixed state (3.18).


## 3.1.2. Lower Bound of Concurrence from “Two-Qubit” Decomposition

In [67] the authors derived an analytical lower bound of concurrence for arbitrary bipartite quantum states by decomposing the joint Hilbert space into many 2⊗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 boundτ2 of concurrence for bipartite states has been obtained in [67]. For a bipartite quantum state ρ in H⊗H, the concurrence C(ρ) satisfies
(3.19)τ2(ρ)≡d2(d-1)∑m,n=1d(d-1)/2Cmn2(ρ)≤C2(ρ),
where Cmn(ρ)=max{0,λmn(1)-λmn(2)-λmn(3)-λmn(4)} with λmn(1),…,λmn(4) being the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃mn with ρ̃mn=(Lm⊗Ln)ρ*(Lm⊗Ln), while Lm and Ln are the generators of SO(d).The lower boundτ2 in (3.19) in fact characterizes all two qubits’ entanglement in a high dimensional bipartite state. One can directly verify that there are at most 4×4=16 nonzero elements in each matrix ρρ̃mn. These elements constitute a 4×4 matrix ϱ(σy⊗σy)ϱ*(σy⊗σy), where σy is the Pauli matrix, the matrix ϱ is a submatrix of the original ρ:(3.20)ϱ=(ρik,ikρik,ilρik,jkρik,jlρil,ikρil,ilρil,jkρil,jlρjk,ikρjk,ilρjk,jkρjk,jlρjl,ikρjl,ilρjl,jkρjl,jl),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 ϱ is not normalized but positive semidefinite. 𝒞mn are just the concurrences of these states (3.20).The boundτ2 provides a much clearer structure of entanglement, which not only yields an effective separability criterion and an easy evaluation of entanglement, but also helps one to classify mixed-state entanglement.(2) Lower Bound of Concurrence for Multipartite States
We first consider tripartite case. A general pure state onH⊗H⊗H is of the form
(3.21)|Ψ〉=∑i,j,k=1daijk|ijk〉,aijk∈ℂ,∑i,j,k=1daijkaijk*=1
with
(3.22)Cd3(|Ψ〉)=d6(d-1)∑(|aijkapqm-aijmapqk|2+|aijkapqm-aiqkapjm|2+|aijkapqm-apjkaiqm|2),
or equivalently
(3.23)Cd3(|Ψ〉)=d6(d-1)(3-(Tr[ρ12]+Tr[ρ22]+Tr[ρ32])),
where ρ1=Tr23[ρ],ρ2=Tr13[ρ],andρ3=Tr12[ρ] are the reduced density matrices of ρ=|Ψ〉〈Ψ|.DefineCαβ12|3(|Ψ〉)=|aijkapqm-aijmapqk|, Cαβ13|2(|Ψ〉)=|aijkapqm-aiqkapjm|, and Cαβ23|1(|Ψ〉)=|aijkapqm-apjkaiqm|, where α and β of Cαβ12|3 (resp., Cαβ13|2, resp., Cαβ23|1) stand for the subindices of a associated with the subspaces 1,2, and 3 (resp., 1,3, and 2; resp., 2,3, and 1). Let Li1i2⋯iN denote the generators of group SO(di1di2⋯diN) associated to the subsystems i1,i2,…,iN. Then for a tripartite pure state (3.21), one has
(3.24)Cd3(|Ψ〉)=d6(d-1)∑αd2(d2-1)/2∑βd(d-1)/2[(Cαβ12|3(|Ψ〉))2+(Cαβ13|2(|Ψ〉))2+(Cαβ23|1(|Ψ〉))2]=d6(d-1)∑αβ[(|〈Ψ|Sαβ12|3|Ψ*〉|)2+(|〈Ψ|Sαβ13|2|Ψ*〉|)2+(|〈Ψ|Sαβ23|2|Ψ*〉|)2],
where Sαβ12|3=(Lα12⊗Lβ3), Sαβ13|2=(Lα13⊗Lβ2), and Sαβ23|1=(Lβ1⊗Lα23).Theorem 3.2.
For an arbitrary mixed stateρ in H⊗H⊗H, the concurrence C(ρ) satisfies [112]
(3.25)τ3(ρ)≡d6(d-1)∑αd2(d2-1)/2∑βd(d-1)/2[(Cαβ12|3(ρ))2+(Cαβ13|2(ρ))2+(Cαβ23|1(ρ))2]≤C2(ρ),
where τ3(ρ) is a lower bound of C(ρ)(3.26)Cαβ12|3(ρ)=max{0,λ(1)αβ12∣3-λ(2)αβ12∣3-λ(3)αβ12∣3-λ(4)αβ12∣3},
where λ(1)αβ12|3,λ(2)αβ12|3,λ(3)αβ12|3,andλ(4)αβ12|3 are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃αβ12|3 with ρ̃αβ12|3=Sαβ12|3ρ*Sαβ12|3. Cαβ13|2(ρ) and Cαβ23|1(ρ) are defined in a similar way to Cαβ12|3(ρ).Proof.
Set|ξi〉=pi|ψi〉, xαβi=|〈ξi|Sαβ12|3|ξi*〉|, yαβi=|〈ξi|Sαβ13|2|ξi*〉|, and zαβi=|〈ξi|Sαβ1|23|ξi*〉|. We have, from Minkowski inequality,
(3.27)C(ρ)=min∑i‍d6(d-1)∑αβ[(xαβi)2+(yαβi)2+(zαβi)2]≥mind6(d-1)∑αβ(∑i[(xαβi)2+(yαβi)2+(zαβi)2]1/2)2.
Noting that for nonnegative real variablesxα, yα, zα and given that X=∑α=1Nxα, Y=∑α=1NYα, and Z=∑α=1Nzα, by using Lagrange multipliers, one obtains that the following inequality holds:
(3.28)∑α=1N(xα2+yα2+zα2)1/2≥(X2+Y2+Z2)1/2.
Therefore we have
(3.29)C(ρ)≥mind6(d-1)∑αβ‍[(∑ixαβi)2+(∑iyαβi)2+(∑izαβi)2]≥d6(d-1)∑αβ[(min∑ixαβi)2+(min∑iyαβi)2+(min∑izαβi)2].
The values ofCαβ12|3(ρ)≡min∑ixαβi, Cαβ13|2(ρ)≡min∑iyαβi, and Cαβ23|1(ρ)≡min∑izαβi can be calculated by using the similar procedure in [53]. Here we compute the value of Cαβ12|3(ρ) in detail. The values of Cαβ13|2(ρ) and Cαβ23|1(ρ) can be obtained analogously.
Letλi and |χi〉 be eigenvalues and eigenvectors of ρ, respectively. Any decomposition of ρ can be obtained from a unitary d3×d3 matrix Vij, |ξj〉=∑i=1d3Vij*(λi|χi〉). Therefore one has 〈ξi|Sαβ12|3|ξj*〉=(VYαβVT)ij, where the matrix Yαβ is defined by (Yαβ)ij=〈χi|Sαβ12|3|χj*〉. Namely, Cαβ12|3(ρ)=min∑i|[VYαβVT]ii|, which has an analytical expression [53], that Cαβ12|3(ρ)=max{0,λ(1)αβ12|3-∑j>1λ(j)αβ12|3}, where λαβ12|3(k) are the square roots of the eigenvalues of the positive Hermitian matrix YαβYαβ†, or equivalently the non-Hermitian matrix ρρ̃αβ, in decreasing order. Here as the matrix Sαβ12|3 has d2-4 rows and d2-4 columns that are identically zero, the matrix ρρ̃αβ has a rank not greater than 4, that is, λαβ12|3(j)=0 for j≥5. From (3.29) we have (3.25).Theorem3.2 can be directly generalized to arbitrary multipartite case.Theorem 3.3.
For an arbitraryN-partite state ρ∈H⊗H⊗⋯⊗H, the concurrence defined in (4.1) satisfies [112],
(3.30)τN(ρ)≡d2m(d-1)∑p∑αβ(Cαβp(ρ))2≤C2(ρ),
where τN(ρ) is the lower bound of C(ρ), ∑p stands for the summation over all possible combinations of the indices of α,β, Cαβp(ρ)=max{0,λ(1)αβp-λ(2)αβp-λ(3)αβp-λ(4)αβp}, and λ(i)αβp, i=1,2,3,4, are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix ρρ̃αβp where ρ̃αβp=Sαβpρ*Sαβp.Lower Bound and Separability
An N-partite quantum stateρ is fully separable if and only if there exist pi with pi≥0,∑ipi=1, and pure states ρij=|ψij〉〈ψij| such that
(3.31)ρ=∑ipiρi1⊗ρi2⊗⋯⊗ρiN.It is easily verified that, for a fully separable multipartite stateρ, τN(ρ)=0. Thus τN(ρ)>0 indicates that there must be some kinds of entanglement inside the quantum state, which shows that the lower bound τN(ρ) can be used to recognize entanglement.As an example, we consider a tripartite quantum state [79] ρ=((1-p)/8)I8+p|W〉〈W|, where I8 is the 8×8 identity matrix, and |W〉=(1/3)(|100〉+|010〉+|001〉) is the tripartite W state. Select an entanglement witness operator to be 𝒲=(1/2)I8-|GHZ〉〈GHZ|, where |GHZ〉=(1/2)(|000〉+|111〉) is to be the tripartite GHZ-state. From the condition Tr[𝒲ρ]<0, the entanglement of ρ is detected for (3/5)<p≤1 in [79]. In [97] the authors have obtained the generalized correlation matrix criterion which says that if an N-qubit quantum state is fully separable, then the inequality ∥𝒯N∥KF≤1 must hold, where ∥𝒯N∥KF=max{∥𝒯nN∥KF}, 𝒯nN is a kind of matrix unfold of tα1α2⋯αN defined by tα1α2⋯αN=Tr[ρσα1(1)σα2(2)⋯σαN(N)], and σαi(i) stands for the Pauli matrix. Now using the generalized correlation, matrix criterion the entanglement of ρ is detected for 0.3068<p≤1. From Theorem 3.2, we have the lower bound τ3(ρ)>0 for 0.2727<p≤1. Therefore the bound (3.71) detects entanglement better than these two criteria in this case. If we replace W with GHZ state in ρ, then the criterion in [97] detects the entanglement of ρ for 0.35355<p≤1, while τ3(ρ) detects, again better, the entanglement for 0.2<p≤1.Nevertheless for PPT statesρ, we have τ3(ρ)=0, which can be seen in the following way. A density matrix ρ is called PPT if the partial transposition of ρ over any subsystem(s) is still positive. Let ρTi denote the partial transposition with respect to the ith subsystem. Assume that there is a PPT state ρ with τ(ρ)>0. Then at least one term in (3.25), say Cα0β012|3(ρ), is not zero. Define ρα0β0=Lα012⊗Lβ03ρ(Lα012⊗Lβ03)†. By using the PPT property of ρ, we have(3.32)ρα0β0T3=Lα012⊗(Lβ03)*ρT3(Lα012)†⊗(Lβ03)T≥0.
Noting that both Lα012 and Lβ03 are projectors to two-dimensional subsystems, ρα0β0 can be considered as a 4×4 density matrix, while a PPT 4×4 density matrix ρα0β0 must be a separable state, which contradicts with Cα0β012|3(ρ)≠0.Relation between Lower Bounds of Bi- and Tripartite Concurrence
τ3 is basically different from τ2 as τ3 characterizes also genuine tripartite entanglement that can not be described by bipartite decompositions. Nevertheless, there are interesting relations between them.Theorem 3.4.
For any pure tripartite state (3.21), the following inequality holds [112]:
(3.33)τ2(ρ12)+τ2(ρ13)+τ2(ρ23)≤3τ3(ρ),
where τ2 is the lower bound of bipartite concurrence (3.19), τ3 is the lower bound of tripartite concurrence (3.25), and ρ12=Tr3[ρ], ρ13=Tr2[ρ], ρ23=Tr1[ρ], and ρ=|Ψ〉123〈Ψ|.Proof.
SinceCαβ2≤(λαβ(1))2≤∑i=14(λαβ(i))2=Tr[ρρ̃αβ] for ρ=ρ12, ρ=ρ13, and ρ=ρ23, we have
(3.34)τ2(ρ12)+τ2(ρ13)+τ2(ρ23)≤d2(d-1)(∑α,β=1d(d-1)/2Tr[ρ12(ρ̃12)αβ]+∑α,β=1d(d-1)/2Tr[ρ13(ρ̃13)αβ]+∑α,β=1d(d-1)/2Tr[ρ23(ρ̃23)αβ])=d2(d-1)(3-Tr[ρ12]-Tr[ρ22]-Tr[ρ32])=3C2(ρ)=3τ3(ρ),
where we have used the similar analysis in [67, 113] to obtain the equalities ∑α,βTr[ρ12(ρ̃12)αβ]=1-Tr[ρ12]-Tr[ρ22]+Tr[ρ32], ∑α,βTr[ρ13(ρ̃13)αβ]=1-Tr[ρ12]+Tr[ρ22]-Tr[ρ32], and ∑α,βTr[ρ23(ρ̃23)αβ]=1+Tr[ρ12]-Tr[ρ22]-Tr[ρ32]. The last equality is due to the fact that ρ is a pure state.In fact, the bipartite entanglement inside a tripartite state is useful for distilling maximally entangled states. Assume that there are two of the qualities{τ(ρ12),τ(ρ13),τ(ρ23)} larger than zero; say τ(ρ12)>0 and τ(ρ13)>0. According to [67], one can distill two maximal entangled states |ψ12〉 and |ψ13〉 which belong to ℋ1⊗ℋ2 and ℋ1⊗ℋ3, respectively. In terms of the result in [114], one can use them to produce a GHZ state.


## 3.1.3. Estimation of Multipartite Entanglement

For a pureN-partite quantum state |ψ〉∈ℋ1⊗ℋ2⊗⋯⊗ℋN, dimℋi=di, i=1,…,N, the concurrence of bipartite decomposition between subsystems 12⋯M and M+1⋯N is defined by(3.35)C2(|ψ〉)=2(1-Tr[ρ12⋯M2])
where ρ12⋯M2=TrM+1⋯N[|ψ〉〈ψ|] is the reduced density matrix of ρ=|ψ〉〈ψ| by tracing over subsystems M+1⋯N. On the other hand, the concurrence of |ψ〉 is defined by (3.9).For a mixed multipartite quantum stateρ=∑ipi|ψi〉〈ψi|∈ℋ1⊗ℋ2⊗⋯⊗ℋN, the corresponding concurrences of (3.35) and (3.9) are then given by the convex roof(3.36)C2(ρ)=min{pi,|ψi}〉∑i‍piC2(|ψi〉〈ψi|),
and (3.12). We now investigate the relation between these two kinds of concurrences.Lemma 3.5.
For a bipartite density matrixρ∈ℋA⊗ℋB, one has
(3.37)1-Tr[ρ2]≤1-Tr[ρA2]+1-Tr[ρB2],
where ρA/B=TrB/A[ρ] are the reduced density matrices of ρ.Proof.
Letρ=∑ijλij|ij〉〈ij| be the spectral decomposition, where λij≥0,∑ijλij=1. Then ρ1=∑ijλij|i〉〈i|, ρ2=∑ijλij|j〉〈j|. Therefore
(3.38)1-Tr[ρA2]+1-Tr[ρB2]-1+Tr[ρ2]=1-Tr[ρA2]-Tr[ρB2]+Tr[ρ2]=(∑ijλij)2-∑i,j,j′λijλij′-∑i,i′,jλijλi′j+∑ijλij2=(∑i=i′,j=j′λij2+∑i=i′,j≠j′λijλij′+∑i≠i′,j=j′λijλi′j+∑i≠i′,j≠j′λijλi′j′)=∑i≠i′,j≠j′λijλi′j′≥0.This lemma can be also derived in another way [46, 115].Theorem 3.6.
For a multipartite quantum stateρ∈ℋ1⊗ℋ2⊗⋯⊗ℋN with N≥3, the following inequality holds [116]:
(3.39)CN(ρ)≥max2(3-N)/2C2(ρ),
where the maximum is taken over all kinds of bipartite concurrence.Proof.
Without loss of generality, we suppose that the maximal bipartite concurrence is attained between subsystems12⋯M and (M+1)⋯N.
For a pure multipartite state|ψ〉∈ℋ1⊗ℋ2⊗⋯⊗ℋN, Tr[ρ12⋯M2]=Tr[ρ(M+1)⋯N2]. From (3.37) we have
(3.40)CN2(|ψ〉〈ψ|)=22-N((2N-2)-∑αTr[ρα2])≥23-N(N-∑k=1NTr[ρk2])≥23-N(1-Tr[ρ12⋯M2]+1-Tr[ρ(M+1)⋯N2])=23-N*2(1-Tr[ρ12⋯M2])=23-NC22(|ψ〉〈ψ|),
that is, CN(|ψ〉〈ψ|)≥2(3-N)/2C2(|ψ〉〈ψ|).
Letρ=∑ipi|ψi〉〈ψi| attain the minimal decomposition of the multipartite concurrence. One has
(3.41)CN(ρ)=∑ipiCN(|ψi〉〈ψi|)≥2(3-N)/2∑ipiC2(|ψi〉〈ψi|)≥2(3-N)/2min{pi,|ψi}∑ipiC2(|ψi〉〈ψi|)=2(3-N)/2C2(ρ).Corollary 3.7.
For a tripartite quantum stateρ∈ℋ1⊗ℋ2⊗ℋ3, the following inequality holds:
(3.42)C3(ρ)≥maxC2(ρ),
where the maximum is taken over all kinds of bipartite concurrence.In [46, 64], from the separability criteria related to local uncertainty relation, covariance matrix, and correlation matrix, the following lower bounds for bipartite concurrence are obtained:(3.43)C2(ρ)≥2∥C(ρ)∥-(1-Tr[ρA2])-(1-Tr[ρB2])2dA(dA-1),(3.44)C2(ρ)≥8dA3dB2(dA-1)(∥T(ρ)∥-dAdB(dA-1)(dB-1)2),
where the entries of the matrix C, Cij=〈λiA⊗λjB〉-〈λiA⊗IdB〉〈IdA⊗λjB〉, Tij=dAdB/2〈λiA⊗λjB〉, λkA/B stand for the normalized generator of SU(dA/dB), that is, Tr[λkA/BλlA/B]=δkl and 〈X〉=Tr[ρX]. It is shown that the lower bounds (3.43) and (3.44) are independent of (3.13).Now we consider a multipartite quantum stateρ∈ℋ1⊗ℋ2⊗⋯⊗ℋN as a bipartite state belonging to ℋA⊗ℋB with the dimensions of the subsystems A and B being dA=ds1ds2⋯dsm and dB=dsm+1dsm+2⋯dsN, respectively. By using Corollary 3.7, (3.13), (3.43), and (3.44), one has the following lower bound.Theorem 3.8.
For any N-partite quantum stateρ [116],
(3.45)CN(ρ)≥2(3-N)/2max{B1,B2,B3},
where
(3.46)B1=max{i}2Mi(Mi-1)[max(∥𝒯A(ρi)∥,∥R(ρi)∥)-1],B2=max{i}2∥C(ρi)∥-(1-Tr[(ρAi)2])-(1-Tr[(ρBi)2])2Mi(Mi-1),B3=max{i}8Mi3Ni2(Mi-1)(∥T(ρi)∥-MiNi(Mi-1)(Ni-1)2),ρi’s are all possible bipartite decompositions of ρ, and
(3.47)Mi=min{ds1ds2⋯dsm,dsm+1dsm+2⋯dsN},Ni=max{ds1ds2⋯dsm,dsm+1dsm+2⋯dsN}.In [46, 106, 107, 117], it is shown that the upper and lower bounds of multipartite concurrence satisfy(3.48)(4-23-N)Tr[ρ2]-22-N∑αTr[ρα2]≤CN(ρ)≤22-N[(2N-2)-∑αTr[ρα2]].In fact one can obtain a more effective upper bound for multipartite concurrence. Letρ=∑iλi|ψi〉〈ψi|∈ℋ1⊗ℋ2⊗⋯⊗ℋN, where |ψi〉’s are the orthogonal pure states and ∑iλi=1. We have(3.49)CN(ρ)=min{pi,|φi}〉∑ipiCN(|φi〉〈φi|)≤∑iλiCN(|ψi〉〈ψi|).
The right side of (3.49) gives a new upper bound of CN(ρ). Since
(3.50)∑iλiCN(|ψi〉〈ψi|)=21-N/2∑iλi(2N-2)-∑αTr[(ραi)2]≤21-N/2(2N-2)-∑αTr[∑iλi(ραi)2]≤21-N/2(2N-2)-∑αTr[(ρα)2],
the upper bound obtained in (3.49) is better than that in (3.48).


## 3.1.4. Bounds of Concurrence and Tangle

In [68], a lower bound for tangle defined in (3.8) has been derived as(3.51)τ(ρ)≥8MN(M+N)(∥T(ρ)∥HS2-MN(M-1)(N-1)4),
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 et al. in [106, 107] and Zhang et al. in [46]:(3.52)2(Tr[ρ2]-Tr[ρA2])≤C(ρ)≤2(1-Tr[ρA2]).Since the convexity ofC2(ρ), we have that τ(ρ)≥C2(ρ) always holds. For two-qubit quantum systems, tangle τ is always equal to the square of concurrence C2 [58, 113], as a decomposition {pi,|ψi〉} achieving the minimum in (3.3) has the property that C(|ψi〉)=C(|ψj〉)∀i,j. For higher dimensional systems we do not have similar relations. Thus it is meaningful to derive valid upper bound for tangle and lower bound for concurrence.Theorem 3.9.
For any quantum stateρ∈ℋA⊗ℋB, one has [118]
(3.53)τ(ρ)≤min{1-Tr[ρA2],1-Tr[ρB2]},(3.54)C(ρ)≥8MN(M+N)(∥T(ρ)∥HS-MN(M-1)(N-1)2),
where ρA is the reduced matrix of ρ, and T(ρ) is the correlation matrix of ρ defined in (3.44).Proof.
We assume that1-Tr[ρA2]≤1-Tr[ρB2] for convenience. By the definition of τ, we have that for a pure state |ψ〉,τ(|ψ〉)=2(1-Tr[(ρA|ψ〉)2]). Let ρ=∑ipiρi be the optimal decomposition such that τ(ρ)=∑ipiτ(ρi). We get
(3.55)τ(ρ)=∑ipiτ(ρi)=∑ipi2[1-Tr[(ρA|ψi〉)2]]=2[1-Tr[∑ipi(ρA|ψi〉)2]]≤2[1-Tr[ρA2]].
Note that, for pure state|ψ〉∈ℋA⊗ℋB [68],
(3.56)C(|ψ〉)=8MN(M+N)(∥T(|ψ〉)∥2-MN(M-1)(N-1)4).
Using the inequality a-b≥a-b for any a≥b, we get
(3.57)C(|ψ〉)≥8MN(M+N)(∥T(|ψ〉)∥HS-MN(M-1)(N-1)2).
Now let ρ=∑ipiρi be the optimal decomposition such that C(ρ)=∑ipiC(ρi). We get
(3.58)C(ρ)=∑ipiC(ρi)≥∑ipi8MN(M+N)(∥T(ρi)∥HS-MN(M-1)(N-1)2)=8MN(M+N)(∑ipi∥T(ρi)∥HS-MN(M-1)(N-1)2)≥8MN(M+N)(∥T(ρ)∥HS-MN(M-1)(N-1)2),
which ends the proof.The upper bound (3.53), together with the lower bounds (3.54), (3.43), (3.44), (3.51), and (3.52), can allow for estimations of entanglement for arbitrary quantum states. Moreover, since the upper bound is exactly the value of tangle for pure states, the upper bound can be a good estimation when the state is very weakly mixed.


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

In this subsection we show that although bound entangled states cannot be distilled, the concurrence and tangle of two entangled states will be always strictly larger than those of one, even if the two entangled states are both bound entangled.Letρ=∑ijklρij,kl|ij〉〈kl|∈ℋA⊗ℋB and σ=∑i′j′k′l′σi′j′,k′l′|i′j′〉〈k′l′|∈ℋA′⊗ℋB′ be two quantum states shared by subsystems AA′ and BB′. We use ρ⊗σ=∑ijkl,i′j′k′l′ρij,klσi′j′,k′l′|ii′〉AA′〈kk′|⊗|jj′〉BB′〈ll′| to denote the state of the whole system.Lemma 3.10.
For pure states|ψ〉∈ℋA⊗ℋB and |φ〉∈ℋA′⊗ℋB′, the inequalities
(3.59)C(|ψ〉⊗|φ〉)≥max{C(|ψ〉),C(|φ〉)},(3.60)τ(|ψ〉⊗|φ〉)≥max{τ(|ψ〉),τ(|φ〉)}
always hold, and  “=” in the two inequalities hold if and only if at least one of {|ψ〉,|φ〉} is separable.Proof.
Without loss of generality we assume thatC(|ψ〉)≥C(|φ〉). First note that
(3.61)ρAA′|ψ〉⊗|φ〉=ρA|ψ〉⊗ρA′|φ〉.
Let ρA|ψ〉=∑iλi|i〉〈i| and ρA′|φ〉=∑jπj|j〉〈j| be the spectral decomposition of ρA|ψ〉 and ρA′|φ〉, with ∑iλi=1 and ∑jπj=1, respectively. By using (3.61) one obtains that
(3.62)Tr[(ρAA′|ψ〉⊗|φ〉)2]=∑λiπjλi′πj′|ij〉〈ij∣i′j′〉〈i′j′|=∑λi2πj2,
while
(3.63)Tr[(ρA|ψ〉)2]=∑iλi2.
Now using the definition of concurrence and the normalization conditions ofλi and πj, one immediately gets
(3.64)C(|ψ〉⊗|φ〉)=2(1-Tr[(ρAA′|ψ〉⊗|φ〉)2])≥2(1-Tr[(ρA|ψ〉)2])=C(|ψ〉).
If one of {|ψ〉,|φ〉} is separable, say |φ〉, then the rank of ρA′|φ〉 must be one, which means that there is only one item in the spectral decomposition in ρA′|φ〉. Using the normalization condition of πj, we obtain Tr[(ρAA′|ψ〉⊗|φ〉)2]=Tr[(ρA|ψ〉)2]. Then inequality (3.64) becomes an equality.
On the other hand, if both|ψ〉 and |φ〉 are entangled (not separable), then there must be at least two items in the decomposition of their reduced density matrices ρA|ψ〉 and ρA′|φ〉, which means that Tr[(ρAA′|ψ〉⊗|φ〉)2] is strictly larger than Tr[(ρA|ψ〉)2].
The inequality (3.60) also holds because, for pure quantum state ρ, τ(ρ)=C2(ρ).From the lemma, we have, for mixed states the following.Theorem 3.11.
For any quantum statesρ∈ℋA⊗ℋB and σ∈ℋA′⊗ℋB′, the inequalities
(3.65)C(ρ⊗σ)≥max{C(ρ),C(σ)},τ(ρ⊗σ)≥max{τ(ρ),τ(σ)}
always hold, and the “=’’ in the two inequalities hold if and only if at least one of {ρ,σ} is separable, that is,  if both ρ and σ are entangled (even if bound entangled), then C(ρ⊗σ)>max{C(ρ),C(σ)} and τ(ρ⊗σ)>max{τ(ρ),τ(σ)} always hold [118].Proof.
We still assume thatC(ρ)≥C(σ) for convenience. Let ρ=∑ipiρi and σ=∑jqjσj be the optimal decomposition such that C(ρ⊗σ)=∑ipiqjC(ρi⊗σj). By using the inequality obtained in Lemma 3.10, we have
(3.66)C(ρ⊗σ)=∑ipiqjC(ρi⊗σj)≥∑ipiqjC(ρi)=∑ipiC(ρi)≥C(ρ).Case 1.
Now let one of{ρ,σ} be separable, say σ, with ensemble representation σ=∑jqjσj, where ∑jqj=1, and σj is the density matrix of separable pure state. Suppose that ρ=∑ipiρi is the optimal decomposition of ρ such that C(ρ)=∑ipiC(ρi). Using Lemma 3.10, we have
(3.67)C(ρ⊗σ)≤∑ipiqjC(ρi⊗σj)=∑ipiqjC(ρi)=∑ipiC(ρi)=C(ρ).
The inequalities (3.66) and (3.67) show that if σ is separable, then C(ρ⊗σ)=C(ρ).Case 2.
If bothρ and σ are inseparable, that is,  there is at least one pure state in the ensemble decomposition of ρ (and σ, resp.), using Lemma 3.10, then we have
(3.68)C(ρ⊗σ)=∑ipiqjC(ρi⊗σj)>∑ipiqjC(ρi)=∑ipiC(ρi)≥C(ρ).
The inequality for tangleτ can be proved in a similar way.Remark 3.12.
In [119] it is shown that any entangled state ρ can enhance the teleportation power of another state σ. This holds even if the state ρ is bound entangled. But if ρ is bound entangled, then the corresponding σ must be free entangled (distillable). By Theorem 3.11, we can see that even if two entangled quantum states ρ and σ are bound entangled, their concurrence and tangle are strictly larger than those of one state.


## 3.3. Subadditivity of Concurrence and Tangle

We now give a proof of the subadditivity of concurrence and tangle, which illustrates that concurrence and tangle may be proper entanglement measurements.Theorem 3.13.
Letρ and σ be quantum states in ℋA⊗ℋB, the one has [118]
(3.69)C(ρ⊗σ)≤C(ρ)+C(σ),τ(ρ⊗σ)≤τ(ρ)+τ(σ).Proof.
We first prove that the theorem holds for pure states, that is,  for|ψ〉 and |ϕ〉 in ℋA⊗ℋB:
(3.70)C(|ψ〉⊗|ϕ〉)≤C(|ψ〉)+C(|ϕ〉),τ(|ψ〉⊗|ϕ〉)≤τ(|ψ〉)+τ(|ϕ〉).
Assume that ρA|ψ〉=∑iλi|i〉〈i| and ρA|ϕ〉=∑jπj|j〉〈j| are the spectral decomposition of the reduced matrices ρA|ψ〉 and ρA|ϕ〉. One has
(3.71)12[C(|ψ〉)+C(|ϕ〉)]2≥1-Tr[(ρA|ψ〉)2]+1-Tr[(ρA|ϕ〉)2]=1-∑iλi2+1-∑jπj2≥1-∑ijλi2πj2=12C2(|ψ〉⊗|ϕ〉).
Now we prove that (3.69) holds for any mixed-quantum states ρ and σ. Let ρ=∑ipiρi and σ=∑jqjσj be the optimal decomposition such that C(ρ)=∑ipiC(ρi) and C(σ)=∑jqjC(σj). We have
(3.72)C(ρ)+C(σ)=∑ijpiqj[C(ρi)+C(σj)]≥∑ijpiqjC(ρi⊗σj)≥C(ρ⊗σ).
The inequality forτ can be derived in a similar way.


## 4. Fidelity of Teleportation and Distillation of Entanglement

Quantum teleportation is an important subject in quantum information processing. In terms of a classical communication channel and a quantum resource (a nonlocal entangled state like an EPR pair of particles), the teleportation protocol gives ways to transmit an unknown quantum state from a sender traditionally named “Alice" to a receiver “Bob" who are spatially separated. These teleportation processes can be viewed as quantum channels. The nature of a quantum channel is determined by the particular protocol and the state used as a teleportation resource. The standard teleportation protocolT0 proposed by Bennett et al. in 1993 uses Bell measurements and Pauli rotations. When the maximally entangled pure state |ϕ〉=(1/n)∑i=0n-1|ii〉 is used as the quantum resource, it provides an ideal noiseless quantum channel ΛT0(|ϕ〉〈ϕ|)(ρ)=ρ. However in realistic situation, instead of the pure maximally entangled states, Alice and Bob usually share a mixed entangled state due to the decoherence. Teleportation using mixed state as an entangled resource is, in general, equivalent to having a noisy quantum channel. An explicit expression for the output state of the quantum channel associated with the standard teleportation protocol T0 with an arbitrary mixed-state resource has been obtained [120, 121].It turns out that, by local quantum operations (including collective actions over all members of pairs in each lab) and classical communication (LOCC) between Alice and Bob, it is possible to obtain a number of pairs in nearly maximally entangled state|ψ+〉 from many pairs of nonmaximally entangled states. Such a procedure proposed in [73–77] is called distillation. In [73] the authors give operational protocol to distill an entangled two-qubit state whose single fraction F, defined by F(ρ)=〈ψ+|ρ|ψ+〉, is larger than 1/2. The protocol is then generalized in [77] to distill any d-dimensional bipartite entangled quantum states with F(ρ)>1/d. It is shown that a quantum state ρ violating the reduction criterion can always be distilled. For such states, if their single fraction of entanglement F(ρ)=〈ψ+|ρ|ψ+〉 is greater than 1/d, one can distill these states directly by using the generalized distillation protocol, otherwise a proper filtering operation has to be used at first to transform ρ to another state ρ′ so that F(ρ′)>1/d.


### 4.1. Fidelity of Quantum Teleportation

Letℋ 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(4.1)ℱ(ρ)=maxU〈ψ+|(I⊗U†)ρ(I⊗U)|ψ+〉
under all unitary transformations U, where |ψ+〉=(1/d)∑i=1d|ii〉 is the maximally entangled state and I is the corresponding identity matrix.In [8, 9], the authors give an optimal teleportation protocol by using a mixed entangled quantum state. The optimal teleportation fidelity is given by
(4.2)fmax(ρ)=dℱ(ρ)d+1+1d+1,
which solely depends on the FEF of the entangled resource state ρ.In fact the fully entangled fraction is tightly related to many quantum information processing such as dense coding [10], teleportation [5–7], entanglement swapping [14–18], and quantum cryptography (Bell inequalities) [11–13]. As the optimal fidelity of teleportation is given by FEF [8, 9], experimentally measurement of FEF can be also used to determine the entanglement of the nonlocal source used in teleportation. Thus an analytic formula for FEF is of great importance. In [122] an elegant formula of FEF for two-qubit system is derived analytically by using the method of Lagrange multipliers. For high-dimensional quantum states the analytical computation of FEF remains formidable and less results have been known. In the following we give an estimation on the values of FEF by giving some upper bounds of FEF.Letλi, i=1,…,d2-1, be the generators of the SU(d) algebra. A bipartite state ρ∈ℋ⊗ℋ can be expressed as
(4.3)ρ=1d2I⊗I+1d∑i=1d2-1ri(ρ)λi⊗I+1d∑j=1d2-1sj(ρ)I⊗λj+∑i,j=1d2-1mij(ρ)λi⊗λj,
where ri(ρ)=(1/2)Tr[ρλi(1)⊗I], sj(ρ)=(1/2)Tr[ρI⊗λj(2)], and mij(ρ)=(1/4)Tr[ρλi(1)⊗λj(2)]. Let M(ρ) denote the correlation matrix with entries mij(ρ).Theorem 4.1.
For anyρ∈ℋ⊗ℋ, the fully entangled fraction ℱ(ρ) satisfies [123]
(4.4)ℱ(ρ)≤1d2+4∥MT(ρ)M(P+)∥KF,
where MT stands for the transpose of M and ∥M∥KF=Tr[MM†] is the Ky Fan norm of M.Proof.
First, we note that(4.5)P+=1d2I⊗I+∑i,j=1d2-1mij(P+)λi⊗λj,
where mij(P+)=(1/4)Tr[P+λi⊗λj]. By definition (4.1), one obtains
(4.6)ℱ(ρ)=maxU〈ψ+|(I⊗U†)ρ(I⊗U)|ψ+〉=maxUTr[ρ(I⊗U)P+(I⊗U†)]=maxU{1d2Tr[ρ]+∑i,j=1d2-1mij(P+)Tr[ρλi⊗UλjU†]}.
SinceUλiU† is a traceless Hermitian operator, it can be expanded according to the SU(d) generators as
(4.7)UλiU†=∑j=1d2-112Tr[UλiU†λj]λj≡∑j=1d2-1Oijλj.
Entries Oij define a real (d2-1)×(d2-1) matrix O. From the completeness relation of SU(d) generators
(4.8)∑j=1d2-1(λj)ki(λj)mn=2δimδkn-2dδkiδmn,
one can show that O is an orthonormal matrix. Using (4.7), we have
(4.9)ℱ(ρ)≤1d2+maxO∑i,j,kmij(P+)OjkTr[ρλi⊗λk]=1d2+4maxO∑i,j,kmij(P+)Ojkmik(ρ)=1d2+4maxOTr[M(ρ)TM(P+)O]=1d2+4∥M(ρ)TM(P+)∥KF.For the cased=2, we can get an exact result from (4.4).Corollary 4.2.
For two-qubit system, one has(4.10)ℱ(ρ)=14+4∥M(ρ)TM(P+)∥KF,
that is,  the upper bound derived in Theorem 4.1 is exactly the FEF.Proof.
We have shown in (4.7) that, given an arbitrary unitary U, one can always obtain an orthonormal matrix O. Now we show that in two-qubit case, for any 3×3 orthonormal matrix O, there always exits 2×2 unitary matrix U such that (4.7) holds.
For any vectort={t1,t2,t3} with unit norm, define an operator X≡∑i=13tiσi, where σi’s are Pauli matrices. Given an orthonormal matrix O, one obtains a new operator X′≡∑i=13ti′σi=∑i,j=13Oijtjσi.
X and X′ are both Hermitian traceless matrices. Their eigenvalues are given by the norms of the vectors t and t′={t1′,t2′,t3′}, respectively. As the norms are invariant under orthonormal transformations O, they have the same eigenvalues: ±t12+t22+t32. Thus there must be a unitary matrix U such that X′=UXU†. Hence the inequality in the proof of Theorem 4.1 becomes an equality. The upper bound (4.4) then becomes exact at this situation, which is in accord with the result in [122].Remark 4.3.
The upper bound of FEF (4.4) and the FEF (4.10) depend on the correlation matrices M(ρ) and M(P+). They can be calculated directly according to a given set of SU(d) generators λi, i=1,…,d2-1. As an example, for d=3, if we choose
(4.11)λ1=(1000-10000),λ2=(1300013000-23),λ3=(010100000),λ4=(001000100),λ5=(000001010),λ6=(0i0-i00000),λ7=(00i000-i00),λ8=(00000i0-i0),
then we have
(4.12)M(P+)=Diag{16,16,16,16,16,-16,-16,-16}.
Nevertheless the FEF and its upper bound do not depend on the choice of the SU(d) generators.The usefulness of the bound depends on detailed states. In the following we give two new upper bounds, which is different from Theorem4.1. These bounds work for different states.Leth and g be n×n matrices such that h|j〉=|(j+1)modn〉, g|j〉=ωj|j〉, with ω=exp{-2iπ/n}. We can introduce n2 linear-independent n×n matrices Ust=htgs, which satisfy(4.13)UstUs't'=ωst'-ts'Us′t′Ust,Tr[Ust]=nδs0δt0.
One can also check that {Ust} satisfy the condition of bases of the unitary operators in the sense of [124], that is,(4.14)Tr[UstUs't'+]=nδtt'δss',UstUst+=In×n,
where In×n is the n×n identity matrix. {Ust} form a complete basis of n×n matrices, namely, for any n×n matrix W, W can be expressed as(4.15)W=1n∑s,tTr[Ust+W]Ust.From{Ust}, we can introduce the generalized Bell states(4.16)|Φst〉=(I⊗Ust*)|ψ+〉=1d∑i,j(Ust)ij*|ij〉,|Φ00〉=|ψ+〉,
where |Φst〉 are all maximally entangled states and form a complete orthogonal normalized basis of ℋd⊗ℋd.Theorem 4.4.
For any quantum stateρ∈ℋd⊗ℋd, the fully entangled fraction defined in (4.1) fulfills the following inequality:
(4.17)ℱ(ρ)≤maxj{λj},
where λjs are the eigenvalues of the real part of matrix M=(TiT-iTT), T is a d2×d2 matrix with entries Tn,m=〈Φn|ρ|Φm〉, and Φj are the maximally entangled basis states defined in (4.16) [125].Proof.
From (4.15), any d×d unitary matrix U can be represented by U=∑k=1d2zkUk, where zk=(1/d)Tr[Uk†U]. Define
(4.18)xl={Re[zl],1≤l≤d2,Im[zl],d2<l≤2d2,Ul′={Ul,1≤l≤d2,i*Ul,d2<l≤2d2.
Then the unitary matrix U can be rewritten as U=∑k=12d2zkUk′. The necessary condition for the unitary property of U implies that ∑kxk2=1. Thus we have
(4.19)F(ρ)≡〈ψ+|(I⊗U†)ρ(I⊗U)|ψ+〉=∑m,n=12d2xmxnMmn,
where Mmn is defined in the theorem. One can deduce that
(4.20)Mmn*=Mnm
from the hermiticity of ρ.
Taking into account the constraint with an undetermined Lagrange multiplierλ, we have
(4.21)∂∂xk{F(ρ)+λ(∑lxl2-1)}=0.
Accounting to (4.20) we have the eigenvalue equation
(4.22)∑n=12d2Re[Mk,n]xn=-λxk.
Inserting (4.22) into (4.19) results in
(4.23)ℱ(ρ)=maxUF≤maxj{ηj},
where ηj=-λj is the corresponding eigenvalues of the real part of the matrix M.Example 4.5.
Horodecki gives a very interesting bound entangled state in [31] as
(4.24)ρ(a)=18a+1(a000a000a0a000000000a000000000a00000a000a000a00000a0000000001+a201-a220000000a0a000a01-a2201+a2).
One can easily compare the upper bound obtained in (4.17) and that in (4.4). From Figure 2 we see that, for 0≤a<0.572, the upper bound in (4.17) is larger than that in (4.4). But for 0.572<a<1 the upper bound in (4.17) is always lower than that in (4.4), which means that the upper bound (4.17) is tighter than (4.4).Figure 2
Upper bound ofℱ(ρ(a)) from (4.17) (solid line) and upper bound from (4.4) (dashed line) for state (4.24).In fact, we can drive another upper bound for FEF which will be very tight for weakly mixed-quantum states.Theorem 4.6.
For any bipartite quantum stateρ∈ℋd⊗ℋd, the following inequality holds [125]:
(4.25)ℱ(ρ)≤1d(Tr[ρA])2,
where ρA is the reduced matrix of ρ.Proof.
Note that in [77] the authors have obtained the FEF for pure state |ψ〉 as
(4.26)ℱ(|ψ〉)=1d(Tr[ρA|ψ〉])2,
where ρA|ψ〉 is the reduced matrix of |ψ〉〈ψ|.
For mixed stateρ=∑ipiρi, we have
(4.27)ℱ(ρ)=maxU〈ψ+|(I⊗U†)ρ(I⊗U)|ψ+〉≤∑ipimaxU〈ψ+|(I⊗U†)ρi(I⊗U)|ψ+〉=1d∑ipi(Tr[ρAi])2=1d∑i(Tr[piρAi])2.
Let λij be the real and nonnegative eigenvalues of the matrix piρAi. Recall that for any function F=∑i(∑jxij2)1/2 subjected to the constraints zj=∑ixij with xij being real and nonnegative, the inequality ∑jzj2≤F2 holds, from which it follows that
(4.28)ℱ(ρ)≤1d∑i(∑jλij)2≤1d(∑j∑iλij)2=1d(Tr[ρA])2,
which ends the proof.


### 4.2. Fully Entangled Fraction and Concurrence

The upper bound of FEF has also interesting relations to the entanglement measure concurrence. As shown in [122], the concurrence of a two-qubit quantum state has some kinds of relation with the optimal teleportation fidelity. For quantum state with high dimension, we have the similar relation between them too.Theorem 4.7.
For any bipartite quantum stateρ∈ℋd⊗ℋd, one has [118]
(4.29)C(ρ)≥2dd-1[ℱ(ρ)-1d].Proof.
In [126], the authors show that, for any pure state |ψ〉∈ℋA⊗ℋB, the following inequality holds:
(4.30)C(|ψ〉)≥2dd-1(max|ϕ〉∈ε|〈ψ∣ϕ〉|2-1d),
where ε denotes the set of d×d dimensional maximally entangled states.
Letρ=∑ipi|ϕi〉〈ϕi| be the optimal decomposition such that C(ρ)=∑ipiC(|ψi〉). We have
(4.31)C(ρ)=∑ipiC(|ψi〉)≥∑ipi2dd-1(max|ϕ〉∈ε|〈ψi∣ϕ〉|2-1d)≥2dd-1(max|ϕ〉∈ε∑ipi|〈ψi∣ϕ〉|2-1d)=2dd-1(max|ϕ〉∈ε〈ϕ|ρ|ϕ〉-1d)=2dd-1(ℱ(ρ)-1d),
which ends the proof.Inequality (4.29) has demonstrated the relation between the lower bound of concurrence and the fully entangled fraction (thus the optimal teleportation fidelity), that is,  the fully entangled fraction of a quantum state ρ is limited by its concurrence.We now consider tripartite case. LetρABC be a state of three-qubit systems denoted by A, B, and C. We study the upper bound of the FEF, ℱ(ρAB), between qubits A and B, and its relations to the concurrence under bipartite partition AB and C. For convenience we normalize ℱ(ρAB) to be(4.32)ℱN(ρAB)=max{2ℱ(ρAB)-1,0}.
Let C(ρAB∣C) denote the concurrence between subsystems AB and C.Theorem 4.8.
For any triqubit stateρABC, ℱN(ρAB) satisfies [123]
(4.33)ℱN(ρAB)≤1-C2(ρAB∣C).Proof.
We first consider the case thatρABC is pure, ρABC=|ψ〉ABC〈ψ|. By using the Schmidt decomposition between qubits A,B, and C, |ψ〉ABC can be written as
(4.34)|ψ〉AB|C=∑i=12ηi|iAB〉|iC〉,η12+η22=1,η1≥η2,
for some orthonormalized bases |iAB〉, |iC〉 of subsystems AB, C, respectively. The reduced density matrix ρAB has the form
(4.35)ρAB=TrC[ρABC]=∑i=12ηi2|iAB〉〈iAB|=UTΛU*,
where Λ is a 4×4 diagonal matrix with diagonal elements {η12,η22,0,0}, U is a unitary matrix, and U* denotes the conjugation of U.
The FEF of the two-qubit stateρAB can be calculated by using formula (4.10) or the one in [122]. Let
(4.36)M=12(100i0i-100i10100-i)
be the 4×4 matrix constituted by the four Bell bases. The FEF of ρAB can be written as
(4.37)ℱ(ρAB)=ηmax(Re{M†ρABM})=12ηmax(M†ρABM+MTρAB*M*)≤12[ηmax(M†UTΛU*M)+ηmax(MTU†ΛUM*)]=η12,
where ηmax(X) stands for the maximal eigenvalues of the matrix X.
For pure state (4.34) in bipartite partition AB and C, we have
(4.38)C(|ψ〉AB|C)=2(1-Tr[ρAB2])=2η1η2.
From (4.32), (4.37), and (4.38) we get
(4.39)ℱN(ρAB)≤1-C2(|ψ〉AB|C).
We now prove that the above inequality (4.39) also holds for mixed state ρABC. Let ρABC=∑ipi|ψi〉ABC〈ψi| be the optimal decomposition of ρABC such that C(ρAB|C)=∑ipiC(|ψi〉)AB|C. We have
(4.40)ℱN(ρAB)≤∑ipiℱN(ρABi)≤∑ipi1-C2(ρAB∣Ci)≤1-∑ipiC2(ρAB∣Ci)≤1-C2(ρAB∣C),
where ρAB|Ci=|ψi〉ABC〈ψi| and ρABi=TrC[ρAB|Ci].From Theorem4.8 we see that the FEF of qubits A and B are bounded by the concurrence between qubits A, B, and qubit C. The upper bound of FEF for ρAB decreases when the entanglement between qubits A,B, and C increases. As an example, we consider the generalized W state defined by |W′〉=α|100〉+β|010〉+γ|001〉, |α|2+|β|2+|γ|2=1. The reduced density matrix is given by
(4.41)ρABW′=(|γ|20000|β|2α*β00αβ*|α|200000).
The FEF of ρABW′ is given by
(4.42)ℱN(ρABW′)=-12+2|α||β|+12||α|2+|β|2-|γ|2|,
while the concurrence of |W′〉 has the form CAB|C(|W′〉)=2|γ||α|2+|β|2. We see that (4.33) always holds. In particular for |α|=|β| and |γ|≤2/2, the inequality (4.33) is saturated (see Figure 3).Figure 3
ℱN(ρABW′) (dashed line) and upper bound 1-C2(|W′〉AB∣C) (solid line) of state |W′〉AB∣C at |α|=|β|.


### 4.3. Improvement of Entanglement Distillation Protocol

The upper bound can give rise to not only an estimation of the fidelity in quantum information processing such as teleportation, but also an interesting application in entanglement distillation of quantum states. In [77] a generalized distillation protocol has been presented. It is shown that a quantum state ρ violating the reduction criterion can always be distilled. For such states if their single fraction of entanglement F(ρ)=〈ψ+|ρ|ψ+〉 is greater than 1/d, then one can distill these states directly by using the generalized distillation protocol. If the FEF (the largest value of single fraction of entanglement under local unitary transformations) is less than or equal to 1/d, then a proper filtering operation has to be used at first to transform ρ to another state ρ′ so that F(ρ′)>1/d. For d=2, one can compute FEF analytically according to the corollary. For d≥3 our upper bound (4.4) can supply a necessary condition in the distillation.Theorem 4.9.
For an entangled stateρ∈ℋ⊗ℋ violating the reduction criterion, if the upper bound (4.4) is less than or equal to 1/d, then the filtering operation has to be applied before using the generalized distillation protocol [123].As an example, we consider a3×3 state(4.43)ρ=89σ+19|ψ+〉〈ψ+|,
where σ=(x|0〉〈0|+(1-x)|1〉〈1|)⊗(x|0〉〈0|+(1-x)|1〉〈1|). It is direct to verify that ρ violates the reduction criterion for 0≤x≤1, as (ρ1⊗I)-ρ has a negative eigenvalue -2/27. Therefore the state is distillable. From Figure 4, we see that for 0≤x<0.0722 and 0.9278<x≤1, the fidelity is already greater than 1/3; thus the generalized distillation protocol can be applied without the filtering operation. However for 0.1188≤x≤0.8811, even the upper bound of the fully entangled fraction is less than or equal to 1/3; hence the filtering operation has to be applied first, before using the generalized distillation protocol.Figure 4
Upper bound ofℱ(ρ)-1/3 from (4.4) (solid line) and fidelity F(ρ)-1/3 (dashed line) for state (4.43).Moreover, the lower bounds of concurrence can be also used to study the distillability of quantum states. Based on the positive partial transpose (PPT) criterion, a necessary and sufficient condition for the distillability was proposed in [127], which is not operational in general. An alternative distillability criterion based on the bound τ2 in (3.19) can be obtained to improve the operationality.Theorem 4.10.
A bipartite quantum stateρ is distillable if and only if τ2(ρ⊗N)>0 for some number N [67].Proof.
It was shown in [127] that a density matrix ρ is distillable if and only if there are some projectors P, Q that map high-dimensional spaces to two-dimensional ones and some number N such that the state P⊗Qρ⊗NP⊗Q is entangled [127]. Thus if τ2(ρ⊗N)>0, then there exists one submatrix of matrix ρ⊗N, similar to (3.20), which has nonzero τ2 and is entangled in a 2⊗2 space; hence ρ is distillable.Corollary 4.11.
(a) The lower boundτ2(ρ)>0 is a sufficient condition for the distillability of any bipartite state ρ.
(b) The lower boundτ2(ρ)=0 is a necessary condition for separability of any bipartite state ρ.Remark 4.12.
Corollary4.11 directly follows from Theorem 4.10 and this case is referred to as one distillable [128]. The problem of whether non-PPT (NPPT) nondistillable states exist is studied numerically in [128, 129]. By using Theorem 4.10, although it seems impossible to solve the problem completely, it is easy to judge the distillability of a state under condition that is one distillable.The lower boundτ2, PPT criterion, separability, and distillability for any bipartite quantum state ρ have the following relations: if τ2(ρ)>0, then ρ is entangled. If ρ is separable, then it is PPT. If τ2(ρ)>0, then ρ is distillable. If ρ is distillable, then it is NPPT. From the last two propositions it follows that if ρ is PPT, then τ2(ρ)=0, that is,  if τ2(ρ)>0, then ρ is NPPT.Theorem 4.13.
For any pure tripartite state|ϕ〉ABC in arbitrary d⊗d⊗d dimensional spaces, bound τ2 satisfies [67]
(4.44)τ2(ρAB)+τ2(ρAC)≤τ2(ρA:BC),
where ρAB=TrC(|ϕ〉ABC〈ϕ|), ρAC=TrB(|ϕ〉ABC〈ϕ|), and ρA:BC=TrBC(|ϕ〉ABC〈ϕ|).Proof.
SinceCmn2≤(λmn(1))2≤∑i=14(λmn(i))2=Tr(ρρ̃mn), one can derive the inequality
(4.45)τ(ρAB)+τ(ρAC)≤∑l,kDTr[ρAB(ρ̃AB)lk]+∑p,qDTr[ρAC(ρ̃AC)pq],
where D=d(d-1)/2. Note that ∑lkTr[ρAB(ρ̃AB)lk]≤1-TrρA2-TrρB2+TrρC2 and ∑pqTr[ρAC(ρ̃AC)pq]≤1-TrρA2+TrρB2-TrρC2, where l,pk,q,=1,…,D. By using the similar analysis in [113], one has that the right-hand side of (4.45) is equal to 2(1-TrρA2)=C2(ρA:BC). Taking into account that τ2(ρA:BC)=𝒞2(ρA:BC) for a pure state, one obtains inequality (4.44).
Generally for any pure multipartite quantum stateρAB1B2⋯bn, one has the following monogamy inequality:
(4.46)τ2(ρAB1)+τ2(ρAB2)+⋯+τ2(ρABn)≤τ2(ρA:B1B2⋯Bn).


## 4.1. Fidelity of Quantum Teleportation

Letℋ 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(4.1)ℱ(ρ)=maxU〈ψ+|(I⊗U†)ρ(I⊗U)|ψ+〉
under all unitary transformations U, where |ψ+〉=(1/d)∑i=1d|ii〉 is the maximally entangled state and I is the corresponding identity matrix.In [8, 9], the authors give an optimal teleportation protocol by using a mixed entangled quantum state. The optimal teleportation fidelity is given by
(4.2)fmax(ρ)=dℱ(ρ)d+1+1d+1,
which solely depends on the FEF of the entangled resource state ρ.In fact the fully entangled fraction is tightly related to many quantum information processing such as dense coding [10], teleportation [5–7], entanglement swapping [14–18], and quantum cryptography (Bell inequalities) [11–13]. As the optimal fidelity of teleportation is given by FEF [8, 9], experimentally measurement of FEF can be also used to determine the entanglement of the nonlocal source used in teleportation. Thus an analytic formula for FEF is of great importance. In [122] an elegant formula of FEF for two-qubit system is derived analytically by using the method of Lagrange multipliers. For high-dimensional quantum states the analytical computation of FEF remains formidable and less results have been known. In the following we give an estimation on the values of FEF by giving some upper bounds of FEF.Letλi, i=1,…,d2-1, be the generators of the SU(d) algebra. A bipartite state ρ∈ℋ⊗ℋ can be expressed as
(4.3)ρ=1d2I⊗I+1d∑i=1d2-1ri(ρ)λi⊗I+1d∑j=1d2-1sj(ρ)I⊗λj+∑i,j=1d2-1mij(ρ)λi⊗λj,
where ri(ρ)=(1/2)Tr[ρλi(1)⊗I], sj(ρ)=(1/2)Tr[ρI⊗λj(2)], and mij(ρ)=(1/4)Tr[ρλi(1)⊗λj(2)]. Let M(ρ) denote the correlation matrix with entries mij(ρ).Theorem 4.1.
For anyρ∈ℋ⊗ℋ, the fully entangled fraction ℱ(ρ) satisfies [123]
(4.4)ℱ(ρ)≤1d2+4∥MT(ρ)M(P+)∥KF,
where MT stands for the transpose of M and ∥M∥KF=Tr[MM†] is the Ky Fan norm of M.Proof.
First, we note that(4.5)P+=1d2I⊗I+∑i,j=1d2-1mij(P+)λi⊗λj,
where mij(P+)=(1/4)Tr[P+λi⊗λj]. By definition (4.1), one obtains
(4.6)ℱ(ρ)=maxU〈ψ+|(I⊗U†)ρ(I⊗U)|ψ+〉=maxUTr[ρ(I⊗U)P+(I⊗U†)]=maxU{1d2Tr[ρ]+∑i,j=1d2-1mij(P+)Tr[ρλi⊗UλjU†]}.
SinceUλiU† is a traceless Hermitian operator, it can be expanded according to the SU(d) generators as
(4.7)UλiU†=∑j=1d2-112Tr[UλiU†λj]λj≡∑j=1d2-1Oijλj.
Entries Oij define a real (d2-1)×(d2-1) matrix O. From the completeness relation of SU(d) generators
(4.8)∑j=1d2-1(λj)ki(λj)mn=2δimδkn-2dδkiδmn,
one can show that O is an orthonormal matrix. Using (4.7), we have
(4.9)ℱ(ρ)≤1d2+maxO∑i,j,kmij(P+)OjkTr[ρλi⊗λk]=1d2+4maxO∑i,j,kmij(P+)Ojkmik(ρ)=1d2+4maxOTr[M(ρ)TM(P+)O]=1d2+4∥M(ρ)TM(P+)∥KF.For the cased=2, we can get an exact result from (4.4).Corollary 4.2.
For two-qubit system, one has(4.10)ℱ(ρ)=14+4∥M(ρ)TM(P+)∥KF,
that is,  the upper bound derived in Theorem 4.1 is exactly the FEF.Proof.
We have shown in (4.7) that, given an arbitrary unitary U, one can always obtain an orthonormal matrix O. Now we show that in two-qubit case, for any 3×3 orthonormal matrix O, there always exits 2×2 unitary matrix U such that (4.7) holds.
For any vectort={t1,t2,t3} with unit norm, define an operator X≡∑i=13tiσi, where σi’s are Pauli matrices. Given an orthonormal matrix O, one obtains a new operator X′≡∑i=13ti′σi=∑i,j=13Oijtjσi.
X and X′ are both Hermitian traceless matrices. Their eigenvalues are given by the norms of the vectors t and t′={t1′,t2′,t3′}, respectively. As the norms are invariant under orthonormal transformations O, they have the same eigenvalues: ±t12+t22+t32. Thus there must be a unitary matrix U such that X′=UXU†. Hence the inequality in the proof of Theorem 4.1 becomes an equality. The upper bound (4.4) then becomes exact at this situation, which is in accord with the result in [122].Remark 4.3.
The upper bound of FEF (4.4) and the FEF (4.10) depend on the correlation matrices M(ρ) and M(P+). They can be calculated directly according to a given set of SU(d) generators λi, i=1,…,d2-1. As an example, for d=3, if we choose
(4.11)λ1=(1000-10000),λ2=(1300013000-23),λ3=(010100000),λ4=(001000100),λ5=(000001010),λ6=(0i0-i00000),λ7=(00i000-i00),λ8=(00000i0-i0),
then we have
(4.12)M(P+)=Diag{16,16,16,16,16,-16,-16,-16}.
Nevertheless the FEF and its upper bound do not depend on the choice of the SU(d) generators.The usefulness of the bound depends on detailed states. In the following we give two new upper bounds, which is different from Theorem4.1. These bounds work for different states.Leth and g be n×n matrices such that h|j〉=|(j+1)modn〉, g|j〉=ωj|j〉, with ω=exp{-2iπ/n}. We can introduce n2 linear-independent n×n matrices Ust=htgs, which satisfy(4.13)UstUs't'=ωst'-ts'Us′t′Ust,Tr[Ust]=nδs0δt0.
One can also check that {Ust} satisfy the condition of bases of the unitary operators in the sense of [124], that is,(4.14)Tr[UstUs't'+]=nδtt'δss',UstUst+=In×n,
where In×n is the n×n identity matrix. {Ust} form a complete basis of n×n matrices, namely, for any n×n matrix W, W can be expressed as(4.15)W=1n∑s,tTr[Ust+W]Ust.From{Ust}, we can introduce the generalized Bell states(4.16)|Φst〉=(I⊗Ust*)|ψ+〉=1d∑i,j(Ust)ij*|ij〉,|Φ00〉=|ψ+〉,
where |Φst〉 are all maximally entangled states and form a complete orthogonal normalized basis of ℋd⊗ℋd.Theorem 4.4.
For any quantum stateρ∈ℋd⊗ℋd, the fully entangled fraction defined in (4.1) fulfills the following inequality:
(4.17)ℱ(ρ)≤maxj{λj},
where λjs are the eigenvalues of the real part of matrix M=(TiT-iTT), T is a d2×d2 matrix with entries Tn,m=〈Φn|ρ|Φm〉, and Φj are the maximally entangled basis states defined in (4.16) [125].Proof.
From (4.15), any d×d unitary matrix U can be represented by U=∑k=1d2zkUk, where zk=(1/d)Tr[Uk†U]. Define
(4.18)xl={Re[zl],1≤l≤d2,Im[zl],d2<l≤2d2,Ul′={Ul,1≤l≤d2,i*Ul,d2<l≤2d2.
Then the unitary matrix U can be rewritten as U=∑k=12d2zkUk′. The necessary condition for the unitary property of U implies that ∑kxk2=1. Thus we have
(4.19)F(ρ)≡〈ψ+|(I⊗U†)ρ(I⊗U)|ψ+〉=∑m,n=12d2xmxnMmn,
where Mmn is defined in the theorem. One can deduce that
(4.20)Mmn*=Mnm
from the hermiticity of ρ.
Taking into account the constraint with an undetermined Lagrange multiplierλ, we have
(4.21)∂∂xk{F(ρ)+λ(∑lxl2-1)}=0.
Accounting to (4.20) we have the eigenvalue equation
(4.22)∑n=12d2Re[Mk,n]xn=-λxk.
Inserting (4.22) into (4.19) results in
(4.23)ℱ(ρ)=maxUF≤maxj{ηj},
where ηj=-λj is the corresponding eigenvalues of the real part of the matrix M.Example 4.5.
Horodecki gives a very interesting bound entangled state in [31] as
(4.24)ρ(a)=18a+1(a000a000a0a000000000a000000000a00000a000a000a00000a0000000001+a201-a220000000a0a000a01-a2201+a2).
One can easily compare the upper bound obtained in (4.17) and that in (4.4). From Figure 2 we see that, for 0≤a<0.572, the upper bound in (4.17) is larger than that in (4.4). But for 0.572<a<1 the upper bound in (4.17) is always lower than that in (4.4), which means that the upper bound (4.17) is tighter than (4.4).Figure 2
Upper bound ofℱ(ρ(a)) from (4.17) (solid line) and upper bound from (4.4) (dashed line) for state (4.24).In fact, we can drive another upper bound for FEF which will be very tight for weakly mixed-quantum states.Theorem 4.6.
For any bipartite quantum stateρ∈ℋd⊗ℋd, the following inequality holds [125]:
(4.25)ℱ(ρ)≤1d(Tr[ρA])2,
where ρA is the reduced matrix of ρ.Proof.
Note that in [77] the authors have obtained the FEF for pure state |ψ〉 as
(4.26)ℱ(|ψ〉)=1d(Tr[ρA|ψ〉])2,
where ρA|ψ〉 is the reduced matrix of |ψ〉〈ψ|.
For mixed stateρ=∑ipiρi, we have
(4.27)ℱ(ρ)=maxU〈ψ+|(I⊗U†)ρ(I⊗U)|ψ+〉≤∑ipimaxU〈ψ+|(I⊗U†)ρi(I⊗U)|ψ+〉=1d∑ipi(Tr[ρAi])2=1d∑i(Tr[piρAi])2.
Let λij be the real and nonnegative eigenvalues of the matrix piρAi. Recall that for any function F=∑i(∑jxij2)1/2 subjected to the constraints zj=∑ixij with xij being real and nonnegative, the inequality ∑jzj2≤F2 holds, from which it follows that
(4.28)ℱ(ρ)≤1d∑i(∑jλij)2≤1d(∑j∑iλij)2=1d(Tr[ρA])2,
which ends the proof.


## 4.2. Fully Entangled Fraction and Concurrence

The upper bound of FEF has also interesting relations to the entanglement measure concurrence. As shown in [122], the concurrence of a two-qubit quantum state has some kinds of relation with the optimal teleportation fidelity. For quantum state with high dimension, we have the similar relation between them too.Theorem 4.7.
For any bipartite quantum stateρ∈ℋd⊗ℋd, one has [118]
(4.29)C(ρ)≥2dd-1[ℱ(ρ)-1d].Proof.
In [126], the authors show that, for any pure state |ψ〉∈ℋA⊗ℋB, the following inequality holds:
(4.30)C(|ψ〉)≥2dd-1(max|ϕ〉∈ε|〈ψ∣ϕ〉|2-1d),
where ε denotes the set of d×d dimensional maximally entangled states.
Letρ=∑ipi|ϕi〉〈ϕi| be the optimal decomposition such that C(ρ)=∑ipiC(|ψi〉). We have
(4.31)C(ρ)=∑ipiC(|ψi〉)≥∑ipi2dd-1(max|ϕ〉∈ε|〈ψi∣ϕ〉|2-1d)≥2dd-1(max|ϕ〉∈ε∑ipi|〈ψi∣ϕ〉|2-1d)=2dd-1(max|ϕ〉∈ε〈ϕ|ρ|ϕ〉-1d)=2dd-1(ℱ(ρ)-1d),
which ends the proof.Inequality (4.29) has demonstrated the relation between the lower bound of concurrence and the fully entangled fraction (thus the optimal teleportation fidelity), that is,  the fully entangled fraction of a quantum state ρ is limited by its concurrence.We now consider tripartite case. LetρABC be a state of three-qubit systems denoted by A, B, and C. We study the upper bound of the FEF, ℱ(ρAB), between qubits A and B, and its relations to the concurrence under bipartite partition AB and C. For convenience we normalize ℱ(ρAB) to be(4.32)ℱN(ρAB)=max{2ℱ(ρAB)-1,0}.
Let C(ρAB∣C) denote the concurrence between subsystems AB and C.Theorem 4.8.
For any triqubit stateρABC, ℱN(ρAB) satisfies [123]
(4.33)ℱN(ρAB)≤1-C2(ρAB∣C).Proof.
We first consider the case thatρABC is pure, ρABC=|ψ〉ABC〈ψ|. By using the Schmidt decomposition between qubits A,B, and C, |ψ〉ABC can be written as
(4.34)|ψ〉AB|C=∑i=12ηi|iAB〉|iC〉,η12+η22=1,η1≥η2,
for some orthonormalized bases |iAB〉, |iC〉 of subsystems AB, C, respectively. The reduced density matrix ρAB has the form
(4.35)ρAB=TrC[ρABC]=∑i=12ηi2|iAB〉〈iAB|=UTΛU*,
where Λ is a 4×4 diagonal matrix with diagonal elements {η12,η22,0,0}, U is a unitary matrix, and U* denotes the conjugation of U.
The FEF of the two-qubit stateρAB can be calculated by using formula (4.10) or the one in [122]. Let
(4.36)M=12(100i0i-100i10100-i)
be the 4×4 matrix constituted by the four Bell bases. The FEF of ρAB can be written as
(4.37)ℱ(ρAB)=ηmax(Re{M†ρABM})=12ηmax(M†ρABM+MTρAB*M*)≤12[ηmax(M†UTΛU*M)+ηmax(MTU†ΛUM*)]=η12,
where ηmax(X) stands for the maximal eigenvalues of the matrix X.
For pure state (4.34) in bipartite partition AB and C, we have
(4.38)C(|ψ〉AB|C)=2(1-Tr[ρAB2])=2η1η2.
From (4.32), (4.37), and (4.38) we get
(4.39)ℱN(ρAB)≤1-C2(|ψ〉AB|C).
We now prove that the above inequality (4.39) also holds for mixed state ρABC. Let ρABC=∑ipi|ψi〉ABC〈ψi| be the optimal decomposition of ρABC such that C(ρAB|C)=∑ipiC(|ψi〉)AB|C. We have
(4.40)ℱN(ρAB)≤∑ipiℱN(ρABi)≤∑ipi1-C2(ρAB∣Ci)≤1-∑ipiC2(ρAB∣Ci)≤1-C2(ρAB∣C),
where ρAB|Ci=|ψi〉ABC〈ψi| and ρABi=TrC[ρAB|Ci].From Theorem4.8 we see that the FEF of qubits A and B are bounded by the concurrence between qubits A, B, and qubit C. The upper bound of FEF for ρAB decreases when the entanglement between qubits A,B, and C increases. As an example, we consider the generalized W state defined by |W′〉=α|100〉+β|010〉+γ|001〉, |α|2+|β|2+|γ|2=1. The reduced density matrix is given by
(4.41)ρABW′=(|γ|20000|β|2α*β00αβ*|α|200000).
The FEF of ρABW′ is given by
(4.42)ℱN(ρABW′)=-12+2|α||β|+12||α|2+|β|2-|γ|2|,
while the concurrence of |W′〉 has the form CAB|C(|W′〉)=2|γ||α|2+|β|2. We see that (4.33) always holds. In particular for |α|=|β| and |γ|≤2/2, the inequality (4.33) is saturated (see Figure 3).Figure 3
ℱN(ρABW′) (dashed line) and upper bound 1-C2(|W′〉AB∣C) (solid line) of state |W′〉AB∣C at |α|=|β|.


## 4.3. Improvement of Entanglement Distillation Protocol

The upper bound can give rise to not only an estimation of the fidelity in quantum information processing such as teleportation, but also an interesting application in entanglement distillation of quantum states. In [77] a generalized distillation protocol has been presented. It is shown that a quantum state ρ violating the reduction criterion can always be distilled. For such states if their single fraction of entanglement F(ρ)=〈ψ+|ρ|ψ+〉 is greater than 1/d, then one can distill these states directly by using the generalized distillation protocol. If the FEF (the largest value of single fraction of entanglement under local unitary transformations) is less than or equal to 1/d, then a proper filtering operation has to be used at first to transform ρ to another state ρ′ so that F(ρ′)>1/d. For d=2, one can compute FEF analytically according to the corollary. For d≥3 our upper bound (4.4) can supply a necessary condition in the distillation.Theorem 4.9.
For an entangled stateρ∈ℋ⊗ℋ violating the reduction criterion, if the upper bound (4.4) is less than or equal to 1/d, then the filtering operation has to be applied before using the generalized distillation protocol [123].As an example, we consider a3×3 state(4.43)ρ=89σ+19|ψ+〉〈ψ+|,
where σ=(x|0〉〈0|+(1-x)|1〉〈1|)⊗(x|0〉〈0|+(1-x)|1〉〈1|). It is direct to verify that ρ violates the reduction criterion for 0≤x≤1, as (ρ1⊗I)-ρ has a negative eigenvalue -2/27. Therefore the state is distillable. From Figure 4, we see that for 0≤x<0.0722 and 0.9278<x≤1, the fidelity is already greater than 1/3; thus the generalized distillation protocol can be applied without the filtering operation. However for 0.1188≤x≤0.8811, even the upper bound of the fully entangled fraction is less than or equal to 1/3; hence the filtering operation has to be applied first, before using the generalized distillation protocol.Figure 4
Upper bound ofℱ(ρ)-1/3 from (4.4) (solid line) and fidelity F(ρ)-1/3 (dashed line) for state (4.43).Moreover, the lower bounds of concurrence can be also used to study the distillability of quantum states. Based on the positive partial transpose (PPT) criterion, a necessary and sufficient condition for the distillability was proposed in [127], which is not operational in general. An alternative distillability criterion based on the bound τ2 in (3.19) can be obtained to improve the operationality.Theorem 4.10.
A bipartite quantum stateρ is distillable if and only if τ2(ρ⊗N)>0 for some number N [67].Proof.
It was shown in [127] that a density matrix ρ is distillable if and only if there are some projectors P, Q that map high-dimensional spaces to two-dimensional ones and some number N such that the state P⊗Qρ⊗NP⊗Q is entangled [127]. Thus if τ2(ρ⊗N)>0, then there exists one submatrix of matrix ρ⊗N, similar to (3.20), which has nonzero τ2 and is entangled in a 2⊗2 space; hence ρ is distillable.Corollary 4.11.
(a) The lower boundτ2(ρ)>0 is a sufficient condition for the distillability of any bipartite state ρ.
(b) The lower boundτ2(ρ)=0 is a necessary condition for separability of any bipartite state ρ.Remark 4.12.
Corollary4.11 directly follows from Theorem 4.10 and this case is referred to as one distillable [128]. The problem of whether non-PPT (NPPT) nondistillable states exist is studied numerically in [128, 129]. By using Theorem 4.10, although it seems impossible to solve the problem completely, it is easy to judge the distillability of a state under condition that is one distillable.The lower boundτ2, PPT criterion, separability, and distillability for any bipartite quantum state ρ have the following relations: if τ2(ρ)>0, then ρ is entangled. If ρ is separable, then it is PPT. If τ2(ρ)>0, then ρ is distillable. If ρ is distillable, then it is NPPT. From the last two propositions it follows that if ρ is PPT, then τ2(ρ)=0, that is,  if τ2(ρ)>0, then ρ is NPPT.Theorem 4.13.
For any pure tripartite state|ϕ〉ABC in arbitrary d⊗d⊗d dimensional spaces, bound τ2 satisfies [67]
(4.44)τ2(ρAB)+τ2(ρAC)≤τ2(ρA:BC),
where ρAB=TrC(|ϕ〉ABC〈ϕ|), ρAC=TrB(|ϕ〉ABC〈ϕ|), and ρA:BC=TrBC(|ϕ〉ABC〈ϕ|).Proof.
SinceCmn2≤(λmn(1))2≤∑i=14(λmn(i))2=Tr(ρρ̃mn), one can derive the inequality
(4.45)τ(ρAB)+τ(ρAC)≤∑l,kDTr[ρAB(ρ̃AB)lk]+∑p,qDTr[ρAC(ρ̃AC)pq],
where D=d(d-1)/2. Note that ∑lkTr[ρAB(ρ̃AB)lk]≤1-TrρA2-TrρB2+TrρC2 and ∑pqTr[ρAC(ρ̃AC)pq]≤1-TrρA2+TrρB2-TrρC2, where l,pk,q,=1,…,D. By using the similar analysis in [113], one has that the right-hand side of (4.45) is equal to 2(1-TrρA2)=C2(ρA:BC). Taking into account that τ2(ρA:BC)=𝒞2(ρA:BC) for a pure state, one obtains inequality (4.44).
Generally for any pure multipartite quantum stateρAB1B2⋯bn, one has the following monogamy inequality:
(4.46)τ2(ρAB1)+τ2(ρAB2)+⋯+τ2(ρABn)≤τ2(ρA:B1B2⋯Bn).


## 5. Summary and Conclusion

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



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