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

Abstract

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

1. Introduction

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

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

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

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

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

There are also some other necessary criteria for separability. Nielsen and Kempe [42] presented a necessary criterion called majorization: the decreasing ordered vector of the eigenvalues for 𝜌 is majorized by that of 𝜌^𝐴1 or 𝜌^𝐴2 alone for a separable state. That is, if a state 𝜌 is separable, then 𝜆^↓_𝜌≺𝜆^↓_𝜌𝐴1, 𝜆^↓_𝜌≺𝜆^↓_𝜌𝐴2. Here 𝜆^↓_𝜌 denotes the decreasing ordered vector of the eigenvalues of 𝜌. A 𝑑-dimensional vector 𝑥^↓ is majorized by 𝑦^↓, 𝑥^↓≺𝑦^↓, if ∑^𝑘_𝑗=1𝑥^↓_𝑗≤∑^𝑘_𝑗=1𝑦^↓_𝑗 for 𝑘=1,…,𝑑−1 and the equality holds for 𝑘=𝑑. Zeros are appended to the vectors 𝜆^↓_{𝜌𝐴12,𝐴} such that their dimensions are equal to the one of 𝜆^↓_𝜌.

In [31], another necessary criterion called range criterion was given. If a bipartite state 𝜌 acting on the space ℋ_𝐴⊗ℋ_𝐵 is separable, then there exists a family of product vectors 𝜓_𝑖⊗𝜙_𝑖 such that (i) they span the range of 𝜌; (ii) the vector {𝜓_𝑖⊗𝜙^∗_𝑖}^𝑘_𝑖=1 spans the range of 𝜌^𝑇𝐵, where ∗ denotes complex conjugation in the basis in which partial transposition was performed and 𝜌^𝑇𝐵 is the partially transposed matrix of 𝜌 with respect to the subspace 𝐵. In particular, any of the vectors 𝜓_𝑖⊗𝜙^∗_𝑖 belongs to the range of 𝜌.

Recently, some elegant results for the separability problem have been derived. In [43–45], a separability criteria based on the local uncertainty relations (LURs) was obtained. The authors show that, for any separable state 𝜌∈ℋ_𝐴⊗ℋ_𝐵, 1−_𝑘⟨𝐺^𝐴_𝑘⊗𝐺^𝐵_𝑘1⟩−2⟨𝐺^𝐴_𝑘⊗𝐼−𝐼⊗𝐺^𝐵_𝑘⟩²≥0, (1.1) where 𝐺^𝐴_𝑘 or 𝐺^𝐵_𝑘 are arbitrary local orthogonal and normalized operators (LOOs) in ℋ_𝐴⊗ℋ_𝐵. This criterion is strictly stronger than the realignment criterion. Thus more bound entangled quantum states can be recognized by the LUR criterion. The criterion is optimized in [46] by choosing the optimal LOOs. In [47] a criterion based on the correlation matrix of a state has been presented. The correlation matrix criterion is shown to be independent of PPT and realignment criterion [48], that is, there exist quantum states that can be recognized by correlation criterion while the PPT and realignment criterion fail. The covariance matrix of a quantum state is also used to study separability in [49]. It has been shown that the LUR criterion, including the optimized one, can be derived from the covariance matrix criterion [50].

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

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

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

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

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

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

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

2. Separability Criteria and Normal Form

A multipartite pure quantum state 𝜌_12⋯𝑁∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁 is said to be fully separable if it can be written as

𝜌_12⋯𝑁=𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁, (2.1) where 𝜌₁ and 𝜌₂,…,𝜌_𝑁 are reduced density matrices defined as 𝜌₁=Tr_23⋯𝑁[𝜌_12⋯𝑁], 𝜌₂=Tr_13⋯𝑁[𝜌_12⋯𝑁],…,𝜌_𝑁=Tr_{12⋯𝑁−1}[𝜌_12⋯𝑁]. This is equivalent to the condition 𝜌_12⋯𝑁=|𝜓₁⟩⟨𝜓₁|⊗|𝜙₂⟩⟨𝜙₂|⊗⋯⊗|𝜇_𝑁⟩⟨𝜇_𝑁|, (2.2) where |𝜓₁⟩∈ℋ₁,|𝜙₂⟩∈ℋ₂,…,|𝜇_𝑁⟩∈ℋ_𝑁.

A multipartite quantum mixed state 𝜌_12⋯𝑁∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁 is said to be fully separable if it can be written as

𝜌_12⋯𝑁=_𝑖𝑞_𝑖𝜌_𝑖¹⊗𝜌_𝑖²⊗⋯⊗𝜌_𝑖^𝑁, (2.3) where 𝜌_𝑖¹,𝜌_𝑖²,…,𝜌_𝑖^𝑁 are the reduced density matrices with respect to the systems 1,2,…,𝑁, respectively, 𝑞_𝑖>0, and ∑_𝑖𝑞_𝑖=1. This is equivalent to the condition 𝜌_12⋯𝑁=_𝑖𝑝_𝑖|𝜓¹_𝑖⟩⟨𝜓¹_𝑖|⊗|𝜙²_𝑖⟩⟨𝜙²_𝑖|⊗⋯⊗|𝜇^𝑁_𝑖⟩⟨𝜇^𝑁_𝑖|, (2.4) where |𝜓¹_𝑖⟩,|𝜙²_𝑖⟩,…,|𝜇^𝑁_𝑖⟩ are normalized pure states of systems 1,2,…,𝑁, respectively, 𝑝_𝑖>0, and ∑_𝑖𝑝_𝑖=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⟩∈ℋ_𝐴⊗ℋ_𝐵 is a pure state of a composite system, 𝐴𝐵, then there exist orthonormal states |𝑖_𝐴⟩ for system 𝐴 and orthonormal states |𝑖_𝐵⟩ for system 𝐵 such that |𝜓⟩=_𝑖𝜆_𝑖|𝑖_𝐴⟩|𝑖_𝐵⟩, (2.5) where 𝜆_𝑖 are nonnegative real numbers satisfying ∑_𝑖𝜆_𝑖²=1, known as Schmidt coefficients.

|𝑖_𝐴⟩ and |𝑖_𝐵⟩ are called Schmidt bases with respect to ℋ_𝐴 and ℋ_𝐵. The number of nonzero values 𝜆_𝑖 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

|Ψ⟩=𝜆₀|000⟩+𝜆₁𝑒^𝑖𝜓|100⟩+𝜆₂|101⟩+𝜆₃|110⟩+𝜆₄|111⟩ (2.6) with normalization condition 𝜆_𝑖≥0,0≤𝜓≤𝜋, where ∑_𝑖𝜇_𝑖=1, 𝜇_𝑖≡𝜆²_𝑖. 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 𝜌_𝐴𝐵∈ℋ_𝐴⊗ℋ_𝐵 is separable, then the new matrix 𝜌^𝑇𝐵_𝐴𝐵 with matrix elements defined in some fixed product basis as ⟨𝑚|⟨𝜇|𝜌^𝑇𝐵_𝐴𝐵|𝑛⟩|𝜈⟩≡⟨𝑚|⟨𝜈|𝜌_𝐴𝐵|𝑛⟩|𝜇⟩ (2.7) is also a density matrix (i.e., it has nonnegative spectrum). The operation 𝑇_𝐵, called partial transpose, just corresponds to the transposition of the indices with respect to the second subsystem 𝐵. It has an interpretation as a partial time reversal [80].

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

2.1.2. Reduced Density Matrix Criterion

Cerf et al. [81] and M. Horodecki and P. Horodecki [82], independently, introduced a map Γ∶𝜌→Tr_𝐵[𝜌_𝐴𝐵]⊗𝐼−𝜌_𝐴𝐵(𝐼⊗Tr_𝐴[𝜌_𝐴𝐵]−𝜌_𝐴𝐵), which gives rise to a simple necessary condition for separability in arbitrary dimensions, called the reduction criterion. If 𝜌_𝐴𝐵 is separable, then 𝜌_𝐴⊗𝐼−𝜌_𝐴𝐵≥0,𝐼⊗𝜌_𝐵−𝜌_𝐴𝐵≥0, (2.8) where 𝜌_𝐴=Tr_𝐵[𝜌_𝐴𝐵], 𝜌_𝐵=Tr_𝐴[𝜌_𝐴𝐵]. 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 𝜌_𝐴𝐵 is separable, then the realigned matrix ℛ(𝜌) with elements ℛ(𝜌)_{𝑖𝑗,𝑘𝑙}=𝜌_{𝑖𝑘,𝑗𝑙} has trace norm not greater than one: ‖ℛ(𝜌)‖_𝐾𝐹≤1. (2.9) Quite remarkably, the realignment criterion can detect some PPT entangled (bound entangled) states [33, 83] and can be used for construction of some nondecomposable maps. It also provides nice lower bound for concurrence [61].

2.1.4. Criteria Based on Bloch Representations

Any Hermitian operator on an 𝑁-dimensional Hilbert space 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 𝑃_𝑗𝑘=|𝑗⟩⟨𝑘|, where |𝑖⟩, 𝑖=1,…,𝑁, are the orthonormal eigenstates of a linear Hermitian operator on ℋ. Set 𝜔_𝑙=−2𝑙(𝑙+1)(𝑃₁₁+𝑃₂₂+⋯+𝑃_𝑙𝑙−𝑙𝑃_{𝑙+1,𝑙+1}𝑢),_𝑗𝑘=𝑃_𝑗𝑘+𝑃_𝑘𝑗,𝑣_𝑗𝑘=𝑖(𝑃_𝑗𝑘−𝑃_𝑘𝑗), (2.10) where 1≤𝑙≤𝑁−1 and 1≤𝑗<𝑘≤𝑁. We get a set of 𝑁²−1 operators Γ≡{𝜔_𝑙,𝜔₂,…,𝜔_𝑁−1,𝑢₁₂,𝑢₁₃,…,𝑣₁₂,𝑣₁₃,…}, (2.11) which satisfies the relations Tr[𝜆_𝑖]=0,Tr[𝜆_𝑖𝜆_𝑗]=2𝛿_𝑖𝑗,∀𝜆_𝑖∈Γ, (2.12) and thus generate the 𝑆𝑈(𝑁) [85].

Any Hermitian operator 𝜌 in ℋ can be represented in terms of these generators of 𝑆𝑈(𝑁) as

1𝜌=𝑁𝐼_𝑁+12^𝑁2−1_𝑗=1𝑟_𝑗𝜆_𝑗, (2.13) where 𝐼_𝑁 is a unit matrix and 𝐫=(𝑟₁,𝑟₂,…,𝑟_𝑁²₋₁)∈ℝ^𝑁2−1 and 𝐫 is called Bloch vector. The set of all the Bloch vectors that constitute a density operator is known as the Bloch vector space 𝐵(ℝ^𝑁2−1).

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

𝐷_𝑟(ℝ^𝑁2−1)⊆𝐵(ℝ^𝑁2−1)⊆𝐷_𝑅(ℝ^𝑁2−1). (2.14)

Let the dimensions of systems 𝐴,𝐵, and 𝐶 be 𝑑_𝐴=𝑁₁,𝑑_𝐵=𝑁₂, and 𝑑_𝐶=𝑁₃, respectively. Any tripartite quantum states 𝜌_𝐴𝐵𝐶∈ℋ_𝐴⊗ℋ_𝐵⊗ℋ_𝐶 can be written as

𝜌_𝐴𝐵𝐶=𝐼_𝑁1⊗𝐼_𝑁2⊗𝑀₀+^𝑁21−1_𝑖=1𝜆_𝑖(1)⊗𝐼_𝑁2⊗𝑀_𝑖+^𝑁22−1_𝑗=1𝐼_𝑁1⊗𝜆_𝑗(2)⊗𝑀_𝑗+^{𝑁21−1𝑁}_𝑖=1²²⁻¹_𝑗=1𝜆_𝑖(1)⊗𝜆_𝑗(2)⊗𝑀_𝑖𝑗, (2.15) where 𝜆_𝑖(1), 𝜆_𝑗(2) are the generators of 𝑆𝑈(𝑁₁) and 𝑆𝑈(𝑁₂); 𝑀_𝑖,𝑀_𝑗, and 𝑀_𝑖𝑗 are operators of ℋ_𝐶.

Theorem 2.2. Let 𝐫∈ℝ^𝑁21−1, 𝐬∈ℝ^𝑁22−1 and |𝐫|≤2/𝑁₁(𝑁₁−1), |𝐬|≤2/𝑁₂(𝑁₂−1). For a tripartite quantum state 𝜌∈ℋ_𝐴⊗ℋ_𝐵⊗ℋ_𝐶 with representation (2.15), one has [87] 𝑀₀−^𝑁21−1_𝑖=1𝑟_𝑖𝑀_𝑖−^𝑁22−1_𝑗=1𝑠_𝑗𝑀_𝑗+^{𝑁21−1𝑁}_𝑖=1²²⁻¹_𝑗=1𝑟_𝑖𝑠_𝑗𝑀_𝑖𝑗≥0. (2.16)

Proof. Since 𝐫∈ℝ^𝑁21−1, 𝐬∈ℝ^𝑁22−1 and |𝐫|≤2/𝑁₁(𝑁₁−1), |𝐬|≤2/𝑁₂(𝑁₂−1), we have that 𝐴₁≡(1/2)((2/𝑁₁)𝐼−∑^𝑁21−1_𝑖=1𝑟_𝑖𝜆_𝑖(1)) and 𝐴₂≡(1/2)((2/𝑁₂)𝐼−∑^𝑁22−1_𝑗=1𝑠_𝑗𝜆_𝑗(2)) are positive Hermitian operators. Let 𝐴=𝐴₁⊗𝐴₂⊗𝐼_𝑁3. Then 𝐴𝜌𝐴≥0 and (𝐴𝜌𝐴)^†=𝐴𝜌𝐴. The partial trace of 𝐴𝜌𝐴 over ℋ_𝐴 (and ℋ_𝐵) should be also positive. Hence 0≤Tr_𝐴𝐵[𝐴𝜌𝐴]=Tr_𝐴𝐵[𝐴₁⊗𝐴₂⊗𝑀₀+_𝑖𝐴₁𝜆_𝑖(1)𝐴₁⊗𝐴₂⊗𝑀_𝑖+_𝑗𝐴₁⊗𝐴₂𝜆_𝑗(2)𝐴₂⊗𝑀_𝑗+_𝑖𝑗𝐴₁𝜆_𝑖(1)𝐴₁⊗𝐴₂𝜆_𝑗(2)𝐴₂⊗𝑀_𝑖𝑗]=𝑀₀−^𝑁21−1_𝑖=1𝑟_𝑖𝑀_𝑖−^𝑁22−1_𝑗=1𝑠_𝑗𝑀_𝑗+^{𝑁21−1𝑁}_𝑖=1²²⁻¹_𝑗=1𝑟_𝑖𝑠_𝑗𝑀_𝑖𝑗. (2.17)

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

Corollary 2.3. Let 𝜌_𝐴𝐵∈ℋ_𝐴⊗ℋ_𝐵, which can be generally written as 𝜌_𝐴𝐵=𝐼_𝑁1⊗𝑀₀+∑^𝑁21−1_𝑗=1𝜆_𝑗⊗𝑀_𝑗, then, for any 𝐫∈ℝ^𝑁21−1 with |𝐫|≤2/𝑁₁(𝑁₁−1), 𝑀₀−∑^𝑁21−1_𝑗=1𝑟_𝑗𝑀_𝑗≥0.

A separable tripartite state 𝜌_𝐴𝐵𝐶 can be written as 𝜌_𝐴𝐵𝐶=_𝑖𝑝_𝑖|𝜓^𝐴_𝑖⟩⟨𝜓^𝐴_𝑖|⊗|𝜙^𝐵_𝑖⟩⟨𝜙^𝐵_𝑖|⊗|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|. (2.18) From (2.13) it can also be represented as

𝜌_𝐴𝐵𝐶=_𝑖𝑝_𝑖12(2𝑁₁𝐼_𝑁1+^𝑁21−1_𝑘=1𝑎_𝑖^(𝑘)𝜆_𝑘1(1))⊗2(2𝑁₂𝐼_𝑁2+^𝑁22−1_𝑙=1𝑏_𝑖^(𝑙)𝜆_𝑙(2))⊗|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|=𝐼_𝑁1⊗𝐼_𝑁2⊗1𝑁₁𝑁_2𝑖𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|+^𝑁21−1_𝑘=1𝜆_𝑘(1)⊗𝐼_𝑁2⊗12𝑁_2𝑖𝑎_𝑖^(𝑘)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|+^𝑁22−1_𝑙=1𝐼_𝑁1⊗𝜆_𝑙1(2)⊗2𝑁_1𝑖𝑏_𝑖^(𝑙)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|+^𝑁21−1_𝑘^𝑁22−1_𝑙𝜆_𝑘(1)⊗𝜆_𝑙1(2)⊗4_𝑖𝑎_𝑖^(𝑘)𝑏_𝑖^(𝑙)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|, (2.19) where (𝑎_𝑖⁽¹⁾,𝑎_𝑖⁽²⁾,…,𝑎^(𝑁21_𝑖⁻¹⁾) and (𝑏_𝑖⁽¹⁾,𝑏_𝑖⁽²⁾,…,𝑏^(𝑁22_𝑖⁻¹⁾) are real vectors on the Bloch sphere satisfying |𝑎_𝑖|²=∑^𝑁21−1_𝑗=1(𝑎_𝑖^(𝑗))²=2(1−1/𝑁₁) and |𝑏_𝑖|²=∑^𝑁22−1_𝑗=1(𝑏_𝑖^(𝑗))²=2(1−1/𝑁₂).

Comparing (2.15) with (2.19), we have

𝑀₀=1𝑁₁𝑁_2𝑖𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|,𝑀_𝑘=12𝑁_2𝑖𝑎_𝑖^(𝑘)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖𝑀|,_𝑙=12𝑁_1𝑖𝑏_𝑖^(𝑙)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|,𝑀_𝑘𝑙=14_𝑖𝑎_𝑖^(𝑘)𝑏_𝑖^(𝑙)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|. (2.20)

For any (𝑁²₁−1)×(𝑁²₁−1) 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

ℛ=𝑅(1)000𝑅(2)000𝑇, (2.21) where 𝑇 is a transformation acting on an (𝑁²₁−1)×(𝑁²₂−1) matrix 𝑀 by

𝑇(𝑀)=𝑅(1)𝑀𝑅^𝑇(2). (2.22)

Using ℛ, we define a new operator 𝛾_ℛ:

𝛾_ℛ(𝜌_𝐴𝐵𝐶)=𝐼_𝑁1⊗𝐼_𝑁2⊗𝑀^′₀+^𝑁21−1_𝑖=1𝜆_𝑖(1)⊗𝐼_𝑁2⊗𝑀^′_𝑖+^𝑁22−1_𝑗=1𝐼_𝑁1⊗𝜆_𝑗(2)⊗𝑀^′_𝑗+^{𝑁21−1𝑁}_𝑖=1²²⁻¹_𝑗=1𝜆_𝑖(1)⊗𝜆_𝑗(2)⊗𝑀^′_𝑖𝑗, (2.23) where 𝑀^′₀=𝑀₀,𝑀^′_𝑘=∑^𝑁21−1_𝑚=1𝑅_𝑘𝑚(1)𝑀_𝑚,𝑀^′_𝑙=∑^𝑁22−1_𝑛=1𝑅_𝑙𝑛(2)𝑀_𝑛, and 𝑀^′_𝑖𝑗=(𝑇(𝑀))_𝑖𝑗=(𝑅(1)𝑀𝑅^𝑇(2))_𝑖𝑗.

Theorem 2.4. If 𝜌_𝐴𝐵𝐶 is separable, then [87] 𝛾_ℛ(𝜌_𝐴𝐵𝐶)≥0.

Proof. From (2.20) and (2.23) we get 𝑀^′₀=𝑀₀=1𝑁₁𝑁_2𝑖𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|,𝑀^′_𝑘=12𝑁_2𝑚𝑖𝑅_𝑘𝑚(1)𝑎_𝑖^(𝑚)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖𝑀|,_′𝑙=12𝑁_1𝑛𝑖𝑅_𝑙𝑛(2)𝑏_𝑖^(𝑛)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|,𝑀^′_𝑘𝑙=14_𝑚𝑛𝑖𝑅_𝑘𝑚(1)𝑎_𝑖^(𝑚)𝑅_𝑙𝑛(2)𝑏_𝑖^(𝑛)𝑝_𝑖|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|. (2.24) A straightforward calculation gives rise to 𝛾_ℛ(𝜌_𝐴𝐵𝐶)=_𝑖𝑝_𝑖12(2𝑁₁𝐼_𝑁1+^{𝑁21−1𝑁}_𝑘=1²¹⁻¹_𝑚=1𝑅_𝑘𝑚(1)𝑎_𝑖^(𝑚)𝜆_𝑘⊗1(1))2(2𝑁₂𝐼_𝑁2+^{𝑁22−1𝑁}_𝑙=1²²⁻¹_𝑛=1𝑅_𝑙𝑛(2)𝑏_𝑖^(𝑛)𝜆_𝑙(2))⊗|𝜔^𝐶_𝑖⟩⟨𝜔^𝐶_𝑖|. (2.25) As (1/(𝑁₁−1)²)𝐼−𝑅(1)^𝑇𝑅(1)≥0 and (1/(𝑁₂−1)²)𝐼−𝑅(2)^𝑇𝑅(2)≥0, we get |𝑎^′_𝑖|²=|𝑅(1)𝑎_𝑖|²≤1(𝑁₁−1)²|𝑎_𝑖|²=2𝑁₁(𝑁₁,|𝑏−1)_′𝑖|²=|𝑅(2)𝑏_𝑖|²≤1(𝑁₂−1)²|𝑏_𝑖|²=2𝑁₂(𝑁₂.−1) (2.26) Therefore 𝛾_ℛ(𝜌_𝐴𝐵𝐶) is still a density operator, that is, 𝛾_ℛ(𝜌_𝐴𝐵𝐶)≥0.

Theorem 2.4 gives a necessary separability criterion for general tripartite systems. The result can be also applied to bipartite systems. Let 𝜌_𝐴𝐵∈ℋ_𝐴⊗ℋ_𝐵, 𝜌_𝐴𝐵=𝐼_𝑁1⊗𝑀₀+∑^𝑁21−1_𝑗=1𝜆_𝑗⊗𝑀_𝑗. For any real (𝑁²₁−1)×(𝑁²₁−1) matrix ℛ satisfying (1/(𝑁₁−1)²)𝐼−ℛ^𝑇ℛ≥0 and any state 𝜌_𝐴𝐵, we define 𝛾_ℛ(𝜌_𝐴𝐵)=𝐼_𝑁1⊗𝑀₀+^𝑁21−1_𝑗=1𝜆_𝑗⊗𝑀^′_𝑗, (2.27) where 𝑀^′_𝑗=∑_𝑘ℛ_𝑗𝑘𝑀_𝑘.

Corollary 2.5. For 𝜌_𝐴𝐵∈ℋ_𝐴⊗ℋ_𝐵, if there exists an ℛ with (1/(𝑁₁−1)²)𝐼−ℛ^𝑇ℛ≥0 such that 𝛾_ℛ(𝜌_𝐴𝐵)<0, then 𝜌_𝐴𝐵 must be entangled.

For 2×𝑁 systems, the above corollary is reduced to the results in [88]. As an example we consider the 3×3 istropic states 𝜌_𝐼=1−𝑝9𝐼₃⊗𝐼₃+𝑝3³_𝑖,𝑗=1|𝑖𝑖⟩⟨𝑗𝑗|=𝐼₃1⊗(9𝐼₃)+⁵_𝑖=1𝜆_𝑖𝑝⊗(6𝜆_𝑖)−⁸_𝑖=6𝜆_𝑖𝑝⊗(6𝜆_𝑖). (2.28) 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 𝜌_𝐼 is entangled for 0.5<𝑝≤1.

For tripartite case, we take the following 3×3×3 mixed state as an example: 𝜌=1−𝑝𝐼27²⁷+𝑝|𝜓⟩⟨𝜓|, (2.29) where |𝜓⟩=(1/√3)(|000⟩+|111⟩+|222⟩)(⟨000|+⟨111|+⟨222|). Taking 𝑅(1)=𝑅(2)=Diag{1/2,1/2,1/2,1/2,1/2,−1/2,−1/2,−1/2}, we have that 𝜌 is entangled for 0.6248<𝑝≤1.

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

2.1.5. Covariance Matrix Criterion

In this subsection we study the separability problem by using the covariance matrix approach. We first give a brief review of covariance matrix criterion proposed in [49]. Let ℋ^𝐴_𝑑 and ℋ^𝐵_𝑑 be 𝑑-dimensional complex vector spaces and 𝜌_𝐴𝐵 a bipartite quantum state in ℋ^𝐴_𝑑⊗ℋ^𝐵_𝑑. Let 𝐴_𝑘 (resp., 𝐵_𝑘) be 𝑑² observables on ℋ^𝐴_𝑑 (resp., ℋ^𝐵_𝑑) such that they form an orthonormal normalized basis of the observable space, satisfying Tr[𝐴_𝑘𝐴_𝑙]=𝛿_𝑘,𝑙 (resp., Tr[𝐵_𝑘𝐵_𝑙]=𝛿_𝑘,𝑙). Consider the total set {𝑀_𝑘}={𝐴_𝑘⊗𝐼,𝐼⊗𝐵_𝑘}. It can be proven that [44]

^𝑁2_𝑘=1(𝑀_𝑘)²=𝑑𝐼,^𝑁2_𝑘=1⟨𝑀_𝑘⟩²=Tr[𝜌²_𝐴𝐵]. (2.30)

The covariance matrix 𝛾 is defined with entries

𝛾_𝑖𝑗(𝜌_𝐴𝐵,{𝑀_𝑘})=⟨𝑀_𝑖𝑀_𝑗⟩+⟨𝑀_𝑗𝑀_𝑖⟩2−⟨𝑀_𝑖⟩⟨𝑀_𝑗⟩, (2.31) which has a block structure [49]

𝐶𝛾=𝐴𝐶_𝑇𝐵, (2.32) where 𝐴=𝛾(𝜌_𝐴,{𝐴_𝑘}),𝐵=𝛾(𝜌_𝐵,{𝐵_𝑘}),𝐶_𝑖𝑗=⟨𝐴_𝑖⊗𝐵_𝑗⟩_𝜌𝐴𝐵−⟨𝐴_𝑖⟩_𝜌𝐴⟨𝐵_𝑗⟩_𝜌𝐵,𝜌_𝐴=Tr_𝐵[𝜌_𝐴𝐵], and 𝜌_𝐵=Tr_𝐴[𝜌_𝐴𝐵]. Such covariance matrix has a concavity property: for a mixed density matrix 𝜌=∑_𝑘𝑝_𝑘𝜌_𝑘 with 𝑝_𝑘≥0 and ∑_𝑘𝑝_𝑘=1, one has 𝛾(𝜌)≥∑_𝑘𝑝_𝑘𝛾(𝜌_𝑘).

For a bipartite product state 𝜌_𝐴𝐵=𝜌_𝐴⊗𝜌_𝐵, 𝐶 in (2.32) is zero. Generally if 𝜌_𝐴𝐵 is separable, then there exist states |𝑎_𝑘⟩⟨𝑎_𝑘| on ℋ^𝐴_𝑑,|𝑏_𝑘⟩⟨𝑏_𝑘| on ℋ^𝐵_𝑑 and 𝑝_𝑘 such that

𝛾(𝜌)≥𝜅_𝐴⊕𝜅_𝐵, (2.33) where 𝜅_𝐴=∑𝑝_𝑘𝛾(|𝑎_𝑘⟩⟨𝑎_𝑘|,{𝐴_𝑘}), 𝜅_𝐵=∑𝑝_𝑘𝛾(|𝑏_𝑘⟩⟨𝑏_𝑘|,{𝐵_𝑘}).

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

^𝑑2_𝑖=1|𝐶_𝑖𝑖|≤(1−Tr[𝜌²_𝐴])+(1−Tr[𝜌²_𝐵])2. (2.34)

Criterion (2.34) depends on the choice of the orthonormal normalized basis of the observables. In fact the term ∑^𝑑2_𝑖=1|𝐶_𝑖𝑖| 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 𝜌_𝐴𝐵 is separable, then

‖𝐶‖_𝐾𝐹≤(1−Tr[𝜌²_𝐴])+(1−Tr[𝜌²_𝐵])2. (2.35)

From the covariance matrix approach, we can also get an alternative criterion. From (2.32) and (2.33) we have that if 𝜌_𝐴𝐵 is separable, then 𝑋≡𝐴−𝜅_𝐴𝐶𝐶^𝑇𝐵−𝜅_𝐵≥0. (2.36) Hence all the 2×2 minor submatrices of 𝑋 must be positive. Namely, one has (𝐴−𝜅_𝐴)_𝑖𝑖𝐶_𝑖𝑗𝐶_𝑗𝑖(𝐵−𝜅_𝐵)_𝑗𝑗≥0, (2.37) that is, (𝐴−𝜅_𝐴)_𝑖𝑖(𝐵−𝜅_𝐵)_𝑗𝑗≥𝐶²_𝑖𝑗. Summing over all 𝑖, 𝑗 and using (2.30), we get

^𝑑2_𝑖,𝑗=1𝐶²_𝑖,𝑗≤(Tr[𝐴]−Tr[𝜅_𝐴])(Tr[𝐵]−Tr[𝜅_𝐵])=(𝑑−Tr[𝜌²_𝐴]−𝑑+1)(𝑑−Tr[𝜌²_𝐵]−𝑑+1)=(1−Tr[𝜌²_𝐴])(1−Tr[𝜌²_𝐵]). (2.38)

That is,

‖𝐶‖²_𝐻𝑆≤(1−Tr[𝜌²_𝐴])(1−Tr[𝜌²_𝐵]), (2.39) 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 (1−Tr[𝜌²_𝐴])(1−Tr[𝜌²_𝐵])<‖𝐶‖_𝐻𝑆≤‖𝐶‖_𝐾𝐹≤(1−Tr[𝜌²_𝐴])+(1−Tr[𝜌²_𝐵])2, (2.40)(2.39) can recognize the entanglement but (2.35) cannot. When ‖𝐶‖_𝐻𝑆≤(1−Tr[𝜌²_𝐴])(1−Tr[𝜌²_𝐵])≤(1−Tr[𝜌²_𝐴])+(1−Tr[𝜌²_𝐵])2<‖𝐶‖_𝐾𝐹, (2.41)(2.35) can recognize the entanglement while (2.39) cannot.

The separability criteria based on covariance matrix approach can be generalized to multipartite systems. We first consider the tripartite case 𝜌_𝐴𝐵𝐶∈ℋ^𝐴_𝑑⊗ℋ^𝐵_𝑑⊗ℋ^𝐶_𝑑. Take 𝑑² observables 𝐴_𝑘 on ℋ_𝐴, respectively, 𝐵_𝑘 on ℋ_𝐵, respectively, 𝐶_𝑘 on ℋ_𝐶. Set {𝑀_𝑘}={𝐴_𝑘⊗𝐼⊗𝐼,𝐼⊗𝐵_𝑘⊗𝐼,𝐼⊗𝐼⊗𝐶_𝑘}. The covariance matrix defined by (2.31) has then the following block structure:

𝛾=𝐷𝐴𝐷𝐸_𝑇𝐸𝐵𝐹_𝑇𝐹^𝑇𝐶, (2.42) where 𝐴=𝛾(𝜌_𝐴,{𝐴_𝑘}), 𝐵=𝛾(𝜌_𝐵,{𝐵_𝑘}), 𝐶=𝛾(𝜌_𝐶,{𝐶_𝑘}), 𝐷_𝑖𝑗=⟨𝐴_𝑖⊗𝐵_𝑗⟩_𝜌𝐴𝐵−⟨𝐴_𝑖⟩_𝜌𝐴⟨𝐵_𝑗⟩_𝜌𝐵, 𝐸_𝑖𝑗=⟨𝐴_𝑖⊗𝐶_𝑗⟩_𝜌𝐴𝐶−⟨𝐴_𝑖⟩_𝜌𝐴⟨𝐶_𝑗⟩_𝜌𝐶, and 𝐹_𝑖𝑗=⟨𝐵_𝑖⊗𝐶_𝑗⟩_𝜌𝐵𝐶−⟨𝐵_𝑖⟩_𝜌𝐵⟨𝐶_𝑗⟩_𝜌𝐶.

Theorem 2.6. If 𝜌_𝐴𝐵𝐶 is fully separable, then [90] ‖𝐷‖²_𝐻𝑆≤(1−Tr[𝜌²_𝐴])(1−Tr[𝜌²_𝐵]),‖𝐸‖²_𝐻𝑆≤(1−Tr[𝜌²_𝐴])(1−Tr[𝜌²_𝐶]),‖𝐹‖²_𝐻𝑆≤(1−Tr[𝜌²_𝐵])(1−Tr[𝜌²_𝐶]),2‖𝐷‖_𝐾𝐹≤(1−Tr[𝜌²_𝐴])+(1−Tr[𝜌²_𝐵]),2‖𝐸‖_𝐾𝐹≤(1−Tr[𝜌²_𝐴])+(1−Tr[𝜌²_𝐶]),2‖𝐹‖_𝐾𝐹≤(1−Tr[𝜌²_𝐵])+(1−Tr[𝜌²_𝐶]). (2.43)(2.44)(2.45)(2.46)(2.47)(2.48)

Proof. For a tripartite product state 𝜌_𝐴𝐵𝐶=𝜌_𝐴⊗𝜌_𝐵⊗𝜌_𝐶, 𝐷,𝐸, and 𝐹 in (2.42) are zero. If 𝜌_𝐴𝐵𝐶 is fully separable, then there exist states |𝑎_𝑘⟩⟨𝑎_𝑘| in ℋ^𝐴_𝑑, |𝑏_𝑘⟩⟨𝑏_𝑘| in ℋ^𝐵_𝑑, and |𝑐_𝑘⟩⟨𝑐_𝑘| in ℋ^𝐶_𝑑, and 𝑝_𝑘 such that 𝛾(𝜌)≥𝜅_𝐴⊕𝜅_𝐵⊕𝜅_𝐶, where 𝜅_𝐴=∑𝑝_𝑘𝛾(|𝑎_𝑘⟩⟨𝑎_𝑘|,{𝐴_𝑘}), 𝜅_𝐵=∑𝑝_𝑘𝛾(|𝑏_𝑘⟩⟨𝑏_𝑘|,{𝐵_𝑘}), and 𝜅_𝐶=∑𝑝_𝑘𝛾(|𝑐_𝑘⟩⟨𝑐_𝑘|,{𝐶_𝑘}), that is, 𝑌≡𝐴−𝜅_𝐴𝐷𝐷𝐸_𝑇𝐵−𝜅_𝐵𝐹𝐸^𝑇𝐹^𝑇𝐶−𝜅_𝐶≥0. (2.49) 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 (𝐴−𝜅_𝐴)_𝑖𝑖𝐷_𝑖𝑗𝐷_𝑗𝑖(𝐵−𝜅_𝐵)_𝑗𝑗≥0, (2.50) that is, (𝐴−𝜅_𝐴)_𝑖𝑖(𝐵−𝜅_𝐵)_𝑗𝑗≥|𝐷_𝑖𝑗|². Summing over all 𝑖, 𝑗 and using (2.30), we get ‖𝐷‖²_𝐻𝑆=^𝑑2_𝑖,𝑗=1𝐷²_𝑖,𝑗≤(Tr[𝐴]−Tr[𝜅_𝐴])(Tr[𝐵]−Tr[𝜅_𝐵]=(𝑑−Tr[𝜌²_𝐴]−𝑑+1)(𝑑−Tr[𝜌²_𝐵]−𝑑+1)=(1−Tr[𝜌²_𝐴])(1−Tr[𝜌²_𝐵]), (2.51) which proves (2.43). Equations (2.44) and (2.45) can be similarly proved.
From (2.50) we also have (𝐴−𝜅_𝐴)_𝑖𝑖+(𝐵−𝜅_𝐵)_𝑖𝑖≥2|𝐷_𝑖𝑖|. Therefore _𝑖|𝐷_𝑖𝑖|≤(Tr[𝐴]−Tr[𝜅_𝐴])+(Tr[𝐵]−Tr[𝜅_𝐵]2=(𝑑−Tr[𝜌²_𝐴]−𝑑+1)+(𝑑−Tr[𝜌²_𝐵]−𝑑+1)2=(1−Tr[𝜌²_𝐴])+(1−Tr[𝜌²_𝐵])2. (2.52) Note that ∑^𝑑2_𝑖=1𝐷_𝑖𝑖≤∑^𝑑2_𝑖=1|𝐷_𝑖𝑖|. By using that Tr[𝑀𝑈]≤‖𝑀‖_𝐾𝐹=Tr[𝑀𝑀^†] for any matrix 𝑀 and any unitary 𝑈 [91], we have ∑^𝑑2_𝑖=1𝐷_𝑖𝑖≤‖𝐷‖_𝐾𝐹.
Let 𝐷=𝑈^†Λ𝑉 be the singular value decomposition of 𝐷. Make a transformation of the orthonormal normalized basis of the local orthonormal observable space 𝐴_𝑖=∑_𝑙𝑈_𝑖𝑙𝐴_𝑙 and 𝐵_𝑗=∑_𝑚𝑉^∗_𝑗𝑚𝐵_𝑚. In the new basis we have 𝐷_𝑖𝑗=_𝑙𝑚𝑈_𝑖𝑙𝐷_𝑙𝑚𝑉_𝑗𝑚=(𝑈𝐷𝑉^†)_𝑖𝑗=Λ_𝑖𝑗. (2.53) Then (2.52) becomes ^𝑑2_𝑖=1𝐷_𝑖𝑖=‖𝐷‖_𝐾𝐹≤(1−Tr[𝜌²_𝐴])+(1−Tr[𝜌²_𝐵])2, (2.54) which proves (2.46). Equations (2.47) and (2.48) can be similarly treated.

We consider now the case that 𝜌_𝐴𝐵𝐶 is bipartite separable.

Theorem 2.7. If 𝜌_𝐴𝐵𝐶 is a bipartite separable state with respect to the bipartite partition of the sub-systems 𝐴 and 𝐵𝐶 (resp., 𝐴𝐵 and 𝐶; resp., 𝐴𝐶 and 𝐵), then (2.43), (2.44) and (2.46), (2.47) (resp., (2.44), (2.45) and (2.47), (2.48); resp., (2.43), (2.45) and (2.46), (2.48)) must hold [90].

Proof. We prove the case that 𝜌_𝐴𝐵𝐶 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. 𝜌_𝐴𝐵𝐶 takes the form 𝜌_𝐴𝐵𝐶=∑_𝑚𝑝_𝑚𝜌^𝑚_𝐴⊗𝜌^𝑚_𝐵𝐶. Define 𝜅_𝐴=∑𝑝_𝑚𝛾(𝜌^𝑚_𝐴,{𝐴_𝑘}), 𝜅_𝐵𝐶=∑𝑝_𝑚𝛾(𝜌^𝑚_𝐵𝐶,{𝐵_𝑘⊗𝐼,𝐼⊗𝐶_𝑘}). 𝜅_𝐵𝐶 has a form 𝜅_𝐵𝐶=𝜅_𝐵𝐹^′(𝐹^′)^𝑇𝜅_𝐶, (2.55) where 𝜅_𝐵=∑𝑝_𝑘𝛾(|𝑏_𝑘⟩⟨𝑏_𝑘|,{𝐵_𝑘}) and 𝜅_𝐶=∑𝑝_𝑘𝛾(|𝑐_𝑘⟩⟨𝑐_𝑘|,{𝐶_𝑘}), (𝐹^′)_𝑖𝑗=∑_𝑚𝑝_𝑚(⟨𝐵_𝑖⊗𝐶_𝑗⟩_𝜌^𝑚_𝐵𝐶−⟨𝐵_𝑖⟩_𝜌^𝑚_𝐵⟨𝐶_𝑗⟩_𝜌^𝑚_𝐶). By using the concavity of covariance matrix we have 𝛾(𝜌_𝐴𝐵𝐶)≥_𝑚𝑝_𝑚𝛾(𝜌^𝑚_𝐴⊗𝜌^𝑚_𝐵C𝜅)=^𝐴000𝜅_𝐵𝐹^′0(𝐹^′)^𝑇𝜅_𝐶. (2.56) Accounting to the method used in proving Theorem 2, we get (2.43), (2.44), and (2.46), (2.47).

From Theorems 2.6 and 2.7 we have the following corollary.

Corollary 2.8. If two of the inequalities (2.43), (2.44), and (2.45) (or (2.46), (2.47), and (2.48)) are violated, then the state must be fully entangled.

The result of Theorem 2.6 can be generalized to general multipartite case 𝜌∈ℋ_𝑑⁽¹⁾⊗ℋ_𝑑⁽²⁾⊗⋯⊗ℋ_𝑑^(𝑁). Define 𝐴^𝑖_𝛼=𝐼⊗𝐼⊗⋯𝜆_𝛼⊗𝐼⊗⋯⊗𝐼, where 𝜆₀=𝐼/𝑑, 𝜆_𝛼 (𝛼=1,2,…,𝑑²−1) are the normalized generators of 𝑆𝑈(𝑑) satisfying Tr[𝜆_𝛼𝜆_𝛽]=𝛿_𝛼𝛽 and acting on the 𝑖th system ℋ_𝑑^(𝑖), 𝑖=1,2,…,𝑁. Denote {𝑀_𝑘} as the set of all 𝐴^𝑖_𝛼. Then the covariance matrix of 𝜌 can be written as

𝒜𝛾(𝜌)=¹¹𝒜₁₂⋯𝒜_1𝑁𝒜^𝑇₁₂𝒜₂₂⋯𝒜_2𝑁𝒜⋮⋮⋮_𝑇1𝑁𝒜^𝑇_2𝑁⋯𝒜_𝑁𝑁, (2.57) where 𝒜_𝑖𝑖=𝛾(𝜌,{𝐴^𝑖_𝑘}) and (𝒜_𝑖𝑗)_𝑚𝑛=⟨𝐴^𝑖_𝑚⊗𝐴^𝑗_𝑛⟩−⟨𝐴^𝑖_𝑚⟩⟨𝐴^𝑗_𝑛⟩ for 𝑖≠𝑗.

For a product state 𝜌_12⋯𝑁, 𝒜_𝑖𝑗, 𝑖≠𝑗, in (2.57) are zero matrices. Define 𝜅_𝒜𝑖𝑖=_𝑘𝑝_𝑘𝛾(|𝜓^𝑖_𝑘⟩⟨𝜓^𝑖_𝑘|,{𝐴^𝑖_𝑙}). (2.58) Then for a fully separable multipartite state 𝜌=∑_𝑘𝑝_𝑘|𝜓¹_𝑘⟩⟨𝜓¹_𝑘|⊗|𝜓²_𝑘⟩⟨𝜓²_𝑘|⊗⋯⊗|𝜓^𝑁_𝑘⟩⟨𝜓^𝑁_𝑘|, one has 𝒜𝑍=¹¹−𝜅_𝒜11𝒜₁₂⋯𝒜_1𝑁𝒜^𝑇₁₂𝒜₂₂−𝜅_𝒜22⋯𝒜_2𝑁𝒜⋮⋮⋮_𝑇1𝑁𝒜^𝑇_2𝑁⋯𝒜_𝑁𝑁−𝜅_𝒜𝑁𝑁≥0, (2.59) from which we have the following separability criterion for multipartite systems.

Theorem 2.9. If a state 𝜌∈ℋ_𝑑⁽¹⁾⊗ℋ_𝑑⁽²⁾⊗⋯⊗ℋ_𝑑^(𝑁) is fully separable, then the following inequalities ‖𝒜_𝑖𝑗‖²_𝐻𝑆≤(1−Tr[𝜌²_𝑖])(1−Tr[𝜌²_𝑗]),‖𝒜_𝑖𝑗‖_𝐾𝐹≤(1−Tr[𝜌²_𝑖])+(1−Tr[𝜌²_𝑗])2 (2.60) 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=ℋ_𝐴⊗ℋ_𝐵 with dimℋ_𝐴=𝑀, dimℋ_𝐵=𝑁, and 𝑀≤𝑁 is mapped to the following form under local filtering transformations [93]:

𝜌→𝜌=(𝐹_𝐴⊗𝐹_𝐵)𝜌(𝐹_𝐴⊗𝐹_𝐵)^†Tr[(𝐹_𝐴⊗𝐹_𝐵)𝜌(𝐹_𝐴⊗𝐹_𝐵)^†], (2.61) where 𝐹_𝐴/𝐵∈𝐺𝐿(𝑀/𝑁,ℂ) are arbitrary invertible matrices. This transformation is also known as stochastic local operations assisted by classical communication (SLOCC). By the definition it is obvious that filtering transformation will preserve the separability of a quantum state.

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

𝜌=(𝐹_𝐴⊗𝐹_𝐵)𝜌(𝐹_𝐴⊗𝐹_𝐵)^†Tr[(𝐹_𝐴⊗𝐹_𝐵)𝜌(𝐹_𝐴⊗𝐹_𝐵)^†]=1𝑀𝑁(𝐼+^𝑀2−1_𝑖=1𝜉_𝑖𝐺^𝐴_𝑖⊗𝐺^𝐵_𝑖), (2.62) where 𝜉_𝑖≥0 and 𝐺^𝐴_𝑖 and 𝐺^𝐵_𝑖 are some traceless orthogonal observables. The matrices 𝐹_𝐴 and 𝐹_𝐵 can be obtained by minimizing the function 𝑓(𝐴,𝐵)=Tr[𝜌(𝐴⊗𝐵)](det𝐴)^1/𝑀(det𝐵)^1/𝑁, (2.63) where 𝐴=𝐹^†_𝐴𝐹_𝐴 and 𝐵=𝐹^†_𝐵𝐹_𝐵. In fact, one can choose 𝐹⁰_𝐴≡|det(𝜌_𝐴)|^1/2𝑀(√𝜌_𝐴)⁻¹, and 𝐹⁰_𝐵≡|det(𝜌^′_𝐵)|^1/2𝑁(𝜌^′_𝐵)⁻¹, where 𝜌^′_𝐵=Tr_𝐴[𝐼⊗(√𝜌_𝐴)⁻¹𝜌𝐼⊗(√𝜌_𝐴)⁻¹]. Then by iterations one can get the optimal 𝐴 and 𝐵. In particular, there is a matlab code available in [95].

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

‖𝑇‖_𝐾𝐹≤𝑀𝑁(𝑀−1)(𝑁−1), (2.64) where 𝑇 is an (𝑀²−1)×(𝑁²−1) matrix with 𝑇_𝑖𝑗=𝑀𝑁⋅Tr[𝜌𝜆^𝐴_𝑖⊗𝜆^𝐵_𝑗], ‖𝑇‖_𝐾𝐹 stands for the trace norm of 𝑇, 𝜆_𝑘^𝐴/𝐵s are the generators of 𝑆𝑈(𝑀/𝑁) and have been chosen to be normalized, and Tr[𝜆_𝑘^(𝐴/𝐵)𝜆_𝑙^(𝐴/𝐵)]=𝛿_𝑘𝑙.

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

_𝑖𝜉_𝑖≤𝑀𝑁(𝑀−1)(𝑁−1). (2.65)

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

1−_𝑘⟨𝐺^𝐴_𝑘⊗𝐺^𝐵_𝑘1⟩−2⟨𝐺^𝐴_𝑘⊗𝐼−𝐼⊗𝐺^𝐵_𝑘⟩²≥0, (2.66) where 𝐺_𝑘^𝐴/𝐵s are LOOs such as the normalized generators of 𝑆𝑈(𝑀/𝑁) and 𝐺^𝐴_𝑘=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 1−_𝑘⟨𝐺^𝐴_𝑘⊗𝐺^𝐵_𝑘1⟩−2⟨𝐺^𝐴_𝑘⊗𝐼−𝐼⊗𝐺^𝐵_𝑘⟩²1=1−𝑀𝑁_𝑘𝜉_𝑘−12(1𝑀+1𝑁)≥0, (2.67) that is,

_𝑘𝜉_𝑘≤𝑀𝑁−𝑀+𝑁2. (2.68) As 𝑀𝑁(𝑀−1)(𝑁−1)≤𝑀𝑁−(𝑀+𝑁)/2 holds for any 𝑀 and 𝑁, from (2.65) and (2.68) it is obvious that the CM criterion recognizes entanglement better when the normal form is taken into account.

We now consider multipartite systems. Let 𝜌 be a strictly positive density matrix in ℋ=ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁 and dimℋ_𝑖=𝑑_𝑖. 𝜌 can be generally expressed in terms of the 𝑆𝑈(𝑛) generators 𝜆_𝛼𝑘 [97] as

1𝜌=Π^𝑁_𝑖𝑑_𝑖(⊗^𝑁_𝑗𝐼_𝑑𝑗+_{𝜇1}𝛼1𝒯^{𝜇1}_𝛼1𝜆^{𝜇1}_𝛼1+_{{𝜇1𝜇2}𝛼1𝛼2}𝒯^{𝜇1𝜇2}_𝛼1𝛼2𝜆^{𝜇1}_𝛼1𝜆^{𝜇2}_𝛼2+_{{𝜇1𝜇2𝜇3}𝛼1𝛼2𝛼3}𝒯^{{𝜇1𝜇2𝜇3}}_{𝛼1𝛼2𝛼3}𝜆^{𝜇1}_𝛼1𝜆^{𝜇2}_𝛼2𝜆^{𝜇3}_𝛼3+⋯+_{{𝜇1𝜇2⋯𝜇𝑀}𝛼1𝛼2⋯𝛼𝑀}𝒯^{{𝜇1𝜇2⋯𝜇𝑀}}_{𝛼1𝛼2⋯𝛼𝑀}𝜆^{𝜇1}_𝛼1𝜆^{𝜇2}_𝛼2⋯𝜆^{𝜇𝑀}_𝛼𝑀+⋯+_{𝛼1𝛼2⋯𝛼𝑁}𝒯_𝛼^{{1,2,…,𝑁}}_{1𝛼2⋯𝛼𝑀}𝜆_𝛼^{1}₁𝜆_𝛼^{2}₂⋯𝜆_𝛼^{𝑁}_𝑁), (2.69) where 𝜆^{𝜇𝑘}_𝛼𝑘=𝐼_𝑑1⊗𝐼_𝑑2⊗⋯⊗𝜆_𝛼𝑘⊗𝐼_{𝑑𝜇𝑘+1}⊗⋯⊗𝐼_𝑑𝑁 with 𝜆_𝛼𝑘 appears at the 𝜇_𝑘th position and

𝒯^{{𝜇1𝜇2⋯𝜇𝑀}}_{𝛼1𝛼2⋯𝛼𝑀}=∏^𝑀_𝑖=1𝑑_𝜇𝑖2^𝑀Tr[𝜌𝜆^{𝜇1}_𝛼1𝜆^{𝜇2}_𝛼2⋯𝜆^{𝜇𝑀}_𝛼𝑀]. (2.70)

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

‖𝒯^{{𝜇1,𝜇2,…,𝜇𝑀}}‖_𝐾𝐹≤12^𝑀𝑀_𝑘=1𝑑_𝜇𝑘(𝑑_𝜇𝑘−1), (2.71) for 2≤𝑀≤𝑁,{𝜇₁,𝜇₂,…,𝜇_𝑀}⊂{1,2,…,𝑁}. The KF norm is defined by ‖𝒯^{{𝜇1,𝜇2,…,𝜇𝑀}}‖_𝐾𝐹=max_{𝑚=1,2,…,𝑀}‖𝒯_(𝑚)‖_𝐾𝐹, (2.72) 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 𝜌=𝐹₁⊗𝐹₂⊗⋯⊗𝐹_𝑁𝜌𝐹^†₁⊗𝐹^†₂⊗𝐹^†_𝑁, (2.73) where 𝐹_𝑖∈𝐺𝐿(𝑑_𝑖,ℂ),𝑖=1,2,…𝑁, followed by normalization, any strictly positive state 𝜌 can be transformed into a normal form [98]: 1𝜌=Π^𝑁_𝑖𝑑_𝑖(⊗^𝑁_𝑗𝐼_𝑑𝑗+_{{𝜇1𝜇2}𝛼1𝛼2}𝒯^{𝜇1𝜇2}_𝛼1𝛼2𝜆^{𝜇1}_𝛼1𝜆^{𝜇2}_𝛼2+_{{𝜇1𝜇2𝜇3}𝛼1𝛼2𝛼3}𝒯^{{𝜇1𝜇2𝜇3}}_{𝛼1𝛼2𝛼3}𝜆^{𝜇1}_𝛼1𝜆^{𝜇2}_𝛼2𝜆^{𝜇3}_𝛼3+⋯+_{{𝜇1𝜇2⋯𝜇𝑀}𝛼1𝛼2⋯𝛼𝑀}𝒯^{{𝜇1𝜇2⋯𝜇𝑀}}_{𝛼1𝛼2⋯𝛼𝑀}𝜆^{𝜇1}_𝛼1𝜆^{𝜇2}_𝛼2⋯𝜆^{𝜇𝑀}_𝛼𝑀+⋯+_{𝛼1𝛼2⋯𝛼𝑁}𝒯_𝛼^{{1,2,…,𝑁}}_{1𝛼2⋯𝛼𝑀}𝜆_𝛼^{1}₁𝜆_𝛼^{2}₂⋯𝜆_𝛼^{𝑁}_𝑁). (2.74)

Proof. Let 𝐷₁,𝐷₂,…,𝐷_𝑁 be the sets of density matrices of the 𝑁 subsystems. The cartesian product 𝐷₁×𝐷₂×⋯×𝐷_𝑁 consisting of all product density matrices 𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁 with normalization Tr[𝜌_𝑖]=1, 𝑖=1,2,…,𝑁, is a compact set of matrices on the full Hilbert space ℋ. For the given density matrix 𝜌 we define the following function of 𝜌_𝑖: 𝑓(𝜌₁,𝜌₂,…,𝜌_𝑁)=Tr[𝜌(𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁)]∏^𝑁_𝑖=1det(𝜌_𝑖)^1/𝑑𝑖. (2.75) The function is well defined on the interior of 𝐷₁×𝐷₂×⋯×𝐷_𝑁 where det𝜌_𝑖>0. As 𝜌 is assumed to be strictly positive, we have Tr[𝜌(𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁)]>0. Since 𝐷₁×𝐷₂×⋯×𝐷_𝑁 is compact, we have Tr[𝜌(𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁)]≥𝐶>0 with a lower bound 𝐶 depending on 𝜌.
It follows that 𝑓→∞ on the boundary of 𝐷₁×𝐷₂×⋯×𝐷_𝑁 where at least one of the 𝜌_𝑖s satisfies det𝜌_𝑖=0. It follows further that 𝑓 has a positive minimum on the interior of 𝐷₁×𝐷₂×⋯×𝐷_𝑁 with the minimum value attained for at least one product density matrix 𝜏₁⊗𝜏₂⊗⋯⊗𝜏_𝑁 with det𝜏_𝑖>0, 𝑖=1,2,…,𝑁. Any positive density matrix 𝜏_𝑖 with det𝜏_𝑖>0 can be factorized in terms of Hermitian matrices 𝐹_𝑖 as 𝜏_𝑖=𝐹^†_𝑖𝐹_𝑖, (2.76) where 𝐹_𝑖∈𝐺𝐿(𝑑_𝑖,ℂ). Denote that 𝐹=𝐹₁⊗𝐹₂⊗⋯⊗𝐹_𝑁, so that 𝜏₁⊗𝜏₂⊗⋯⊗𝜏_𝑁=𝐹^†𝐹. Set 𝜌=𝐹𝜌𝐹^† and define 𝑓(𝜌₁,𝜌₂,…,𝜌_𝑁)=Tr[𝜌(𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁)]∏^𝑁_𝑖=1det(𝜌_𝑖)^1/𝑑𝑖=^𝑁_𝑖=1det(𝜏_𝑖)^1/𝑑𝑖⋅Tr[𝜌(𝐹^†₁𝜌₁𝐹₁⊗𝐹^†₂𝜌₂𝐹₂⊗⋯⊗𝐹^†_𝑁𝜌_𝑁𝐹_𝑁)]∏^𝑁_𝑖=1det(𝜏_𝑖)^1/𝑑𝑖det(𝜌_𝑖)^1/𝑑𝑖=^𝑁_𝑖=1det(𝜏_𝑖)^1/𝑑𝑖⋅𝑓(𝐹^†₁𝜌₁𝐹₁,𝐹^†₂𝜌₂𝐹₂,…,𝐹^†_𝑁𝜌_𝑁𝐹_𝑁). (2.77)
We see that, when 𝐹^†_𝑖𝜌_𝑖𝐹_𝑖=𝜏_𝑖, 𝑓 has a minimum and 𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁=(𝐹^†)⁻¹𝜏₁⊗𝜏₂⊗⋯⊗𝜏_𝑁𝐹⁻¹=𝐼. (2.78)
Since 𝑓 is stationary under infinitesimal variations about the minimum, it follows that Tr[𝜌𝛿(𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁)]=0 (2.79) for all infinitesimal variations 𝛿(𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁)=𝛿𝜌₁⊗𝐼_𝑑2⊗⋯⊗𝐼_𝑑𝑁+𝐼_𝑑1⊗𝛿_𝜌2⊗𝐼_𝑑3⊗⋯⊗𝐼_𝑑𝑁+⋯+𝐼_𝑑1⊗𝐼_𝑑2⊗⋯⊗𝐼_𝑑𝑁−1⊗𝛿𝜌_𝑁, (2.80) subjected to the constraint det(𝐼_𝑑𝑖+𝛿𝜌_𝑖)=1, which is equivalent to Tr[𝛿𝜌_𝑖]=0, 𝑖=1,2,…,𝑁, using det(𝑒^𝐴)=𝑒^Tr[𝐴] for a given matrix 𝐴. Thus, 𝛿𝜌_𝑖 can be represented by the 𝑆𝑈 generators, 𝛿𝜌_𝑖=∑_𝑘𝛿𝑐^𝑖_𝑘𝜆^𝑖_𝑘. It follows that Tr[𝜌𝜆^{𝜇𝑘}_𝛼𝑘]=0 for any 𝛼_𝑘 and 𝜇_𝑘. Hence the terms proportional to 𝜆^{𝜇𝑘}_𝛼𝑘 in (2.69) disappear.

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

Proof. Let 𝜌 be such a state. From (2.74), we get that 𝜌_𝑖=Tr_{1,2,…,𝑖−1,𝑖+1,…,𝑁}1[𝜌]=𝑑_𝑖𝐼_𝑑𝑖. (2.81) Therefore for a product state 𝜌 we have 𝜌=𝜌₁⊗𝜌₂⊗⋯⊗𝜌_𝑁=1∏^𝑁_𝑖=1𝑑_𝑖⊗^𝑁_𝑖=1𝐼_𝑑𝑖. (2.82)

As an example for separability of multipartite states in terms of their normal forms (2.74), we consider the PPT entangled edge state [79] 𝜌=1100000010𝑎00000000𝑏00000000𝑐00000000𝑐100000000𝑏100000000𝑎010000001 (2.83) mixed with noises 𝜌_𝑝=𝑝𝜌+(1−𝑝)8𝐼₈. (2.84) Select 𝑎=2,𝑏=3, and 𝑐=0.6. Using the criterion in [97] we get that 𝜌_𝑝 is entangled for 0.92744<𝑝≤1. But after transforming 𝜌_𝑝 to its normal form (2.74), the criterion can detect entanglement for 0.90285<𝑝≤1.

Here we indicate that the filtering transformation does not change the PPT property. Let 𝜌∈ℋ_𝐴⊗ℋ_𝐵 be PPT, that is, 𝜌^𝑇𝐴≥0, and 𝜌^𝑇𝐵≥0. Let 𝜌 be the normal form of 𝜌. From (2.61) we have 𝜌^𝑇𝐴=(𝐹^∗_𝐴⊗𝐹_𝐵)𝜌^𝑇𝐴(𝐹^𝑇_𝐴⊗𝐹^†_𝐵)Tr[(𝐹_𝐴⊗𝐹_𝐵)𝜌(𝐹_𝐴⊗𝐹_𝐵)^†]. (2.85) For any vector |𝜓⟩, we have ⟨𝜓|𝜌^𝑇𝐴|𝜓⟩=⟨𝜓|(𝐹^∗_𝐴⊗𝐹_𝐵)𝜌^𝑇𝐴(𝐹^𝑇_𝐴⊗𝐹^†_𝐵)|𝜓⟩Tr[(𝐹_𝐴⊗𝐹_𝐵)𝜌(𝐹_𝐴⊗𝐹_𝐵)^†]≡⟨𝜓^′|𝜌^𝑇𝐴|𝜓^′⟩≥0, (2.86) where |𝜓^′⟩=(𝐹^𝑇_𝐴⊗𝐹^†_𝐵)|𝜓⟩/Tr[(𝐹_𝐴⊗𝐹_𝐵)𝜌(𝐹_𝐴⊗𝐹_𝐵)^†].  𝜌^𝑇𝐵≥0 can be proved similarly. This property is also valid for multipartite case. Hence a bound entangled state will be bound entangled under filtering transformations.

2.3. Entanglement Witness Based on Correlation Matrix Criterion

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

Theorem 2.12. For any properly selected LOOs 𝐺^𝐴_𝑘 and 𝐺^𝐵_𝑘, 𝑊=𝐼−𝛼^𝑁2−1_𝑘=0𝐺^𝐴_𝑘⊗𝐺^𝐵_𝑘 (2.87) is an EW [98], where 𝛼=√𝑀𝑁/((𝑀−1)(𝑁−1)+1) and 𝐺^𝐴₀=1√𝑀𝐼_𝑀,𝐺^𝐵₀=1√𝑁𝐼_𝑁. (2.88)

Proof. Let 𝜌=∑^𝑁2−1_𝑙,𝑚=0𝑇_𝑙𝑚𝜆^𝐴_𝑙⊗𝜆^𝐵_𝑚 be a separable state, where 𝜆_𝑘^𝐴/𝐵 are normalized generators of 𝑆𝑈(𝑀/𝑁) with 𝜆^𝐴₀=(1/√𝑀)𝐼_𝑀, 𝜆^𝐵₀=(1/√𝑁)𝐼_𝑁. Any other LOOs 𝐺_𝑘^𝐴/𝐵 fulfilling (2.88) can be obtained from these 𝜆s through orthogonal transformations 𝒪^𝐴/𝐵,𝐺_𝑘^𝐴/𝐵=∑^𝑁2−1_𝑙=0𝒪^𝐴/𝐵_𝑘𝑙𝜆_𝑙, where 𝒪^𝐴/𝐵=(¹⁰_{0ℛ𝐴/𝐵}) and ℛ^𝐴/𝐵 are (𝑁²−1)×(𝑁²−1) orthogonal matrices. We have 1Tr[𝜌𝑊]=1−𝛼√𝑀𝑁−𝛼^{𝑁2−1𝑁}_𝑘=1²⁻¹_𝑙,𝑚=1ℛ^𝐴_𝑘𝑙ℛ^𝐵_𝑘𝑚Tr[𝜌(𝜆^𝐴_𝑙⊗𝜆^𝐵_𝑚=)](𝑀−1)(𝑁−1)−1(𝑀−1)(𝑁−1)+1√𝑀𝑁((𝑀−1)(𝑁−1)+1)^{𝑁2−1𝑁}_𝑘=1²⁻¹_𝑙,𝑚=1ℛ^𝐴_𝑘𝑙𝑇_𝑙𝑚ℛ^𝐵_𝑘𝑚≥𝑀𝑁(𝑀−1)(𝑁−1)−‖𝑇‖_𝐾𝐹√𝑀𝑁((𝑀−1)(𝑁−1)+1)≥0, (2.89) 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 let 𝜌=(1/𝑀𝑁)(𝐼_𝑀𝑁+∑^𝑀2−1_𝑖=1𝑠_𝑖𝜆^𝐴_𝑖⊗𝐼_𝑁+∑^𝑁2−1_𝑗=1𝑟_𝑗𝐼_𝑀⊗𝜆^𝐵_𝑗+∑^𝑀2−1_𝑖=1∑^𝑁2−1_𝑗=1𝑇_𝑖𝑗𝜆^𝐴_𝑖⊗𝜆^𝐵_𝑗) be a state in ℋ^𝑀_𝐴⊗ℋ^𝑁_𝐵 which violates the CM criterion. Denote 𝜎_𝑘(𝑇) as the singular values of 𝑇. By singular value decomposition, one has 𝑇=𝑈^†Λ𝑉^∗, where Λ is a diagonal matrix with Λ_𝑘𝑘=𝜎_𝑘(𝑇). Now choose LOOs to be 𝐺^𝐴_𝑘=∑_𝑙𝑈_𝑘𝑙𝜆^𝐴_𝑙, 𝐺^𝐵_𝑘=∑_𝑚𝑉_𝑘𝑚𝜆^𝐵_𝑚 for 𝑘=1,2,…,𝑁²−1 and 𝐺^𝐴₀=(1/𝑀)𝐼_𝑀,𝐺^𝐵₀=(1/𝑁)𝐼_𝑁. We obtain 1Tr[𝜌𝑊]=1−𝛼√𝑀𝑁−𝛼^{𝑁2−1𝑁}_𝑘=1²⁻¹_𝑙,𝑚=1𝑈_𝑘𝑙𝑉_𝑘𝑚Tr[𝜌(𝜆^𝐴_𝑙⊗𝜆^𝐵_𝑚=)](𝑀−1)(𝑁−1)−1(𝑀−1)(𝑁−1)+1√𝑀𝑁((𝑀−1)(𝑁−1)+1)Tr[𝑈𝑇𝑉^𝑇]=𝑀𝑁(𝑀−1)(𝑁−1)−‖𝑇‖_𝐾𝐹√𝑀𝑁((𝑀−1)(𝑁−1)+1)<0, (2.90) 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 ℋ^𝑑1₁⊗ℋ^𝑑2₂⊗⋯⊗ℋ^𝑑𝑁_𝑁. Set 𝑑(𝑀)=max{𝑑_𝜇𝑖,𝑖=1,2,…,𝑀}. Choose LOOs 𝐺^{𝜇𝑖}_𝑘 for 0≤𝑘≤𝑑(𝑀)²−1 with 𝐺^{𝜇𝑖}₀=(1/𝑑_𝜇𝑖)𝐼_𝑑𝜇𝑖 and define

𝑊^(𝑀)=𝐼−𝛽^{(𝑀)𝑑(𝑀)2−1}_𝑘=0𝐺^{𝜇1}_𝑘⊗𝐺^{𝜇2}_𝑘⊗⋯⊗𝐺^{𝜇𝑀}_𝑘, (2.91) where 𝛽^(𝑀)=∏^𝑀_𝑖=1𝑑_𝜇𝑖/(1+∏^𝑀_𝑖=1(𝑑_𝜇𝑖−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 ∑^{𝑑(𝑀)2−1}_𝑘=1𝒯^{{𝜇1𝜇2⋯𝜇𝑀}}_{𝑘𝑘⋯𝑘}=Tr[𝒯_(𝑚0)𝑃]. Then for an N-partite separable state we have Tr[𝜌𝑊^(𝑀)]=1−𝛽^(𝑀)1∏^𝑀_𝑖=1𝑑_𝜇𝑖−𝛽^(𝑀)1∏^𝑀_𝑖=1𝑑_𝜇𝑖Tr[𝒯_(𝑚0)𝑃]≥1−𝛽^(𝑀)1∏^𝑀_𝑖=1𝑑_𝜇𝑖−𝛽^(𝑀)1∏^𝑀_𝑖=1𝑑_𝜇𝑖‖𝒯_(𝑚0)‖_𝐾𝐹≥1−𝛽^(𝑀)1∏^𝑀_𝑖=1𝑑_𝜇𝑖−𝛽^(𝑀)1∏^𝑀_𝑖=1𝑑_𝜇𝑖^𝑀_𝑘=1𝑑_𝜇𝑘(𝑑_𝜇𝑘−1)=0 (2.92) for any 2≤𝑀≤𝑁, where we have taken into account that 𝑃 is orthogonal and Tr[𝑀𝑈]≤‖𝑀‖_𝐾𝐹 for any unitary 𝑈 at the first inequality. The second inequality is due to the generalized CM criterion.

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

3. Concurrence and Tangle

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

Let ℋ_𝐴 (resp., ℋ_𝐵) be an 𝑀-(resp., 𝑁-) dimensional complex vector space with |𝑖⟩, 𝑖=1,…,𝑀 (resp., |𝑗⟩,𝑗=1,…,𝑁), as an orthonormal basis. A general pure state on ℋ_𝐴⊗ℋ_𝐵 is of the form

|Ψ⟩=^𝑀𝑁_𝑖=1𝑗=1𝑎_𝑖𝑗|𝑖⟩⊗|𝑗⟩, (3.1) where 𝑎_𝑖𝑗∈ℂ satisfy the normalization ∑^𝑀_𝑖=1∑^𝑁_𝑗=1𝑎_𝑖𝑗𝑎^∗_𝑖𝑗=1.

The concurrence of (3.1) is defined by [27, 108, 109] 𝐶(|𝜓⟩)=2(1−Tr[𝜌²_𝐴]), (3.2) where 𝜌_𝐴=Tr_𝐵[|𝜓⟩⟨𝜓|]. The definition is extended to general mixed states 𝜌=∑_𝑖𝑝_𝑖|𝜓_𝑖⟩⟨𝜓_𝑖| by the convex roof

𝐶(𝜌)=min_{{𝑝𝑖,|𝜓𝑖⟩}𝑖}𝑝_𝑖𝐶(|𝜓_𝑖⟩). (3.3)

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

𝐶(|Ψ⟩)=|⟨Ψ∣Ψ⟩|=2|𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁|, (3.4) where |Ψ⟩=𝜎_𝑦⊗𝜎_𝑦|Ψ^∗⟩, |Ψ^∗⟩ is the complex conjugate of |Ψ⟩, and 𝜎_𝑦 is the Pauli matrix, 𝜎_𝑦=(^0−𝑖_𝑖0).

For a mixed two-qubit quantum state 𝜌, the entanglement of formation 𝐸(𝜌) has a simple relation with the concurrence [53, 101] 𝐸(𝜌)=ℎ(1+1−𝐶(𝜌)²2), (3.5) where ℎ(𝑥)=−𝑥log₂𝑥−(1−𝑥)log₂(1−𝑥), 𝐶(𝜌)=max{𝜆₁−𝜆₂−𝜆₃−𝜆₄,0}, (3.6) where the 𝜆_𝑖s are the eigenvalues, in decreasing order, of the Hermitian matrix √𝜌𝜌√𝜌 and 𝜌=(𝜎_𝑦⊗𝜎_𝑦)𝜌^∗(𝜎_𝑦⊗𝜎_𝑦).

Another entanglement measure called tangle is defined by

𝜏(|𝜓⟩)=𝐶²(|𝜓⟩)=2(1−Tr[𝜌²_𝐴]) (3.7) for a pure state |𝜓⟩. For mixed state 𝜌=∑_𝑖𝑝_𝑖|𝜓_𝑖⟩⟨𝜓_𝑖|, the definition is given by

𝜏(𝜌)=min_{{𝑝𝑖,|𝜓𝑖⟩}𝑖}𝑝_𝑖𝜏(|𝜓_𝑖⟩). (3.8)

For multipartite state |𝜓⟩∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁, dimℋ_𝑖=𝑑_𝑖,𝑖=1,…,𝑁, the concurrence of |𝜓⟩ is defined by [110, 111]

𝐶_𝑁(|𝜓⟩⟨𝜓|)=2^1−𝑁/2(2^𝑁−2)−_𝛼Tr[𝜌²_𝛼], (3.9) 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

|Ψ⟩=^𝑑_{𝑖1,𝑖2,…𝑖𝑁=1}𝑎_{𝑖1,𝑖2,…𝑖𝑁}|𝑖₁,𝑖₂,…𝑖_𝑁⟩,𝑎_{𝑖1,𝑖2,…𝑖𝑁}∈ℂ. (3.10)

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

𝐶^𝑁_𝑑(|Ψ⟩)=𝑑2𝑚(𝑑−1)_𝑝^𝑑_{{𝛼,𝛼′,𝛽,𝛽′}}|𝑎_𝛼𝛽𝑎_{𝛼′𝛽′}−𝑎_𝛼𝛽′𝑎_𝛼′𝛽|², (3.11) where 𝑚=2^𝑁−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=∑_𝑖𝑝_𝑖|𝜓_𝑖⟩⟨𝜓_𝑖| in ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁, the corresponding concurrence is given by the convex roof:

𝐶_𝑁(𝜌)=min_{{𝑝𝑖,|𝜓𝑖}⟩𝑖}𝑝_𝑖𝐶_𝑁(|𝜓_𝑖⟩). (3.12)

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

𝐶(𝜌)≥2𝑀(𝑀−1)[Max(‖𝑇_𝐴(𝜌)‖,‖𝑅(𝜌)‖)−1], (3.13) where 𝑇_𝐴 and 𝑅 stand for partial transpose with respect to subsystem 𝐴 and the realignment, respectively. This bound is further improved based on local uncertainty relations [64]

𝐶(𝜌)≥𝑀+𝑁−2−∑_𝑖Δ²_𝜌(𝐺^𝐴_𝑖⊗𝐼+𝐼⊗𝐺^𝐵_𝑖),2𝑀(𝑀−1) (3.14) where 𝐺^𝐴_𝑖 and 𝐺^𝐵_𝑖 are any set of local orthonormal observables, Δ²_𝜌(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. Let 𝜌 be a bipartite state in ℋ^𝐴_𝑀⊗ℋ^𝐵_𝑁. Then 𝐶(𝜌) satisfies [90] 𝐶(𝜌)≥2‖𝐶‖_𝐾𝐹−(1−Tr[𝜌²_𝐴])−(1−Tr[𝜌²_𝐵]).2𝑀(𝑀−1) (3.15)

Proof. The other orthonormal normalized basis of the local orthonormal observable space can be obtained from 𝐴_𝑖 and 𝐵_𝑖 by unitary transformations 𝑈 and 𝑉: 𝐴_𝑖=∑_𝑙𝑈_𝑖𝑙𝐴_𝑙 and 𝐵_𝑗=∑_𝑚𝑉^∗_𝑗𝑚𝐵_𝑚. Select 𝑈 and 𝑉 so that 𝐶=𝑈^†Λ𝑉 is the singular value decomposition of 𝐶. Then the new observables can be written as 𝐴_𝑖=∑_𝑙𝑈_𝑖𝑙𝐴_𝑙, 𝐵_𝑗=−∑_𝑚𝑉^∗_𝑗𝑚𝐵_𝑚. We have _𝑖Δ²_𝜌(𝐴_𝑖⊗𝐼+𝐼⊗𝐵_𝑖)=_𝑖[Δ²_𝜌𝐴(𝐴_𝑖)+Δ²_𝜌𝐴(𝐵_𝑖)+2(⟨𝐴_𝑖⊗𝐵_𝑖⟩−⟨𝐴_𝑖⟩⟨𝐵_𝑖⟩)]=𝑀−Tr[𝜌²_𝐴]+𝑁−Tr[𝜌²_𝐵]−2_𝑖(𝑈𝐶𝑉^†)_𝑖𝑖=𝑀−Tr[𝜌²_𝐴]+𝑁−Tr[𝜌²_𝐵]−2‖𝐶‖_𝐾𝐹. (3.16) 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]

1𝜌=4(𝐼₉−⁴_𝑖=0|𝜉_𝑖⟩⟨𝜉_𝑖|), (3.17) 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)∑²_𝑖=0|𝑖𝑖⟩, that is,

𝜌^′=(1−𝑥)𝜌+𝑥|𝜓⟩⟨𝜓|, (3.18) 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 bound 𝜏₂ of concurrence for bipartite states has been obtained in [67]. For a bipartite quantum state 𝜌 in 𝐻⊗𝐻, the concurrence 𝐶(𝜌) satisfies 𝜏₂𝑑(𝜌)≡2(𝑑−1)^{𝑑(𝑑−1)/2}_𝑚,𝑛=1𝐶²_𝑚𝑛(𝜌)≤𝐶²(𝜌), (3.19) where 𝐶_𝑚𝑛(𝜌)=max{0,𝜆⁽¹⁾_𝑚𝑛−𝜆⁽²⁾_𝑚𝑛−𝜆⁽³⁾_𝑚𝑛−𝜆⁽⁴⁾_𝑚𝑛} with 𝜆⁽¹⁾_𝑚𝑛,…,𝜆⁽⁴⁾_𝑚𝑛 being the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix 𝜌𝜌_𝑚𝑛 with 𝜌_𝑚𝑛=(𝐿_𝑚⊗𝐿_𝑛)𝜌^∗(𝐿_𝑚⊗𝐿_𝑛), while 𝐿_𝑚 and 𝐿_𝑛 are the generators of 𝑆𝑂(𝑑).

The lower bound 𝜏₂ in (3.19) in fact characterizes all two qubits’ entanglement in a high dimensional bipartite state. One can directly verify that there are at most 4×4=16 nonzero elements in each matrix 𝜌𝜌_𝑚𝑛. These elements constitute a 4×4 matrix 𝜚(𝜎_𝑦⊗𝜎_𝑦)𝜚^∗(𝜎_𝑦⊗𝜎_𝑦), where 𝜎_𝑦 is the Pauli matrix, the matrix 𝜚 is a submatrix of the original 𝜌:

𝜌𝜚=^{𝑖𝑘,𝑖𝑘}𝜌_{𝑖𝑘,𝑖𝑙}𝜌_{𝑖𝑘,𝑗𝑘}𝜌_{𝑖𝑘,𝑗𝑙}𝜌_{𝑖𝑙,𝑖𝑘}𝜌_{𝑖𝑙,𝑖𝑙}𝜌_{𝑖𝑙,𝑗𝑘}𝜌_{𝑖𝑙,𝑗𝑙}𝜌_{𝑗𝑘,𝑖𝑘}𝜌_{𝑗𝑘,𝑖𝑙}𝜌_{𝑗𝑘,𝑗𝑘}𝜌_{𝑗𝑘,𝑗𝑙}𝜌_{𝑗𝑙,𝑖𝑘}𝜌_{𝑗𝑙,𝑖𝑙}𝜌_{𝑗𝑙,𝑗𝑘}𝜌_{𝑗𝑙,𝑗𝑙}, (3.20)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. 𝒞_𝑚𝑛 are just the concurrences of these states (3.20).

The bound 𝜏₂ provides a much clearer structure of entanglement, which not only yields an effective separability criterion and an easy evaluation of entanglement, but also helps one to classify mixed-state entanglement.

(2) Lower Bound of Concurrence for Multipartite States
We first consider tripartite case. A general pure state on 𝐻⊗𝐻⊗𝐻 is of the form |Ψ⟩=^𝑑_{𝑖,𝑗,𝑘=1}𝑎_𝑖𝑗𝑘|𝑖𝑗𝑘⟩,𝑎_𝑖𝑗𝑘∈ℂ,^𝑑_{𝑖,𝑗,𝑘=1}𝑎_𝑖𝑗𝑘𝑎^∗_𝑖𝑗𝑘=1 (3.21) with 𝐶³_𝑑=(|Ψ⟩)𝑑6(𝑑−1)(|𝑎_𝑖𝑗𝑘𝑎_𝑝𝑞𝑚−𝑎_𝑖𝑗𝑚𝑎_𝑝𝑞𝑘|²+|𝑎_𝑖𝑗𝑘𝑎_𝑝𝑞𝑚−𝑎_𝑖𝑞𝑘𝑎_𝑝𝑗𝑚|²+|𝑎_𝑖𝑗𝑘𝑎_𝑝𝑞𝑚−𝑎_𝑝𝑗𝑘𝑎_𝑖𝑞𝑚|²), (3.22) or equivalently 𝐶³_𝑑(|Ψ⟩)=𝑑6(𝑑−1)(3−(Tr[𝜌²₁]+Tr[𝜌²₂]+Tr[𝜌²₃])), (3.23) where 𝜌₁=Tr₂₃[𝜌],𝜌₂=Tr₁₃[𝜌],and𝜌₃=Tr₁₂[𝜌] are the reduced density matrices of 𝜌=|Ψ⟩⟨Ψ|.

Define 𝐶^12|3_𝛼𝛽(|Ψ⟩)=|𝑎_𝑖𝑗𝑘𝑎_𝑝𝑞𝑚−𝑎_𝑖𝑗𝑚𝑎_𝑝𝑞𝑘|, 𝐶^13|2_𝛼𝛽(|Ψ⟩)=|𝑎_𝑖𝑗𝑘𝑎_𝑝𝑞𝑚−𝑎_𝑖𝑞𝑘𝑎_𝑝𝑗𝑚|, and 𝐶^23|1_𝛼𝛽(|Ψ⟩)=|𝑎_𝑖𝑗𝑘𝑎_𝑝𝑞𝑚−𝑎_𝑝𝑗𝑘𝑎_𝑖𝑞𝑚|, where 𝛼 and 𝛽 of 𝐶^12|3_𝛼𝛽 (resp., 𝐶^13|2_𝛼𝛽, resp., 𝐶^23|1_𝛼𝛽) stand for the subindices of 𝑎 associated with the subspaces 1,2, and 3 (resp., 1,3, and 2; resp., 2,3, and 1). Let 𝐿^{𝑖1𝑖2⋯𝑖𝑁} denote the generators of group 𝑆𝑂(𝑑_𝑖1𝑑_𝑖2⋯𝑑_𝑖𝑁) associated to the subsystems 𝑖₁,𝑖₂,…,𝑖_𝑁. Then for a tripartite pure state (3.21), one has 𝐶³_𝑑(|Ψ⟩)=𝑑6(𝑑−1)^{𝑑2(𝑑2−1)/2}_𝛼^{𝑑(𝑑−1)/2}_𝛽[(𝐶^12|3_𝛼𝛽(|Ψ⟩))²+(𝐶^13|2_𝛼𝛽(|Ψ⟩))²+(𝐶^23|1_𝛼𝛽(|Ψ⟩))²]=𝑑6(𝑑−1)_𝛼𝛽[(|⟨Ψ|𝑆^12|3_𝛼𝛽|Ψ^∗⟩|)²+(|⟨Ψ|𝑆^13|2_𝛼𝛽|Ψ^∗⟩|)²+(|⟨Ψ|𝑆^23|2_𝛼𝛽|Ψ^∗⟩|)²], (3.24) where 𝑆^12|3_𝛼𝛽=(𝐿_𝛼¹²⊗𝐿³_𝛽), 𝑆^13|2_𝛼𝛽=(𝐿_𝛼¹³⊗𝐿²_𝛽), and 𝑆^23|1_𝛼𝛽=(𝐿¹_𝛽⊗𝐿_𝛼²³).

Theorem 3.2. For an arbitrary mixed state 𝜌 in 𝐻⊗𝐻⊗𝐻, the concurrence 𝐶(𝜌) satisfies [112] 𝜏₃𝑑(𝜌)≡6(𝑑−1)^{𝑑2(𝑑2−1)/2}_𝛼^{𝑑(𝑑−1)/2}_𝛽[(𝐶^12|3_𝛼𝛽(𝜌))²+(𝐶^13|2_𝛼𝛽(𝜌))²+(𝐶^23|1_𝛼𝛽(𝜌))²]≤𝐶²(𝜌), (3.25) where 𝜏₃(𝜌) is a lower bound of 𝐶(𝜌)𝐶^12|3_𝛼𝛽(𝜌)=max{0,𝜆(1)^12∣3_𝛼𝛽−𝜆(2)^12∣3_𝛼𝛽−𝜆(3)^12∣3_𝛼𝛽−𝜆(4)^12∣3_𝛼𝛽}, (3.26) where 𝜆(1)^12|3_𝛼𝛽,𝜆(2)^12|3_𝛼𝛽,𝜆(3)^12|3_𝛼𝛽,and𝜆(4)^12|3_𝛼𝛽 are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix 𝜌𝜌^12|3_𝛼𝛽 with 𝜌^12|3_𝛼𝛽=𝑆^12|3_𝛼𝛽𝜌^∗𝑆^12|3_𝛼𝛽. 𝐶^13|2_𝛼𝛽(𝜌) and 𝐶^23|1_𝛼𝛽(𝜌) are defined in a similar way to 𝐶^12|3_𝛼𝛽(𝜌).

Proof. Set |𝜉_𝑖⟩=𝑝_𝑖|𝜓_𝑖⟩, 𝑥^𝑖_𝛼𝛽=|⟨𝜉_𝑖|𝑆^12|3_𝛼𝛽|𝜉^∗_𝑖⟩|, 𝑦^𝑖_𝛼𝛽=|⟨𝜉_𝑖|𝑆^13|2_𝛼𝛽|𝜉^∗_𝑖⟩|, and 𝑧^𝑖_𝛼𝛽=|⟨𝜉_𝑖|𝑆^1|23_𝛼𝛽|𝜉^∗_𝑖⟩|. We have, from Minkowski inequality, 𝐶(𝜌)=min_𝑖𝑑6(𝑑−1)_𝛼𝛽[(𝑥^𝑖_𝛼𝛽)²+(𝑦^𝑖_𝛼𝛽)²+(𝑧^𝑖_𝛼𝛽)²]≥min𝑑6(𝑑−1)_𝛼𝛽(_𝑖[(𝑥^𝑖_𝛼𝛽)²+(𝑦^𝑖_𝛼𝛽)²+(𝑧^𝑖_𝛼𝛽)²]^1/2)². (3.27)
Noting that for nonnegative real variables 𝑥_𝛼, 𝑦_𝛼, 𝑧_𝛼 and given that 𝑋=∑^𝑁_𝛼=1𝑥_𝛼, 𝑌=∑^𝑁_𝛼=1𝑌_𝛼, and 𝑍=∑^𝑁_𝛼=1𝑧_𝛼, by using Lagrange multipliers, one obtains that the following inequality holds: ^𝑁_𝛼=1(𝑥²_𝛼+𝑦²_𝛼+𝑧²_𝛼)^1/2≥(𝑋²+𝑌²+𝑍²)^1/2. (3.28) Therefore we have 𝐶(𝜌)≥min𝑑6(𝑑−1)_𝛼𝛽[(_𝑖𝑥^𝑖_𝛼𝛽)²+(_𝑖𝑦^𝑖_𝛼𝛽)²+(_𝑖𝑧^𝑖_𝛼𝛽)²]≥𝑑6(𝑑−1)_𝛼𝛽[(min_𝑖𝑥^𝑖_𝛼𝛽)²+(min_𝑖𝑦^𝑖_𝛼𝛽)²+(min_𝑖𝑧^𝑖_𝛼𝛽)²]. (3.29)
The values of 𝐶^12|3_𝛼𝛽(𝜌)≡min∑_𝑖𝑥^𝑖_𝛼𝛽, 𝐶^13|2_𝛼𝛽(𝜌)≡min∑_𝑖𝑦^𝑖_𝛼𝛽, and 𝐶^23|1_𝛼𝛽(𝜌)≡min∑_𝑖𝑧^𝑖_𝛼𝛽 can be calculated by using the similar procedure in [53]. Here we compute the value of 𝐶^12|3_𝛼𝛽(𝜌) in detail. The values of 𝐶^13|2_𝛼𝛽(𝜌) and 𝐶^23|1_𝛼𝛽(𝜌) can be obtained analogously.
Let 𝜆_𝑖 and |𝜒_𝑖⟩ be eigenvalues and eigenvectors of 𝜌, respectively. Any decomposition of 𝜌 can be obtained from a unitary 𝑑³×𝑑³ matrix 𝑉_𝑖𝑗, |𝜉_𝑗⟩=∑^𝑑3_𝑖=1𝑉^∗_𝑖𝑗(𝜆_𝑖|𝜒_𝑖⟩). Therefore one has ⟨𝜉_𝑖|𝑆^12|3_𝛼𝛽|𝜉^∗_𝑗⟩=(𝑉𝑌_𝛼𝛽𝑉^𝑇)_𝑖𝑗, where the matrix 𝑌_𝛼𝛽 is defined by (𝑌_𝛼𝛽)_𝑖𝑗=⟨𝜒_𝑖|𝑆^12|3_𝛼𝛽|𝜒^∗_𝑗⟩. Namely, 𝐶^12|3_𝛼𝛽(𝜌)=min∑_𝑖|[𝑉𝑌_𝛼𝛽𝑉^𝑇]_𝑖𝑖|, which has an analytical expression [53], that 𝐶^12|3_𝛼𝛽(𝜌)=max{0,𝜆(1)^12|3_𝛼𝛽−∑_𝑗>1𝜆(𝑗)^12|3_𝛼𝛽}, where 𝜆^12|3_𝛼𝛽(𝑘) are the square roots of the eigenvalues of the positive Hermitian matrix 𝑌_𝛼𝛽𝑌^†_𝛼𝛽, or equivalently the non-Hermitian matrix 𝜌𝜌_𝛼𝛽, in decreasing order. Here as the matrix 𝑆^12|3_𝛼𝛽 has 𝑑²−4 rows and 𝑑²−4 columns that are identically zero, the matrix 𝜌𝜌_𝛼𝛽 has a rank not greater than 4, that is, 𝜆^12|3_𝛼𝛽(𝑗)=0 for 𝑗≥5. From (3.29) we have (3.25).

Theorem 3.2 can be directly generalized to arbitrary multipartite case.

Theorem 3.3. For an arbitrary 𝑁-partite state 𝜌∈𝐻⊗𝐻⊗⋯⊗𝐻, the concurrence defined in (4.1) satisfies [112], 𝜏_𝑁𝑑(𝜌)≡2𝑚(𝑑−1)_𝑝𝛼𝛽(𝐶^𝑝_𝛼𝛽(𝜌))²≤𝐶²(𝜌), (3.30) where 𝜏_𝑁(𝜌) is the lower bound of 𝐶(𝜌), ∑_𝑝 stands for the summation over all possible combinations of the indices of 𝛼,𝛽, 𝐶^𝑝_𝛼𝛽(𝜌)=max{0,𝜆(1)^𝑝_𝛼𝛽−𝜆(2)^𝑝_𝛼𝛽−𝜆(3)^𝑝_𝛼𝛽−𝜆(4)^𝑝_𝛼𝛽}, and 𝜆(𝑖)^𝑝_𝛼𝛽, 𝑖=1,2,3,4, are the square roots of the four nonzero eigenvalues, in decreasing order, of the non-Hermitian matrix 𝜌𝜌^𝑝_𝛼𝛽 where 𝜌^𝑝_𝛼𝛽=𝑆^𝑝_𝛼𝛽𝜌^∗𝑆^p_𝛼𝛽.

Lower Bound and Separability
An N-partite quantum state 𝜌 is fully separable if and only if there exist 𝑝_𝑖 with 𝑝_𝑖≥0,∑_𝑖𝑝_𝑖=1, and pure states 𝜌^𝑗_𝑖=|𝜓^𝑗_𝑖⟩⟨𝜓^𝑗_𝑖| such that 𝜌=_𝑖𝑝_𝑖𝜌¹_𝑖⊗𝜌²_𝑖⊗⋯⊗𝜌^𝑁_𝑖. (3.31)

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

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 ‖𝒯^𝑁‖_𝐾𝐹≤1 must hold, where ‖𝒯^𝑁‖_𝐾𝐹=max{‖𝒯^𝑁_𝑛‖_𝐾𝐹}, 𝒯^𝑁_𝑛 is a kind of matrix unfold of 𝑡_{𝛼1𝛼2⋯𝛼𝑁} defined by 𝑡_{𝛼1𝛼2⋯𝛼𝑁}=Tr[𝜌𝜎_𝛼⁽¹⁾₁𝜎_𝛼⁽²⁾₂⋯𝜎_𝛼^(𝑁)_𝑁], and 𝜎_𝛼^(𝑖)_𝑖 stands for the Pauli matrix. Now using the generalized correlation, matrix criterion the entanglement of 𝜌 is detected for 0.3068<𝑝≤1. From Theorem 3.2, we have the lower bound 𝜏₃(𝜌)>0 for 0.2727<𝑝≤1. Therefore the bound (3.71) detects entanglement better than these two criteria in this case. If we replace 𝑊 with GHZ state in 𝜌, then the criterion in [97] detects the entanglement of 𝜌 for 0.35355<𝑝≤1, while 𝜏₃(𝜌) detects, again better, the entanglement for 0.2<𝑝≤1.

Nevertheless for PPT states 𝜌, we have 𝜏₃(𝜌)=0, which can be seen in the following way. A density matrix 𝜌 is called PPT if the partial transposition of 𝜌 over any subsystem(s) is still positive. Let 𝜌^𝑇𝑖 denote the partial transposition with respect to the 𝑖th subsystem. Assume that there is a PPT state 𝜌 with 𝜏(𝜌)>0. Then at least one term in (3.25), say 𝐶_𝛼^12|3_0𝛽0(𝜌), is not zero. Define 𝜌_𝛼0𝛽0=𝐿_𝛼¹²₀⊗𝐿³_𝛽0𝜌(𝐿_𝛼¹²₀⊗𝐿³_𝛽0)^†. By using the PPT property of 𝜌, we have

𝜌^𝑇3_𝛼0𝛽0=𝐿_𝛼¹²₀⊗(𝐿³_𝛽0)^∗𝜌^𝑇3(𝐿_𝛼¹²₀)^†⊗(𝐿³_𝛽0)^𝑇≥0. (3.32) Noting that both 𝐿_𝛼¹²₀ and 𝐿³_𝛽0 are projectors to two-dimensional subsystems, 𝜌_𝛼0𝛽0 can be considered as a 4×4 density matrix, while a PPT 4×4 density matrix 𝜌_𝛼0𝛽0 must be a separable state, which contradicts with C_𝛼^12|3_0𝛽0(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]: 𝜏₂(𝜌₁₂)+𝜏₂(𝜌₁₃)+𝜏₂(𝜌₂₃)≤3𝜏₃(𝜌), (3.33) 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. Since 𝐶²_𝛼𝛽≤(𝜆_𝛼𝛽(1))²≤∑⁴_𝑖=1(𝜆_𝛼𝛽(𝑖))²=Tr[𝜌𝜌_𝛼𝛽] for 𝜌=𝜌₁₂, 𝜌=𝜌₁₃, and 𝜌=𝜌₂₃, we have 𝜏₂(𝜌₁₂)+𝜏₂(𝜌₁₃)+𝜏₂(𝜌₂₃)≤𝑑(2(𝑑−1)^{𝑑(𝑑−1)/2}_𝛼,𝛽=1Tr[𝜌₁₂(𝜌₁₂)_𝛼𝛽]+^{𝑑(𝑑−1)/2}_𝛼,𝛽=1Tr[𝜌₁₃(𝜌₁₃)_𝛼𝛽]+^{𝑑(𝑑−1)/2}_𝛼,𝛽=1Tr[𝜌₂₃(𝜌₂₃)_𝛼𝛽=𝑑])2(𝑑−1)(3−Tr[𝜌²₁]−Tr[𝜌²₂]−Tr[𝜌²₃])=3𝐶²(𝜌)=3𝜏₃(𝜌), (3.34) where we have used the similar analysis in [67, 113] to obtain the equalities ∑_𝛼,𝛽Tr[𝜌₁₂(𝜌₁₂)_𝛼𝛽]=1−Tr[𝜌²₁]−Tr[𝜌²₂]+Tr[𝜌²₃], ∑_𝛼,𝛽Tr[𝜌₁₃(𝜌₁₃)_𝛼𝛽]=1−Tr[𝜌²₁]+Tr[𝜌²₂]−Tr[𝜌²₃], and ∑_𝛼,𝛽Tr[𝜌₂₃(𝜌₂₃)_𝛼𝛽]=1+Tr[𝜌²₁]−Tr[𝜌²₂]−Tr[𝜌²₃]. 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 pure 𝑁-partite quantum state |𝜓⟩∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁, dimℋ_𝑖=𝑑_𝑖, 𝑖=1,…,𝑁, the concurrence of bipartite decomposition between subsystems 12⋯𝑀 and 𝑀+1⋯𝑁 is defined by

𝐶₂(|𝜓⟩)=2(1−Tr[𝜌²_12⋯𝑀]) (3.35) where 𝜌²_12⋯𝑀=Tr_{𝑀+1⋯𝑁}[|𝜓⟩⟨𝜓|] is the reduced density matrix of 𝜌=|𝜓⟩⟨𝜓| by tracing over subsystems 𝑀+1⋯𝑁. On the other hand, the concurrence of |𝜓⟩ is defined by (3.9).

For a mixed multipartite quantum state 𝜌=∑_𝑖𝑝_𝑖|𝜓_𝑖⟩⟨𝜓_𝑖|∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁, the corresponding concurrences of (3.35) and (3.9) are then given by the convex roof

𝐶₂(𝜌)=min_{{𝑝𝑖,|𝜓𝑖}⟩𝑖}𝑝_𝑖𝐶₂(|𝜓_𝑖⟩⟨𝜓_𝑖|), (3.36) and (3.12). We now investigate the relation between these two kinds of concurrences.

Lemma 3.5. For a bipartite density matrix 𝜌∈ℋ_𝐴⊗ℋ_𝐵, one has 1−Tr[𝜌²]≤1−Tr[𝜌²_𝐴]+1−Tr[𝜌²_𝐵], (3.37) where 𝜌_𝐴/𝐵=Tr_𝐵/𝐴[𝜌] are the reduced density matrices of 𝜌.

Proof. Let 𝜌=∑_𝑖𝑗𝜆_𝑖𝑗|𝑖𝑗⟩⟨𝑖𝑗| be the spectral decomposition, where 𝜆_𝑖𝑗≥0,∑_𝑖𝑗𝜆_𝑖𝑗=1. Then 𝜌₁=∑_𝑖𝑗𝜆_𝑖𝑗|𝑖⟩⟨𝑖|, 𝜌₂=∑_𝑖𝑗𝜆_𝑖𝑗|𝑗⟩⟨𝑗|. Therefore 1−Tr[𝜌²_𝐴]+1−Tr[𝜌²_𝐵]−1+Tr[𝜌²]=1−Tr[𝜌²_𝐴]−Tr[𝜌²_𝐵]+Tr[𝜌²]=(_𝑖𝑗𝜆_𝑖𝑗)²−_{𝑖,𝑗,𝑗′}𝜆_𝑖𝑗𝜆_𝑖𝑗′−_{𝑖,𝑖′,𝑗}𝜆_𝑖𝑗𝜆_𝑖′𝑗+_𝑖𝑗𝜆²_𝑖𝑗=(_{𝑖=𝑖′,𝑗=𝑗′}𝜆²_𝑖𝑗+_{𝑖=𝑖′,𝑗≠𝑗′}𝜆_𝑖𝑗𝜆_𝑖𝑗′+_{𝑖≠𝑖′,𝑗=𝑗′}𝜆_𝑖𝑗𝜆_𝑖′𝑗+_{𝑖≠𝑖′,𝑗≠𝑗′}𝜆_𝑖𝑗𝜆_{𝑖′𝑗′})=_{𝑖≠𝑖′,𝑗≠𝑗′}𝜆_𝑖𝑗𝜆_{𝑖′𝑗′}≥0. (3.38)

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

Theorem 3.6. For a multipartite quantum state 𝜌∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁 with 𝑁≥3, the following inequality holds [116]: 𝐶_𝑁(𝜌)≥max2^(3−𝑁)/2𝐶₂(𝜌), (3.39) where the maximum is taken over all kinds of bipartite concurrence.

Proof. Without loss of generality, we suppose that the maximal bipartite concurrence is attained between subsystems 12⋯𝑀 and (𝑀+1)⋯𝑁.
For a pure multipartite state |𝜓⟩∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁, Tr[𝜌²_12⋯𝑀]=Tr[𝜌²_{(𝑀+1)⋯𝑁}]. From (3.37) we have 𝐶²_𝑁(|𝜓⟩⟨𝜓|)=2^2−𝑁((2^𝑁−2)−_𝛼Tr[𝜌²_𝛼])≥2^3−𝑁(𝑁−^𝑁_𝑘=1Tr[𝜌²_𝑘])≥2^3−𝑁(1−Tr[𝜌²_12⋯𝑀]+1−Tr[𝜌²_{(𝑀+1)⋯𝑁}])=2^3−𝑁∗2(1−Tr[𝜌²_12⋯𝑀])=2^3−𝑁𝐶²₂(|𝜓⟩⟨𝜓|), (3.40) that is, 𝐶_𝑁(|𝜓⟩⟨𝜓|)≥2^(3−𝑁)/2𝐶₂(|𝜓⟩⟨𝜓|).
Let 𝜌=∑_𝑖𝑝_𝑖|𝜓_𝑖⟩⟨𝜓_𝑖| attain the minimal decomposition of the multipartite concurrence. One has 𝐶_𝑁(𝜌)=_𝑖𝑝_𝑖𝐶_𝑁(|𝜓_𝑖⟩⟨𝜓_𝑖|)≥2^(3−𝑁)/2_𝑖𝑝_𝑖𝐶₂(|𝜓_𝑖⟩⟨𝜓_𝑖|)≥2^(3−𝑁)/2min_{{𝑝𝑖,|𝜓𝑖}𝑖}𝑝_𝑖𝐶₂(|𝜓_𝑖⟩⟨𝜓_𝑖|)=2^(3−𝑁)/2𝐶₂(𝜌). (3.41)

Corollary 3.7. For a tripartite quantum state 𝜌∈ℋ₁⊗ℋ₂⊗ℋ₃, the following inequality holds: 𝐶₃(𝜌)≥max𝐶₂(𝜌), (3.42) 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:

𝐶₂(𝜌)≥2‖𝐶(𝜌)‖−(1−Tr[𝜌²_𝐴])−(1−Tr[𝜌²_𝐵])2𝑑_𝐴(𝑑_𝐴,𝐶−1)₂(𝜌)≥8𝑑³_𝐴𝑑²_𝐵(𝑑_𝐴−1)(‖𝑇(𝜌)‖−𝑑_𝐴𝑑_𝐵(𝑑_𝐴−1)(𝑑_𝐵−1)2), (3.43)(3.44) where the entries of the matrix 𝐶, 𝐶_𝑖𝑗=⟨𝜆^𝐴_𝑖⊗𝜆^𝐵_𝑗⟩−⟨𝜆^𝐴_𝑖⊗𝐼_𝑑𝐵⟩⟨𝐼_𝑑𝐴⊗𝜆^𝐵_𝑗⟩, 𝑇_𝑖𝑗=𝑑_𝐴𝑑_𝐵/2⟨𝜆^𝐴_𝑖⊗𝜆^𝐵_𝑗⟩, 𝜆_𝑘^𝐴/𝐵 stand for the normalized generator of 𝑆𝑈(𝑑_𝐴/𝑑_𝐵), that is, Tr[𝜆_𝑘^𝐴/𝐵𝜆_𝑙^𝐴/𝐵]=𝛿_𝑘𝑙 and ⟨𝑋⟩=Tr[𝜌𝑋]. It is shown that the lower bounds (3.43) and (3.44) are independent of (3.13).

Now we consider a multipartite quantum state 𝜌∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁 as a bipartite state belonging to ℋ^𝐴⊗ℋ^𝐵 with the dimensions of the subsystems 𝐴 and 𝐵 being 𝑑_𝐴=𝑑_𝑠1𝑑_𝑠2⋯𝑑_𝑠𝑚 and 𝑑_𝐵=𝑑_𝑠𝑚+1𝑑_𝑠𝑚+2⋯𝑑_𝑠𝑁, respectively. By using Corollary 3.7, (3.13), (3.43), and (3.44), one has the following lower bound.

Theorem 3.8. For any N-partite quantum state 𝜌 [116], 𝐶_𝑁(𝜌)≥2^(3−𝑁)/2max{𝐵₁,𝐵₂,𝐵₃}, (3.45) where 𝐵₁=max_{𝑖}2𝑀_𝑖(𝑀_𝑖−1)[max(‖𝒯_𝐴(𝜌^𝑖)‖,‖𝑅(𝜌^𝑖𝐵)‖)−1],₂=max_{𝑖}2‖𝐶(𝜌^𝑖)‖−(1−Tr[(𝜌^𝑖_𝐴)²])−(1−Tr[(𝜌^𝑖_𝐵)²])2𝑀_𝑖(𝑀_𝑖,𝐵−1)₃=max_{𝑖}8𝑀³_𝑖𝑁²_𝑖(𝑀_𝑖−1)(‖𝑇(𝜌^𝑖)‖−𝑀_𝑖𝑁_𝑖(𝑀_𝑖−1)(𝑁_𝑖−1)2), (3.46)𝜌^𝑖’s are all possible bipartite decompositions of 𝜌, and 𝑀_𝑖=min{𝑑_𝑠1𝑑_𝑠2⋯𝑑_𝑠𝑚,𝑑_𝑠𝑚+1𝑑_𝑠m+2⋯𝑑_𝑠𝑁𝑁},_𝑖=max{𝑑_𝑠1𝑑_𝑠2⋯𝑑_𝑠𝑚,𝑑_𝑠𝑚+1𝑑_𝑠𝑚+2⋯𝑑_𝑠𝑁}. (3.47)

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

(4−2^3−𝑁)Tr[𝜌²]−2^2−𝑁_𝛼Tr[𝜌²_𝛼]≤𝐶_𝑁(𝜌)≤2^2−𝑁[(2^𝑁−2)−_𝛼Tr[𝜌²_𝛼]]. (3.48)

In fact one can obtain a more effective upper bound for multipartite concurrence. Let 𝜌=∑_𝑖𝜆_𝑖|𝜓_𝑖⟩⟨𝜓_𝑖|∈ℋ₁⊗ℋ₂⊗⋯⊗ℋ_𝑁, where |𝜓_𝑖⟩’s are the orthogonal pure states and ∑_𝑖𝜆_𝑖=1. We have

𝐶_𝑁(𝜌)=min_{{𝑝𝑖,|𝜑𝑖}⟩𝑖}𝑝_𝑖𝐶_𝑁(|𝜑_𝑖⟩⟨𝜑_𝑖|)≤_𝑖𝜆_𝑖𝐶_𝑁(|𝜓_𝑖⟩⟨𝜓_𝑖|). (3.49) The right side of (3.49) gives a new upper bound of 𝐶_𝑁(𝜌). Since _𝑖𝜆_𝑖𝐶_𝑁(|𝜓_𝑖⟩⟨𝜓_𝑖|)=2^1−𝑁/2_𝑖𝜆_𝑖(2^𝑁−2)−_𝛼Tr[(𝜌^𝑖_𝛼)²]≤2^1−𝑁/2(2^𝑁−2)−_𝛼Tr[_𝑖𝜆_𝑖(𝜌^𝑖_𝛼)²]≤2^1−𝑁/2(2^𝑁−2)−_𝛼Tr[(𝜌_𝛼)²], (3.50) 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

8𝜏(𝜌)≥𝑀𝑁(𝑀+𝑁)(‖𝑇(𝜌)‖²_𝐻𝑆−𝑀𝑁(𝑀−1)(𝑁−1)4), (3.51) 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]:

2(Tr[𝜌²]−Tr[𝜌²_𝐴])≤𝐶(𝜌)≤2(1−Tr[𝜌²_𝐴]). (3.52)

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

Theorem 3.9. For any quantum state 𝜌∈ℋ_𝐴⊗ℋ_𝐵, one has [118] 𝜏(𝜌)≤min{1−Tr[𝜌²_𝐴],1−Tr[𝜌²_𝐵]},𝐶(𝜌)≥8𝑀𝑁(𝑀+𝑁)(‖𝑇(𝜌)‖_𝐻𝑆−𝑀𝑁(𝑀−1)(𝑁−1)2), (3.53)(3.54) where 𝜌_𝐴 is the reduced matrix of 𝜌, and 𝑇(𝜌) is the correlation matrix of 𝜌 defined in (3.44).

Proof. We assume that 1−Tr[𝜌²_𝐴]≤1−Tr[𝜌²_𝐵] for convenience. By the definition of 𝜏, we have that for a pure state |𝜓⟩,𝜏(|𝜓⟩)=2(1−Tr[(𝜌_𝐴^|𝜓⟩)²]). Let 𝜌=∑_𝑖𝑝_𝑖𝜌_𝑖 be the optimal decomposition such that 𝜏(𝜌)=∑_𝑖𝑝_𝑖𝜏(𝜌_𝑖). We get 𝜏(𝜌)=_𝑖𝑝_𝑖𝜏(𝜌_𝑖)=_𝑖𝑝_𝑖2[1−Tr[(𝜌^|𝜓𝑖⟩_𝐴)²]]=2[1−Tr[_𝑖𝑝_𝑖(𝜌^|𝜓𝑖⟩_𝐴)²]]≤2[1−Tr[𝜌²_𝐴]]. (3.55)
Note that, for pure state |𝜓⟩∈ℋ_𝐴⊗ℋ_𝐵 [68], 𝐶(|𝜓⟩)=8𝑀𝑁(𝑀+𝑁)(‖𝑇(|𝜓⟩)‖²−𝑀𝑁(𝑀−1)(𝑁−1)4). (3.56) Using the inequality √𝑎−𝑏≥√𝑎−√𝑏 for any 𝑎≥𝑏, we get 𝐶(|𝜓⟩)≥8𝑀𝑁(𝑀+𝑁)(‖𝑇(|𝜓⟩)‖_𝐻𝑆−𝑀𝑁(𝑀−1)(𝑁−1)2). (3.57) Now let 𝜌=∑_𝑖𝑝_𝑖𝜌_𝑖 be the optimal decomposition such that 𝐶(𝜌)=∑_𝑖𝑝_𝑖𝐶(𝜌_𝑖). We get 𝐶(𝜌)=_𝑖𝑝_𝑖𝐶(𝜌_𝑖)≥_𝑖𝑝_𝑖8𝑀𝑁(𝑀+𝑁)(‖𝑇(𝜌_𝑖)‖_𝐻𝑆−𝑀𝑁(𝑀−1)(𝑁−1)2)=8𝑀𝑁(𝑀+𝑁)(_𝑖𝑝_𝑖‖𝑇(𝜌_𝑖)‖_𝐻𝑆−𝑀𝑁(𝑀−1)(𝑁−1)2)≥8𝑀𝑁(𝑀+𝑁)(‖𝑇(𝜌)‖_𝐻𝑆−𝑀𝑁(𝑀−1)(𝑁−1)2), (3.58) which ends the proof.

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

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

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

Let 𝜌=∑_{𝑖𝑗𝑘𝑙}𝜌_{𝑖𝑗,𝑘𝑙}|𝑖𝑗⟩⟨𝑘𝑙|∈ℋ_𝐴⊗ℋ_𝐵 and 𝜎=∑_{𝑖′𝑗′𝑘′𝑙′}𝜎_{𝑖′𝑗′,𝑘′𝑙′}|𝑖^′𝑗^′⟩⟨𝑘^′𝑙^′|∈ℋ_𝐴′⊗ℋ_𝐵′ be two quantum states shared by subsystems 𝐴𝐴^′ and 𝐵𝐵^′. We use 𝜌⊗𝜎=∑_{𝑖𝑗𝑘𝑙,𝑖′𝑗′𝑘′𝑙′}𝜌_{𝑖𝑗,𝑘𝑙}𝜎_{𝑖′𝑗′,𝑘′𝑙′}|𝑖𝑖^′⟩_𝐴𝐴′⟨𝑘𝑘^′|⊗|𝑗𝑗^′⟩_𝐵𝐵′⟨𝑙𝑙^′| to denote the state of the whole system.

Lemma 3.10. For pure states |𝜓⟩∈ℋ_𝐴⊗ℋ_𝐵 and |𝜑⟩∈ℋ_𝐴′⊗ℋ_𝐵′, the inequalities 𝐶(|𝜓⟩⊗|𝜑⟩)≥max{𝐶(|𝜓⟩),𝐶(|𝜑⟩)},𝜏(|𝜓⟩⊗|𝜑⟩)≥max{𝜏(|𝜓⟩),𝜏(|𝜑⟩)} (3.59)(3.60) 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 that 𝐶(|𝜓⟩)≥𝐶(|𝜑⟩). First note that 𝜌^{|𝜓⟩⊗|𝜑⟩}_𝐴𝐴′=𝜌_𝐴^|𝜓⟩⊗𝜌_𝐴^|𝜑⟩_′. (3.61) Let 𝜌_𝐴^|𝜓⟩=∑_𝑖𝜆_𝑖|𝑖⟩⟨𝑖| and 𝜌_𝐴^|𝜑⟩_′=∑_𝑗𝜋_𝑗|𝑗⟩⟨𝑗| be the spectral decomposition of 𝜌_𝐴^|𝜓⟩ and 𝜌_𝐴^|𝜑⟩_′, with ∑_𝑖𝜆_𝑖=1 and ∑_𝑗𝜋_𝑗=1, respectively. By using (3.61) one obtains that Tr[(𝜌^{|𝜓⟩⊗|𝜑⟩}_𝐴𝐴′)²]=𝜆_𝑖𝜋_𝑗𝜆_𝑖′𝜋_𝑗′|𝑖𝑗⟩⟨𝑖𝑗∣𝑖^′𝑗^′⟩⟨𝑖^′𝑗^′|=𝜆²_𝑖𝜋²_𝑗, (3.62) while Tr[(𝜌_𝐴^|𝜓⟩)²]=_𝑖𝜆²_𝑖. (3.63)
Now using the definition of concurrence and the normalization conditions of 𝜆_𝑖 and 𝜋_𝑗, one immediately gets 𝐶(|𝜓⟩⊗|𝜑⟩)=2(1−Tr[(𝜌^{|𝜓⟩⊗|𝜑⟩}_𝐴𝐴′)²])≥2(1−Tr[(𝜌_𝐴^|𝜓⟩)²])=𝐶(|𝜓⟩). (3.64) If one of {|𝜓⟩,|𝜑⟩} is separable, say |𝜑⟩, then the rank of 𝜌_𝐴^|𝜑⟩_′ must be one, which means that there is only one item in the spectral decomposition in 𝜌_𝐴^|𝜑⟩_′. Using the normalization condition of 𝜋_𝑗, we obtain Tr[(𝜌^{|𝜓⟩⊗|𝜑⟩}_𝐴𝐴′)²]=Tr[(𝜌_𝐴^|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 𝜌_𝐴^|𝜓⟩ and 𝜌_𝐴^|𝜑⟩_′, which means that Tr[(𝜌^{|𝜓⟩⊗|𝜑⟩}_𝐴𝐴′)²] is strictly larger than Tr[(𝜌_𝐴^|𝜓⟩)²].
The inequality (3.60) also holds because, for pure quantum state 𝜌, 𝜏(𝜌)=𝐶²(𝜌).

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

Theorem 3.11. For any quantum states 𝜌∈ℋ_𝐴⊗ℋ_𝐵 and 𝜎∈ℋ_𝐴′⊗ℋ_𝐵′, the inequalities 𝐶(𝜌⊗𝜎)≥max{𝐶(𝜌),𝐶(𝜎)},𝜏(𝜌⊗𝜎)≥max{𝜏(𝜌),𝜏(𝜎)} (3.65) 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 that 𝐶(𝜌)≥𝐶(𝜎) for convenience. Let 𝜌=∑_𝑖𝑝_𝑖𝜌_𝑖 and 𝜎=∑_𝑗𝑞_𝑗𝜎_𝑗 be the optimal decomposition such that 𝐶(𝜌⊗𝜎)=∑_𝑖𝑝_𝑖𝑞_𝑗𝐶(𝜌_𝑖⊗𝜎_𝑗). By using the inequality obtained in Lemma 3.10, we have 𝐶(𝜌⊗𝜎)=_𝑖𝑝_𝑖𝑞_𝑗𝐶(𝜌_𝑖⊗𝜎_𝑗)≥_𝑖𝑝_𝑖𝑞_𝑗𝐶(𝜌_𝑖)=_𝑖𝑝_𝑖𝐶(𝜌_𝑖)≥𝐶(𝜌). (3.66)Case 1. Now let one of {𝜌,𝜎} be separable, say 𝜎, with ensemble representation 𝜎=∑_𝑗𝑞_𝑗𝜎_𝑗, where ∑_𝑗𝑞_𝑗=1, and 𝜎_𝑗 is the density matrix of separable pure state. Suppose that 𝜌=∑_𝑖𝑝_𝑖𝜌_𝑖 is the optimal decomposition of 𝜌 such that 𝐶(𝜌)=∑_𝑖𝑝_𝑖𝐶(𝜌_𝑖). Using Lemma 3.10, we have 𝐶(𝜌⊗𝜎)≤_𝑖𝑝_𝑖𝑞_𝑗𝐶(𝜌_𝑖⊗𝜎_𝑗)=_𝑖𝑝_𝑖𝑞_𝑗𝐶(𝜌_𝑖)=_𝑖𝑝_𝑖𝐶(𝜌_𝑖)=𝐶(𝜌). (3.67) The inequalities (3.66) and (3.67) show that if 𝜎 is separable, then 𝐶(𝜌⊗𝜎)=𝐶(𝜌).Case 2. If both 𝜌 and 𝜎 are inseparable, that is, there is at least one pure state in the ensemble decomposition of 𝜌 (and 𝜎, resp.), using Lemma 3.10, then we have 𝐶(𝜌⊗𝜎)=_𝑖𝑝_𝑖𝑞_𝑗𝐶(𝜌_𝑖⊗𝜎_𝑗)>_𝑖𝑝_𝑖𝑞_𝑗𝐶(𝜌_𝑖)=_𝑖𝑝_𝑖𝐶(𝜌_𝑖)≥𝐶(𝜌). (3.68)
The inequality for tangle 𝜏 can be proved in a similar way.

Remark 3.12. In [119] it is shown that any entangled state 𝜌 can enhance the teleportation power of another state 𝜎. This holds even if the state 𝜌 is bound entangled. But if 𝜌 is bound entangled, then the corresponding 𝜎 must be free entangled (distillable). By Theorem 3.11, we can see that even if two entangled quantum states 𝜌 and 𝜎 are bound entangled, their concurrence and tangle are strictly larger than those of one state.

3.3. Subadditivity of Concurrence and Tangle

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

Theorem 3.13. Let 𝜌 and 𝜎 be quantum states in ℋ_𝐴⊗ℋ_𝐵, the one has [118] 𝐶(𝜌⊗𝜎)≤𝐶(𝜌)+𝐶(𝜎),𝜏(𝜌⊗𝜎)≤𝜏(𝜌)+𝜏(𝜎). (3.69)

Proof. We first prove that the theorem holds for pure states, that is, for |𝜓⟩ and |𝜙⟩ in ℋ_𝐴⊗ℋ_𝐵: 𝐶(|𝜓⟩⊗|𝜙⟩)≤𝐶(|𝜓⟩)+𝐶(|𝜙⟩),𝜏(|𝜓⟩⊗|𝜙⟩)≤𝜏(|𝜓⟩)+𝜏(|𝜙⟩). (3.70) Assume that 𝜌_𝐴^|𝜓⟩=∑_𝑖𝜆_𝑖|𝑖⟩⟨𝑖| and 𝜌_𝐴^|𝜙⟩=∑_𝑗𝜋_𝑗|𝑗⟩⟨𝑗| are the spectral decomposition of the reduced matrices 𝜌_𝐴^|𝜓⟩ and 𝜌_𝐴^|𝜙⟩. One has 12[𝐶(|𝜓⟩)+𝐶(|𝜙⟩)]²≥1−Tr[(𝜌_𝐴^|𝜓⟩)²]+1−Tr[(𝜌_𝐴^|𝜙⟩)²]=1−_𝑖𝜆²_𝑖+1−_𝑗𝜋²_𝑗≥1−_𝑖𝑗𝜆²_𝑖𝜋²_𝑗=12𝐶²(|𝜓⟩⊗|𝜙⟩). (3.71)
Now we prove that (3.69) holds for any mixed-quantum states 𝜌 and 𝜎. Let 𝜌=∑_𝑖𝑝_𝑖𝜌_𝑖 and 𝜎=∑_𝑗𝑞_𝑗𝜎_𝑗 be the optimal decomposition such that 𝐶(𝜌)=∑_𝑖𝑝_𝑖𝐶(𝜌_𝑖) and 𝐶(𝜎)=∑_𝑗𝑞_𝑗𝐶(𝜎_𝑗). We have 𝐶(𝜌)+𝐶(𝜎)=_𝑖𝑗𝑝_𝑖𝑞_𝑗[𝐶(𝜌_𝑖)+𝐶(𝜎_𝑗)]≥_𝑖𝑗𝑝_𝑖𝑞_𝑗𝐶(𝜌_𝑖⊗𝜎_𝑗)≥𝐶(𝜌⊗𝜎). (3.72)
The inequality for 𝜏 can be derived in a similar way.

4. Fidelity of Teleportation and Distillation of Entanglement

Quantum teleportation is an important subject in quantum information processing. In terms of a classical communication channel and a quantum resource (a nonlocal entangled state like an EPR pair of particles), the teleportation protocol gives ways to transmit an unknown quantum state from a sender traditionally named “Alice" to a receiver “Bob" who are spatially separated. These teleportation processes can be viewed as quantum channels. The nature of a quantum channel is determined by the particular protocol and the state used as a teleportation resource. The standard teleportation protocol 𝑇₀ proposed by Bennett et al. in 1993 uses Bell measurements and Pauli rotations. When the maximally entangled pure state |𝜙⟩=(1/√𝑛)∑^𝑛−1_𝑖=0|𝑖𝑖⟩ is used as the quantum resource, it provides an ideal noiseless quantum channel Λ_𝑇^{(|𝜙⟩⟨𝜙|)}₀(𝜌)=𝜌. However in realistic situation, instead of the pure maximally entangled states, Alice and Bob usually share a mixed entangled state due to the decoherence. Teleportation using mixed state as an entangled resource is, in general, equivalent to having a noisy quantum channel. An explicit expression for the output state of the quantum channel associated with the standard teleportation protocol 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

Let ℋ be a 𝑑-dimensional complex vector space with computational basis |𝑖⟩, 𝑖=1,…,𝑑. The fully entangled fraction (FEF) of a density matrix 𝜌∈ℋ⊗ℋ is defined by

ℱ(𝜌)=max_𝑈⟨𝜓₊|(𝐼⊗𝑈^†)𝜌(𝐼⊗𝑈)|𝜓₊⟩ (4.1) under all unitary transformations 𝑈, where |𝜓₊⟩=(1/√𝑑)∑^𝑑_𝑖=1|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 𝑓_max(𝜌)=𝑑ℱ(𝜌)+1𝑑+1,𝑑+1 (4.2) which solely depends on the FEF of the entangled resource state 𝜌.

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

Let 𝜆_𝑖, 𝑖=1,…,𝑑²−1, be the generators of the 𝑆𝑈(𝑑) algebra. A bipartite state 𝜌∈ℋ⊗ℋ can be expressed as 1𝜌=𝑑²1𝐼⊗𝐼+𝑑^𝑑2−1_𝑖=1𝑟_𝑖(𝜌)𝜆_𝑖1⊗𝐼+𝑑^𝑑2−1_𝑗=1𝑠_𝑗(𝜌)𝐼⊗𝜆_𝑗+^𝑑2−1_𝑖,𝑗=1𝑚_𝑖𝑗(𝜌)𝜆_𝑖⊗𝜆_𝑗, (4.3) where 𝑟_𝑖(𝜌)=(1/2)Tr[𝜌𝜆_𝑖(1)⊗𝐼], 𝑠_𝑗(𝜌)=(1/2)Tr[𝜌𝐼⊗𝜆_𝑗(2)], and 𝑚_𝑖𝑗(𝜌)=(1/4)Tr[𝜌𝜆_𝑖(1)⊗𝜆_𝑗(2)]. Let 𝑀(𝜌) denote the correlation matrix with entries 𝑚_𝑖𝑗(𝜌).

Theorem 4.1. For any 𝜌∈ℋ⊗ℋ, the fully entangled fraction ℱ(𝜌) satisfies [123] 1ℱ(𝜌)≤𝑑²+4‖𝑀^𝑇(𝜌)𝑀(𝑃₊)‖_𝐾𝐹, (4.4) where 𝑀^𝑇 stands for the transpose of 𝑀 and ‖𝑀‖_𝐾𝐹=Tr[𝑀𝑀^†] is the Ky Fan norm of 𝑀.

Proof. First, we note that 𝑃₊=1𝑑²𝐼⊗𝐼+^𝑑2−1_𝑖,𝑗=1𝑚_𝑖𝑗(𝑃₊)𝜆_𝑖⊗𝜆_𝑗, (4.5) where 𝑚_𝑖𝑗(𝑃₊)=(1/4)Tr[𝑃₊𝜆_𝑖⊗𝜆_𝑗]. By definition (4.1), one obtains ℱ(𝜌)=max_𝑈⟨𝜓₊|(𝐼⊗𝑈^†)𝜌(𝐼⊗𝑈)|𝜓₊⟩=max_𝑈Tr[𝜌(𝐼⊗𝑈)𝑃₊(𝐼⊗𝑈^†)]=max_𝑈{1𝑑²Tr[𝜌]+^𝑑2−1_𝑖,𝑗=1𝑚_𝑖𝑗(𝑃₊)Tr[𝜌𝜆_𝑖⊗𝑈𝜆_𝑗𝑈^†]}. (4.6)
Since 𝑈𝜆_𝑖𝑈^† is a traceless Hermitian operator, it can be expanded according to the 𝑆𝑈(𝑑) generators as 𝑈𝜆_𝑖𝑈^†=^𝑑2−1_𝑗=112Tr[𝑈𝜆_𝑖𝑈^†𝜆_𝑗]𝜆_𝑗≡^𝑑2−1_𝑗=1𝑂_𝑖𝑗𝜆_𝑗. (4.7) Entries 𝑂_𝑖𝑗 define a real (𝑑²−1)×(𝑑²−1) matrix 𝑂. From the completeness relation of 𝑆𝑈(𝑑) generators ^𝑑2−1_𝑗=1(𝜆_𝑗)_𝑘𝑖(𝜆_𝑗)_𝑚𝑛=2𝛿_𝑖𝑚𝛿_𝑘𝑛−2𝑑𝛿_𝑘𝑖𝛿_𝑚𝑛, (4.8) one can show that 𝑂 is an orthonormal matrix. Using (4.7), we have 1ℱ(𝜌)≤𝑑²+max_{𝑂𝑖,𝑗,𝑘}𝑚_𝑖𝑗(𝑃₊)𝑂_𝑗𝑘Tr[𝜌𝜆_𝑖⊗𝜆_𝑘]=1𝑑²+4max_{𝑂𝑖,𝑗,𝑘}𝑚_𝑖𝑗(𝑃₊)𝑂_𝑗𝑘𝑚_𝑖𝑘1(𝜌)=𝑑²+4max_𝑂Tr[𝑀(𝜌)^𝑇𝑀(𝑃₊=1)𝑂]𝑑²+4‖𝑀(𝜌)^𝑇𝑀(𝑃₊)‖_𝐾𝐹. (4.9)

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

Corollary 4.2. For two-qubit system, one has 1ℱ(𝜌)=4+4‖𝑀(𝜌)^𝑇𝑀(𝑃₊)‖_𝐾𝐹, (4.10) that is, the upper bound derived in Theorem 4.1 is exactly the FEF.

Proof. We have shown in (4.7) that, given an arbitrary unitary 𝑈, one can always obtain an orthonormal matrix 𝑂. Now we show that in two-qubit case, for any 3×3 orthonormal matrix 𝑂, there always exits 2×2 unitary matrix 𝑈 such that (4.7) holds.
For any vector 𝐭={𝑡₁,𝑡₂,𝑡₃} with unit norm, define an operator 𝑋≡∑³_𝑖=1𝑡_𝑖𝜎_𝑖, where 𝜎_𝑖’s are Pauli matrices. Given an orthonormal matrix 𝑂, one obtains a new operator 𝑋^′≡∑³_𝑖=1𝑡^′_𝑖𝜎_𝑖=∑³_𝑖,𝑗=1𝑂_𝑖𝑗𝑡_𝑗𝜎_𝑖.
𝑋 and 𝑋^′ are both Hermitian traceless matrices. Their eigenvalues are given by the norms of the vectors 𝐭 and 𝐭^′={𝑡^′₁,𝑡^′₂,𝑡^′₃}, respectively. As the norms are invariant under orthonormal transformations 𝑂, they have the same eigenvalues: ±𝑡²₁+𝑡²₂+𝑡²₃. 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 𝜆_𝑖, 𝑖=1,…,𝑑²−1. As an example, for 𝑑=3, if we choose 𝜆₁=1000−10000,𝜆₂=1√30100√30200−√3,𝜆₃=010100000,𝜆₄=,𝜆001000100₅=000001010,𝜆₆=0𝑖0−𝑖00000,𝜆₇=00𝑖000−𝑖00,𝜆₈=,00000𝑖0−𝑖0 (4.11) then we have 𝑀(𝑃₊1)=Diag{6,16,16,16,161,−61,−61,−6}. (4.12) Nevertheless the FEF and its upper bound do not depend on the choice of the 𝑆𝑈(𝑑) generators.

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

Let ℎ and 𝑔 be 𝑛×𝑛 matrices such that ℎ|𝑗⟩=|(𝑗+1)mod𝑛⟩, 𝑔|𝑗⟩=𝜔^𝑗|𝑗⟩, with 𝜔=exp{−2𝑖𝜋/𝑛}. We can introduce 𝑛² linear-independent 𝑛×𝑛 matrices 𝑈_𝑠𝑡=ℎ^𝑡𝑔^𝑠, which satisfy

𝑈_𝑠𝑡𝑈_{𝑠′𝑡′}=𝜔^{𝑠𝑡′−𝑡𝑠′}𝑈_{𝑠′𝑡′}𝑈_𝑠𝑡,Tr[𝑈_𝑠𝑡]=𝑛𝛿_𝑠0𝛿_𝑡0. (4.13) One can also check that {𝑈_𝑠𝑡} satisfy the condition of bases of the unitary operators in the sense of [124], that is,

Tr[𝑈_𝑠𝑡𝑈⁺_{𝑠′𝑡′}]=𝑛𝛿_𝑡𝑡′𝛿_𝑠𝑠′,𝑈_𝑠𝑡𝑈⁺_𝑠𝑡=𝐼_𝑛×𝑛, (4.14) where 𝐼_𝑛×𝑛 is the 𝑛×𝑛 identity matrix. {𝑈_𝑠𝑡} form a complete basis of 𝑛×𝑛 matrices, namely, for any 𝑛×𝑛 matrix 𝑊, 𝑊 can be expressed as

1𝑊=𝑛_𝑠,𝑡Tr[𝑈⁺_𝑠𝑡𝑊]𝑈_𝑠𝑡. (4.15)

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

|Φ_𝑠𝑡⟩=(𝐼⊗𝑈^∗_𝑠𝑡)|𝜓₊1⟩=√𝑑_𝑖,𝑗(𝑈_𝑠𝑡)^∗_𝑖𝑗|𝑖𝑗⟩,|Φ₀₀⟩=|𝜓₊⟩, (4.16) where |Φ_𝑠𝑡⟩ are all maximally entangled states and form a complete orthogonal normalized basis of ℋ_𝑑⊗ℋ_𝑑.

Theorem 4.4. For any quantum state 𝜌∈ℋ_𝑑⊗ℋ_𝑑, the fully entangled fraction defined in (4.1) fulfills the following inequality: ℱ(𝜌)≤max_𝑗{𝜆_𝑗}, (4.17) where 𝜆_𝑗s are the eigenvalues of the real part of matrix 𝑀=(^𝑇𝑖𝑇_{−𝑖𝑇𝑇}), 𝑇 is a 𝑑²×𝑑² matrix with entries 𝑇_𝑛,𝑚=⟨Φ_𝑛|𝜌|Φ_𝑚⟩, and Φ_𝑗 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=∑^𝑑2_𝑘=1𝑧_𝑘𝑈_𝑘, where 𝑧_𝑘=(1/𝑑)Tr[𝑈^†_𝑘𝑈]. Define 𝑥_𝑙=Re[𝑧_𝑙],1≤𝑙≤𝑑²,Im[𝑧_𝑙],𝑑²<𝑙≤2𝑑²,𝑈^′_𝑙=𝑈_𝑙,1≤𝑙≤𝑑²,𝑖∗𝑈_𝑙,𝑑²<𝑙≤2𝑑². (4.18) Then the unitary matrix 𝑈 can be rewritten as 𝑈=∑^2𝑑2_𝑘=1𝑧_𝑘𝑈^′_𝑘. The necessary condition for the unitary property of 𝑈 implies that ∑_𝑘𝑥²_𝑘=1. Thus we have 𝐹(𝜌)≡⟨𝜓₊|(𝐼⊗𝑈^†)𝜌(𝐼⊗𝑈)|𝜓₊⟩=^2𝑑2_𝑚,𝑛=1𝑥_𝑚𝑥_𝑛𝑀_𝑚𝑛, (4.19) where 𝑀_𝑚𝑛 is defined in the theorem. One can deduce that 𝑀^∗_𝑚𝑛=𝑀_𝑛𝑚 (4.20) from the hermiticity of 𝜌.
Taking into account the constraint with an undetermined Lagrange multiplier 𝜆, we have 𝜕𝜕𝑥_𝑘{𝐹(𝜌)+𝜆(_𝑙𝑥²_𝑙−1)}=0. (4.21) Accounting to (4.20) we have the eigenvalue equation ^2𝑑2_𝑛=1Re[𝑀_𝑘,𝑛]𝑥_𝑛=−𝜆𝑥_𝑘. (4.22)
Inserting (4.22) into (4.19) results in ℱ(𝜌)=max_𝑈𝐹≤max_𝑗{𝜂_𝑗}, (4.23) where 𝜂_𝑗=−𝜆_𝑗 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 1𝜌(𝑎)=8𝑎+1𝑎000𝑎000𝑎0𝑎000000000𝑎000000000𝑎00000𝑎000𝑎000𝑎00000𝑎0000000001+𝑎20√1−𝑎²2√0000000𝑎0𝑎000𝑎01−𝑎²201+𝑎2. (4.24) One can easily compare the upper bound obtained in (4.17) and that in (4.4). From Figure 2 we see that, for 0≤𝑎<0.572, the upper bound in (4.17) is larger than that in (4.4). But for 0.572<𝑎<1 the upper bound in (4.17) is always lower than that in (4.4), which means that the upper bound (4.17) is tighter than (4.4).

Figure 2 

Upper bound of ℱ(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 state 𝜌∈ℋ_𝑑⊗ℋ_𝑑, the following inequality holds [125]: 1ℱ(𝜌)≤𝑑(Tr[√𝜌_𝐴])², (4.25) where 𝜌_𝐴 is the reduced matrix of 𝜌.

Proof. Note that in [77] the authors have obtained the FEF for pure state |𝜓⟩ as 1ℱ(|𝜓⟩)=𝑑(Tr[𝜌_𝐴^|𝜓⟩])², (4.26) where 𝜌_𝐴^|𝜓⟩ is the reduced matrix of |𝜓⟩⟨𝜓|.
For mixed state 𝜌=∑_𝑖𝑝_𝑖𝜌^𝑖, we have ℱ(𝜌)=max_𝑈⟨𝜓₊|(𝐼⊗𝑈^†)𝜌(𝐼⊗𝑈)|𝜓₊⟩≤_𝑖𝑝_𝑖max_𝑈⟨𝜓₊|(𝐼⊗𝑈^†)𝜌^𝑖(𝐼⊗𝑈)|𝜓₊⟩=1𝑑_𝑖𝑝_𝑖(Tr[𝜌^𝑖_𝐴])²=1𝑑_𝑖(Tr[𝑝_𝑖𝜌^𝑖_𝐴])². (4.27) Let 𝜆_𝑖𝑗 be the real and nonnegative eigenvalues of the matrix 𝑝_𝑖𝜌^𝑖_𝐴. Recall that for any function 𝐹=∑_𝑖(∑_𝑗𝑥²_𝑖𝑗)^1/2 subjected to the constraints 𝑧_𝑗=∑_𝑖𝑥_𝑖𝑗 with 𝑥_𝑖𝑗 being real and nonnegative, the inequality ∑_𝑗𝑧²_𝑗≤𝐹² holds, from which it follows that 1ℱ(𝜌)≤𝑑_𝑖(_𝑗𝜆_𝑖𝑗)²≤1𝑑(_𝑗𝑖𝜆_𝑖𝑗)²=1𝑑(Tr[√𝜌_𝐴])², (4.28) which ends the proof.

4.2. Fully Entangled Fraction and Concurrence

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

Theorem 4.7. For any bipartite quantum state 𝜌∈ℋ_𝑑⊗ℋ_𝑑, one has [118] 𝐶(𝜌)≥2𝑑1𝑑−1[ℱ(𝜌)−𝑑]. (4.29)

Proof. In [126], the authors show that, for any pure state |𝜓⟩∈ℋ_𝐴⊗ℋ_𝐵, the following inequality holds: 𝐶(|𝜓⟩)≥2𝑑𝑑−1(max_{|𝜙⟩∈𝜀}|⟨𝜓∣𝜙⟩|²−1𝑑), (4.30) where 𝜀 denotes the set of 𝑑×𝑑 dimensional maximally entangled states.
Let 𝜌=∑_𝑖𝑝_𝑖|𝜙_𝑖⟩⟨𝜙_𝑖| be the optimal decomposition such that 𝐶(𝜌)=∑_𝑖𝑝_𝑖𝐶(|𝜓_𝑖⟩). We have 𝐶(𝜌)=_𝑖𝑝_𝑖𝐶(|𝜓_𝑖⟩)≥_𝑖𝑝_𝑖2𝑑𝑑−1(max_{|𝜙⟩∈𝜀}|⟨𝜓_𝑖∣𝜙⟩|²−1d)≥2𝑑𝑑−1(max_{|𝜙⟩∈𝜀𝑖}𝑝_𝑖|⟨𝜓_𝑖∣𝜙⟩|²−1𝑑)=2𝑑𝑑−1(max_{|𝜙⟩∈𝜀}1⟨𝜙|𝜌|𝜙⟩−𝑑)=2𝑑1𝑑−1(ℱ(𝜌)−𝑑), (4.31) which ends the proof.

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

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

ℱ_𝑁(𝜌_𝐴𝐵)=max{2ℱ(𝜌_𝐴𝐵)−1,0}. (4.32) Let 𝐶(𝜌_{𝐴𝐵∣𝐶}) denote the concurrence between subsystems 𝐴𝐵 and 𝐶.

Theorem 4.8. For any triqubit state 𝜌_𝐴𝐵𝐶, ℱ_𝑁(𝜌_𝐴𝐵) satisfies [123] ℱ_𝑁(𝜌_𝐴𝐵)≤1−𝐶²(𝜌_{𝐴𝐵∣𝐶}). (4.33)

Proof. We first consider the case that 𝜌_A𝐵𝐶 is pure, 𝜌_𝐴𝐵𝐶=|𝜓⟩_𝐴𝐵𝐶⟨𝜓|. By using the Schmidt decomposition between qubits 𝐴,𝐵, and 𝐶, |𝜓⟩_𝐴𝐵𝐶 can be written as |𝜓⟩_{𝐴𝐵|𝐶}=²_𝑖=1𝜂_𝑖|𝑖_𝐴𝐵⟩|𝑖_𝐶⟩,𝜂²₁+𝜂²₂=1,𝜂₁≥𝜂₂, (4.34) for some orthonormalized bases |𝑖_𝐴𝐵⟩, |𝑖_𝐶⟩ of subsystems 𝐴𝐵, 𝐶, respectively. The reduced density matrix 𝜌_𝐴𝐵 has the form 𝜌_𝐴𝐵=Tr_𝐶[𝜌_𝐴𝐵𝐶]=²_𝑖=1𝜂²_𝑖|𝑖_𝐴𝐵⟩⟨𝑖_𝐴𝐵|=𝑈^𝑇Λ𝑈^∗, (4.35) where Λ is a 4×4 diagonal matrix with diagonal elements {𝜂²₁,𝜂²₂,0,0}, 𝑈 is a unitary matrix, and 𝑈^∗ denotes the conjugation of 𝑈.
The FEF of the two-qubit state 𝜌_𝐴𝐵 can be calculated by using formula (4.10) or the one in [122]. Let 1𝑀=√2100𝑖0𝑖−100𝑖10100−𝑖 (4.36) be the 4×4 matrix constituted by the four Bell bases. The FEF of 𝜌_𝐴𝐵 can be written as ℱ(𝜌_𝐴𝐵)=𝜂_max(Re{𝑀^†𝜌_𝐴𝐵1𝑀})=2𝜂_max(𝑀^†𝜌_𝐴𝐵𝑀+𝑀^𝑇𝜌^∗_𝐴𝐵𝑀^∗)≤12[𝜂_max(𝑀^†𝑈^𝑇Λ𝑈^∗𝑀)+𝜂_max(𝑀^𝑇𝑈^†Λ𝑈𝑀^∗)]=𝜂²₁, (4.37) where 𝜂_max(𝑋) stands for the maximal eigenvalues of the matrix 𝑋.
For pure state (4.34) in bipartite partition 𝐴𝐵 and 𝐶, we have 𝐶(|𝜓⟩_{𝐴𝐵|𝐶})=2(1−Tr[𝜌²_𝐴𝐵])=2𝜂₁𝜂₂. (4.38) From (4.32), (4.37), and (4.38) we get ℱ_𝑁(𝜌_𝐴𝐵)≤1−𝐶²(|𝜓⟩_{𝐴𝐵|𝐶}). (4.39)
We now prove that the above inequality (4.39) also holds for mixed state 𝜌_𝐴𝐵𝐶. Let 𝜌_𝐴𝐵𝐶=∑_𝑖𝑝_𝑖|𝜓_𝑖⟩_𝐴𝐵𝐶⟨𝜓_𝑖| be the optimal decomposition of 𝜌_𝐴𝐵𝐶 such that 𝐶(𝜌_{𝐴𝐵|𝐶})=∑_𝑖𝑝_𝑖𝐶(|𝜓_𝑖⟩)_{𝐴𝐵|𝐶}. We have ℱ_𝑁(𝜌_𝐴𝐵)≤_𝑖𝑝_𝑖ℱ_𝑁(𝜌^𝑖_𝐴𝐵)≤_𝑖𝑝_𝑖1−𝐶²(𝜌^𝑖_{𝐴𝐵∣𝐶})≤1−_𝑖𝑝_𝑖𝐶²(𝜌^𝑖_{𝐴𝐵∣𝐶})≤1−𝐶²(𝜌_{𝐴𝐵∣𝐶}), (4.40) where 𝜌^𝑖_{𝐴𝐵|𝐶}=|𝜓_𝑖⟩_𝐴𝐵𝐶⟨𝜓_𝑖| and 𝜌^𝑖_𝐴𝐵=T𝑟_𝐶[𝜌^𝑖_{𝐴𝐵|𝐶}].

From Theorem 4.8 we see that the FEF of qubits 𝐴 and 𝐵 are bounded by the concurrence between qubits 𝐴, 𝐵, and qubit 𝐶. The upper bound of FEF for 𝜌_𝐴𝐵 decreases when the entanglement between qubits 𝐴,𝐵, and 𝐶 increases. As an example, we consider the generalized 𝑊 state defined by |𝑊^′⟩=𝛼|100⟩+𝛽|010⟩+𝛾|001⟩, |𝛼|²+|𝛽|²+|𝛾|²=1. The reduced density matrix is given by 𝜌^𝑊′_𝐴𝐵=|𝛾|²0000|𝛽|²𝛼^∗𝛽00𝛼𝛽^∗|𝛼|²00000. (4.41) The FEF of 𝜌^𝑊′_𝐴𝐵 is given by ℱ_𝑁(𝜌^𝑊′_𝐴𝐵1)=−21+2|𝛼||𝛽|+2||𝛼|²+|𝛽|²−|𝛾|²|, (4.42) while the concurrence of |𝑊^′⟩ has the form 𝐶_{𝐴𝐵|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(𝜌𝑊′𝐴𝐵) (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 state 𝜌∈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 a 3×3 state

8𝜌=91𝜎+9|𝜓₊⟩⟨𝜓₊|, (4.43) 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 of ℱ(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 state 𝜌 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 bound 𝜏₂(𝜌)>0 is a sufficient condition for the distillability of any bipartite state 𝜌.
(b) The lower bound 𝜏₂(𝜌)=0 is a necessary condition for separability of any bipartite state 𝜌.

Remark 4.12. Corollary 4.11 directly follows from Theorem 4.10 and this case is referred to as one distillable [128]. The problem of whether non-PPT (NPPT) nondistillable states exist is studied numerically in [128, 129]. By using Theorem 4.10, although it seems impossible to solve the problem completely, it is easy to judge the distillability of a state under condition that is one distillable.

The lower bound 𝜏₂, PPT criterion, separability, and distillability for any bipartite quantum state 𝜌 have the following relations: if 𝜏₂(𝜌)>0, then 𝜌 is entangled. If 𝜌 is separable, then it is PPT. If 𝜏₂(𝜌)>0, then 𝜌 is distillable. If 𝜌 is distillable, then it is NPPT. From the last two propositions it follows that if 𝜌 is PPT, then 𝜏₂(𝜌)=0, that is, if 𝜏₂(𝜌)>0, then 𝜌 is NPPT.

Theorem 4.13. For any pure tripartite state |𝜙⟩_𝐴𝐵𝐶 in arbitrary 𝑑⊗𝑑⊗𝑑 dimensional spaces, bound 𝜏₂ satisfies [67] 𝜏₂(𝜌_𝐴𝐵)+𝜏₂(𝜌_𝐴𝐶)≤𝜏₂(𝜌_{𝐴∶𝐵𝐶}), (4.44) where 𝜌_𝐴𝐵=Tr_𝐶(|𝜙⟩_𝐴𝐵𝐶⟨𝜙|), 𝜌_𝐴𝐶=Tr_𝐵(|𝜙⟩_𝐴𝐵𝐶⟨𝜙|), and 𝜌_{𝐴∶𝐵𝐶}=Tr_𝐵𝐶(|𝜙⟩_𝐴𝐵𝐶⟨𝜙|).

Proof. Since 𝐶²_𝑚𝑛≤(𝜆⁽¹⁾_𝑚𝑛)²≤∑⁴_𝑖=1(𝜆^(𝑖)_𝑚𝑛)²=Tr(𝜌𝜌_𝑚𝑛), one can derive the inequality 𝜏(𝜌_𝐴𝐵)+𝜏(𝜌_𝐴𝐶)≤^𝐷_𝑙,𝑘Tr[𝜌_𝐴𝐵(𝜌_𝐴𝐵)_𝑙𝑘]+^𝐷_𝑝,𝑞Tr[𝜌_𝐴𝐶(𝜌_𝐴𝐶)_𝑝𝑞], (4.45) where 𝐷=𝑑(𝑑−1)/2. Note that ∑_𝑙𝑘Tr[𝜌_𝐴𝐵(𝜌_𝐴𝐵)_𝑙𝑘]≤1−Tr𝜌²_𝐴−Tr𝜌²_𝐵+Tr𝜌²_𝐶 and ∑_𝑝𝑞Tr[𝜌_𝐴𝐶(𝜌_𝐴𝐶)_𝑝𝑞]≤1−Tr𝜌²_𝐴+Tr𝜌²_𝐵−Tr𝜌²_𝐶, 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−Tr𝜌²_𝐴)=𝐶²(𝜌_{𝐴∶𝐵𝐶}). Taking into account that 𝜏₂(𝜌_{𝐴∶𝐵𝐶})=𝒞²(𝜌_{𝐴∶𝐵𝐶}) for a pure state, one obtains inequality (4.44).
Generally for any pure multipartite quantum state 𝜌_{𝐴𝐵1𝐵2⋯𝑏𝑛}, one has the following monogamy inequality: 𝜏₂(𝜌_𝐴𝐵1)+𝜏₂(𝜌_𝐴𝐵2)+⋯+𝜏₂(𝜌_𝐴𝐵𝑛)≤𝜏₂(𝜌_{𝐴∶𝐵1𝐵2⋯𝐵𝑛}). (4.46)

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.