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