Introduction

Quantum entanglement is a fascinating phenomenon in quantum physics. It can be seen as a physical resource like energy with applications from quantum teleportation to quantum cryptography1,2,3,4,5. In the last years, much work has been devoted to understanding entanglement, but there are still many problems unsolved. One of them is to determine whether a given quantum state is entangled or separable. This problem is extremely difficult to solve, and has been proved as a nondeterministic polynomial-time hard problem6. Nevertheless, a variety of operational criteria for separability of quantum states have been proposed in the last decades. Among them are the positive partial transpose (PPT) criterion or Peres-Horodecki criterion7,8, realignment criteria9,10,11,12,13, covariance matrix criteria14,15,16 and so on; see, e.g.17,18, for a comprehensive survey.

The Bloch representation19,20,21 of density matrices stands as an important role in quantum information. The correlation matrices or tensors in the Bloch representation are encoded with entanglement properties22,23, which can be exploited to study quantum entanglement. In ref. 24, by making use of correlation matrices, Vicente obtained the correlation matrix criterion for bipartite quantum states, which can be more efficient than the PPT criterion7,8 and the computable cross norm or realignment (CCNR) criterion9,10 in many different situations. After that, this criterion was used to give the analytical lower bounds for the entanglement measures: concurrence and tangle25,26, which are good supplement to the lower bounds based on PPT and CCNR criteria. By the matricizations of tensors, the correlation matrix criterion was generalized to detect non-full-separability of multipartite states27. Later, this multipartite criterion was extended and improved to be a much more general case28. Meanwhile, by the standard tensor norm and the norms of matricizations of tensors, some genuine entanglement conditions were derived. In refs 22, 23, some simple geometrical methods based on correlation tensors were presented to detect various multipartite entanglement. By bounding tensor norms for partially separable states and states of limited dimension, Klöckl and Huber29 studied the detection of multipartite entanglement in an experimentally feasible way. In many cases, only few definite measurements are needed. Recently, Li et al.30 presented some separability criteria under the combination of correlation matrices and the Bloch vectors of reduced density matrices, which can be stronger than the correlation matrix criterion24 by examples.

This paper is further devoted to an investigation of entanglement detection in terms of Bloch representations of density matrices. On the one hand, by adding some parameters, a more general separability criterion for bipartite states is presented, which can outperform the corresponding criteria given in24,30. On the another hand, the presented bipartite separability criterion is extended to the multipartite case. An example shows that the new multipartite separability criterion can be better than the corresponding criteria obtained in refs 27, 28 and 30.

Results

Separability criteria for bipartite states

Let be the traceless Hermitian generators of SU(d) satisfying the orthogonality relation . Then any state ρ in can be represented as21

where Id denotes the d × d identity matrix,

Denote by ||·||tr, ||·||2 and Ep×q the trace norm (the sum of singular values), the spectral norm (the maximum singular value) and the p × q matrix with all entries being 1, respectively. By defining , and T = (tij), we construct the following matrix

where α and β are nonnegative real numbers, m is a given natural number, t stands for transpose, and for any column vector x,

Using , we can get the following separability criterion for bipartite states.

Theorem 1. If the state ρ in is separable, then

See Methods for the proof of Theorem 1.

When α and β are chosen to be 0, Theorem 1 reduces to the correlation matrix criterion in ref. 24: if ρ in is separable, then

If we choose α = β = m = 1, then Theorem 1 becomes the separability criterion given in [30, Corollary 2]: any separable state ρ in must satisfy

For simplicity, we call these criteria in (6) and (7) the V-B and L-B criteria, respectively. The following result can help us find that our separability criterion from Theorem 1 is stronger than the V-B and L-B criteria.

Proposition 1. If α and β are selected to satisfy

then Theorem 1 becomes more effective when m gets larger.

See Methods for the proof of Proposition 1.

From Proposition 1, Theorem 1 with the condition (8) is stronger than the V-B criterion.

For the case d1 = d2 and α = β, it follows from Proposition 1 that Theorem 1 is more efficient when m gets larger. In particular, Theorem 1 is better than the L-B criterion, and the L-B criterion is better than the V-B criterion. For the case d1 ≠ d2, let us consider the following example. The following 2 × 4 bound entangled state is due to31:

where 0 < b < 1. To verify the efficiency of the present criteria, we consider the state

where . For simplicity, we choose

Then Theorem 1 can detect the entanglement in ρx for 0.2235 ≤ x ≤ 1, while the V-B criterion and L-B criterion can only detect the entanglement in ρx for 0.2293 ≤ x ≤ 1 and 0.2841 ≤ x ≤ 1, respectively. Thus, Theorem 1 is better than the V-B and L-B criteria.

Separability criteria for multipartite states

Let be an f1 × ··· × fN tensor, A and be two nonempty subsets of {1, ···, N} satisfying . Then we denote by the matricization of ; see28 for detail. This matricization is a generalization of mode-n matricization in the multilinear algebra32.

For any state ρ in , we import a natural number m and nonnegative real parameters α1, ···, αN, and define

We define the tensor with elements

Clearly, if m = 0, the tensor reduces to the correlation tensor in ref. 27. When m = α1 = ··· = αN = 1, the tensor becomes the tensor with a constant multiple in ref. 30.

An n partite sate ρ in is (fully) separable33 if and only if it can be written in the form

where the probabilities , and are pure states of the subsystems.

In the following, we give the full separability criterion based on .

Theorem 2. If the state ρ in is fully separable, then, for any subset A of {1, ···, N}, we have

See Methods for the proof of Theorem 2.

For the case α1 = ··· = αN = 0, Theorem 2 reduces to the criterion given in [28, Theorem 4], which has an important improvement on the corresponding criterion given in ref. 27. If α1 = ··· = αN = 1 and m = 1, then Theorem 2 becomes [30, Corollary 3]. For simplicity, we call these criteria in refs 27, 28 and 30 V-M, H-M and L-M criteria, respectively. In the following we give a tripartite example to demonstrate the efficiency of Theorem 2. Consider a perturbation of the tripartite GHZ state16:

where is a given real parameter, and γ denotes the normalization. We consider the mixture of this state with the maximally mixed state:

In the tripartite case, the V-M criterion is equivalent to the H-M criterion obviously. By taking m = 1 and α1 = α2 = α3 = 0.1, Table 1 displays the detection results with different values of . Clearly, Theorem 2 is more efficient than the V-M, H-M and L-M criteria.

Table 1 Entanglement conditions of with different values of from the V-M (H-M) criterion, the L-M criterion and Theorem 2 with α1 = α2 = α3 = 0.1 and m = 1.
Full size table

Discussions

Correlation matrices or tensors in the Bloch representation of quantum states contain the information of entanglement of the quantum states. Based on the Bloch representation of quantum states, we have given some new separability criteria including the V-B, L-B, V-M, H-M and L-M criteria as special cases. For bipartite cases, by choosing some special parameters involved, our criteria are stronger than the V-B and L-B criteria. For multipartite cases, by a simple example it has been also shown that our criterion can be more efficient than the V-M, H-M and L-M criteria.

Nevertheless, the problem of how to choose the involved parameters such that Theorems 1–2 can detect more entangled states needs to be further studied in the future. In the Bloch representation (1), the traceless Hermitian generators of SU(d) come from Gell-Mann matrices. But this is by far not the only possible choice. Maybe the new basis of observables34 constructed from Heisenberg-Weyl operators can be used to obtain better separable criteria, since the Heisenberg-Weyl based observables can outperform the canonical basis of generalized Gell-Mann operators in entanglement detection34. Thus, this problem is worth studying in the coming days.

It should be noted that the separability criteria Theorems 1–2 presented in30 for bipartite and multipartite states are at most as good as the corresponding V-B, L-B, V-M and L-M criteria, respectively. For example, set . It was shown by [30, Theorem 1] that any separable state ρ in satisfies

where M = (mij) is any real matrix. From (18) and35, we get

which implies that the L-B criterion is at least as good as the criterion (18). Other cases can be proved similarly.

Methods

Proof of Theorem 1. Since ρ is separable, from [24, (17)], it follows that there exist vectors and such that

where

Thus, the matrix can be written as

and then

where we have used the following equality, for any vectors |a〉 and |b〉,

■

Proof of Proposition 1. For any state ρ, from [24, Lemma 1], we get

If the inequality from (5),

holds, then from (25) we have

where the equality (24) has been used in the first and fifth equalities, and, in the third and fourth equalities, we have employed the fact that the trace norm of a Hermitian positive semidefinite matrix is equal to its trace.

Proof of Theorem 2. Without loss of generality, we assume that

Since ρ is fully separable, then from27 there exist vectors such that

where

Thus,

where we have used the equality (24).

Additional Information

How to cite this article: Shen, S.-Q. et al. Improved Separability Criteria Based on Bloch Representation of Density Matrices. Sci. Rep. 6, 28850; doi: 10.1038/srep28850 (2016).