Improved Separability Criteria Based on Bloch Representation of Density Matrices

The correlation matrices or tensors in the Bloch representation of density matrices are encoded with entanglement properties. In this paper, based on the Bloch representation of density matrices, we give some new separability criteria for bipartite and multipartite quantum states. Theoretical analysis and some examples show that the proposed criteria can be more efficient than the previous related criteria.


Results
Separability criteria for bipartite states. Let  Denote by ||·|| tr , ||·|| 2 and E p×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 = −  r r r ( , , ) and T = (t ij ), 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, , we can get the following separability criterion for bipartite states.
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 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 For the case d 1 = d 2 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 d 1 ≠ d 2 , let us consider the following example. The following 2 × 4 bound entangled state is due to 31 : To verify the efficiency of the present criteria, we consider the state Scientific RepoRts | 6:28850 | DOI: 10.1038/srep28850 x where ξ = + ( 00 11 ) 1 2 . For simplicity, we choose Then Theorem 1 can detect the entanglement in ρ x for 0.2235 ≤ x ≤ 1, while the V-B criterion and L-B criterion can only detect the entanglement in ρ x for 0.2293 ≤ x ≤ 1 and 0.2841 ≤ x ≤ 1, respectively. Thus, Theorem 1 is better than the V-B and L-B criteria. 28 for detail. This matricization is a generalization of mode-n matricization in the multilinear algebra 32 .

Separability criteria for multipartite states. Let
For , we import a natural number m and nonnegative real parameters α 1 , ···, α N , and define reduces to the correlation tensor in ref. 27

Theorem 2. If the state ρ in
is fully separable, then, for any subset A of {1, ···, N}, we have 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 α 1 = α 2 = α 3 = 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.

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 observables 34 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 detection 34 . Thus, this problem is worth studying in the coming days.
It should be noted that the separability criteria Theorems 1-2 presented in 30 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   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   where we have used the following equality, for any vectors |a〉 and |b〉 ,  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 from 27 there exist vectors  where we have used the equality (24).