We propose an $E$ -truncated $K$ -moment and semidefinite relaxations (ETKM-SDR) method to check whether an $m$ -partite quantum mixed state is separable and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Then, employing the $E$ -truncated $K$ -moment method, we obtain an optimization model for discriminating separability. Moreover, by applying the semidefinite relaxation method, we get a hierarchy of semidefinite relaxation optimization models and propose an algorithm for detecting the separability of mixed states. The algorithm can also be used for symmetric and nonsymmetric decomposition of separable mixed states. By numerical examples, we find that not all symmetric separable states have symmetric decompositions.We propose an $E$-truncated $K$-moment and semidefinite relaxations (ETKM-SDR) method to check whether an $m$-partite quantum mixed state is separable and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Then, employing the $E$-truncated $K$-moment method, we obtain an optimization model for discriminating separability. Moreover, by applying the semidefinite relaxation method, we get a hierarchy of semidefinite relaxation optimization models and propose an algorithm for detecting the separability of mixed states. The algorithm can also be used for symmetric and nonsymmetric decomposition of separable mixed states. By numerical examples, we find that not all symmetric separable states have symmetric decompositions.

arXiv:1912.08985Physical Review AJul 6, 202015 citations

Lewenstein-Sanpera decomposition For Bell Decomposable

Seyed Javad Akhtarshenas, M. A. Jafarizadeh

We propose a simple geometrical approach for finding the Lewenstein-Sanpera decomposition of Bell decomposable states of 2 otimes 2 quantum systems. We show that in these systems, the weight of the pure entangled part in the decomposition is equal to the concurrence of the state. It is also shown that the optimized separable part of L-S decomposition minimizes the von Neumann relative entropy. We also obtain the decomposition for a class of mixed states by using some LQCC actions. It is also shown that for these states the average concurrence of L-S decomposition is equal to their concurrence.We propose a simple geometrical approach for finding the Lewenstein-Sanpera decomposition of Bell decomposable states of 2 otimes 2 quantum systems. We show that in these systems, the weight of the pure entangled part in the decomposition is equal to the concurrence of the state. It is also shown that the optimized separable part of L-S decomposition minimizes the von Neumann relative entropy. We also obtain the decomposition for a class of mixed states by using some LQCC actions. It is also shown that for these states the average concurrence of L-S decomposition is equal to their concurrence.

arXiv:quant-ph/0207161Jul 28, 20020 citations

Separable States with Unique Decompositions

Kil-Chan Ha, Seung-Hyeok Kye

arXiv:1210.1088Communications in Mathematical PhysicsMar 10, 201427 citations

We propose an

E

-truncated

K

-moment and semidefinite relaxations (ETKM-SDR) method to check whether an

m

-partite quantum mixed state is separable and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Then, employing the

E

-truncated

K

-moment method, we obtain an optimization model for discriminating separability. Moreover, by applying the semidefinite relaxation method, we get a hierarchy of semidefinite relaxation optimization models and propose an algorithm for detecting the separability of mixed states. The algorithm can also be used for symmetric and nonsymmetric decomposition of separable mixed states. By numerical examples, we find that not all symmetric separable states have symmetric decompositions.