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Physical Review A

Separability discrimination and decomposition of quantum mixed states based on the Broyden-Fletcher-Goldfarb-Shanno algorithm

Bo Yang, Ying Li, Guyan Ni

Published Nov 16, 2023• 6 citations
Publisher

Abstract

In this paper, we consider separability discrimination and decomposition problems of quantum mixed states based on the tensor optimization method. We first convert the separability determination problem of quantum mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Then we study a rank-$R$ positive Hermitian approximation model of Hermitian tensors and introduce a Broyden--Fletcher--Goldfarb--Shanno (BFGS) algorithm for rank-$R$ positive Hermitian approximation. We prove that the BFGS algorithm can be used to compute a decomposition of a mixed state if it is separable. Finally, we propose a BFGS algorithm for separability determination and decomposition of quantum mixed states. Numerical examples show the effectiveness of the algorithm.

DOI
10.1103/physreva.108.052217

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We propose an EEE-truncated KKK-moment and semidefinite relaxations (ETKM-SDR) method to check whether an mmm-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 EEE-truncated KKK-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.
Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e. it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect the separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect the separability for much larger-sized Hermitian tensors. This method is a good starting point to determine psd ranks of separable Hermitian tensors.
Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e., it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect separability for much larger sized Hermitian tensors. This method is a good start on determining psd ranks of separable Hermitian tensors.
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