Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum entanglement and separable matrices

Jiawang Nie

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

A real symmetric matrix is separable if it can be written as a sum of Kronecker products of positive semidefinite matrices. This talk concerns how to check if a matrix is separable or not. We propose a numerical algorithm, based on Lasserre type semidefinite relaxations, for solving the question. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability. This is a joint work with Xinzhen Zhang.

Related papers

Positive Maps and Separable Matrices

Jiawang Nie, Xinzhen Zhang

A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite matrices. This paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not. We propose numerical algorithms, based on Lasserre's type of semidefinite relaxations, for solving such questions. To check the positivity of a linear map, we construct a hierarchy of semidefinite relaxations for minimizing the associated biquadratic form over the unit spheres. We show that the positivity can be detected by solving a finite number of such semidefinite relaxations. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability.

arXiv:1504.06595•SIAM Journal on Optimization•Jan 1, 2016•23 citations

Entanglement and the truncated moment problem

Fabian Bohnet-Waldraff, Daniel Braun, O. Giraud

We map the quantum entanglement problem onto the mathematically well-studied truncated moment problem. This yields a necessary and sufficient condition for separability that can be checked by a hierarchy of semidefinite programs. The algorithm always gives a certificate of entanglement if the state is entangled. If the state is separable, typically a certificate of separability is obtained in a finite number of steps and an explicit decomposition into separable pure states can be extracted.

arXiv:1704.02277•Physical Review A•Sep 11, 2017•22 citations

Separability of symmetric states and vandermonde decomposition

Lilong Qian, Lin Chen, Delin Chu

Abstract Symmetry is one of the central mysteries of quantum mechanics and plays an essential role in multipartite entanglement. In this paper, we consider the separability problem of quantum states in the symmetric space. We establish the relation between the separability of multiqubit symmetric states and the decomposability of Hermitian positive semidefinite matrices. This relation allows us to exchange concepts and ideas between quantum entanglement and Vandermonde decomposition. As an application, we build a suite of tools to investigate the decomposability and show the power of this relation both in theoretical and numerical aspects. For theoretical results, we establish the witness for the decomposability similar to the entanglement witness and characterize the decomposability of some subclasses of matrices. Furthermore, we provide the necessary conditions for the decomposability. Besides, we suggest a numerical algorithm to check whether a given matrix is decomposable. The numerical examples are tested to show the effectiveness.

New Journal of Physics•Feb 10, 2020•4 citations

Separability discrimination and decomposition of m -partite quantum mixed states

Ying Li, Guyan Ni

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:1912.08985•Physical Review A•Jul 6, 2020•15 citations

Separability of Hermitian Tensors and PSD Decompositions

Mareike Dressler, Jiawang Nie, Zi Yang

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.

arXiv:2011.08132•Nov 16, 2020•1 citation

Separability of Hermitian tensors and PSD decompositions

Mareike Dressler, Jiawang Nie, Zi Yang

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.

Linear and Multilinear Algebra•Aug 27, 2021•10 citations
A real symmetric matrix is separable if it can be written as a sum of Kronecker products of positive semidefinite matrices. This talk concerns how to check if a matrix is separable or not. We propose a numerical algorithm, based on Lasserre type semidefinite relaxations, for solving the question. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability. This is a joint work with Xinzhen Zhang.
A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite matrices. This paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not. We propose numerical algorithms, based on Lasserre's type of semidefinite relaxations, for solving such questions. To check the positivity of a linear map, we construct a hierarchy of semidefinite relaxations for minimizing the associated biquadratic form over the unit spheres. We show that the positivity can be detected by solving a finite number of such semidefinite relaxations. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability.
Abstract Symmetry is one of the central mysteries of quantum mechanics and plays an essential role in multipartite entanglement. In this paper, we consider the separability problem of quantum states in the symmetric space. We establish the relation between the separability of multiqubit symmetric states and the decomposability of Hermitian positive semidefinite matrices. This relation allows us to exchange concepts and ideas between quantum entanglement and Vandermonde decomposition. As an application, we build a suite of tools to investigate the decomposability and show the power of this relation both in theoretical and numerical aspects. For theoretical results, we establish the witness for the decomposability similar to the entanglement witness and characterize the decomposability of some subclasses of matrices. Furthermore, we provide the necessary conditions for the decomposability. Besides, we suggest a numerical algorithm to check whether a given matrix is decomposable. The numerical examples are tested to show the effectiveness.
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 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.
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.