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New Journal of Physics

Separability of symmetric states and vandermonde decomposition

Lilong Qian, Lin Chen, Delin Chu

Published Feb 10, 2020• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1367-2630/ab74a3

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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.
In this paper, we consider a subclass of quantum states in the multipartite system, namely, the supersymmetric states. We investigate the problem whether they admit the symmetrically separable decomposition, i.e., each term in this decomposition is a supersymmetric pure product state ∣x,x⟩⟨x,x∣|x,x\rangle\langle x,x|∣x,x⟩⟨x,x∣, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or NNN. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to 444 or N+1N+1N+1. We also propose a numerical algorithm which is able to detect S-separability. Besides, we analysis the convergence behavior of this algorithm. Some numerical examples are tested to show the effectiveness of the algorithm.
The structural relation between multipartite entanglement and symmetry is one of the central mysteries of quantum mechanics. In this paper, we study the separability of quantum states in the bosonic system. We show that a mixture of multiqubit Dicke states is separable if and only if its partial transpose is positive semidefinite, which confirms the hypothesis of Wolfe and Yelin [E. Wolfe and S. F. Yelin, Phys. Rev. Lett. 112, 140402 (2014)]. We generalize this result to a class of bosonic states in the $d\ensuremath{\bigotimes}d$ system; and for general ddd, we determine its separability is NP-hard although verifiable conditions for separability are easily derived when d=3,4d=3,4d=3,4.
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.
In this paper, we consider a subclass of quantum states in the multipartite system, namely, the supersymmetric states. We investigate the problem whether they admit the symmetrically separable decomposition, i.e., each term in this decomposition is a supersymmetric pure product state ∣x,x⟩⟨x,x∣|x,x\rangle\langle x,x|∣x,x⟩⟨x,x∣, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or NNN. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It was proved to be true when the rank is less than or equal to 444 or N+1N+1N+1. We also propose a numerical algorithm which is able to detect S-separability. Besides, we analysis the convergence behavior of this algorithm. Some numerical examples are tested to show the effectiveness of the algorithm.
Quantum entanglement is a key property of quantum information theory, that is at the heart of numerous promising applications in fields such as quantum cryptography, quantum computing or quantum sensing. In the past decades, the advent of such innovative technologies has reinforced the need for a better understanding of entanglement. The aim of this thesis is to contribute to this effort through the development of new tools targeting the characterization of several features of entanglement. Concerning the issue of entanglement detection, we present an optimization of the approach that exploits the concept of generalized concurrences to solve the separability problem for pure states. We then reformulate the separability question of mixed states into a matrix analysis problem, from which we obtain general separability criteria for multipartite states of ranks two and three. We also briefly discuss some properties of separable states. In particular, we characterize optimal separable decompositions of symmetric (i.e. permutation invariant) states of two and three qubits with maximal rank properties. Regarding the quantification of entanglement, we propose a function to quantify the entanglement of symmetric multiqubit states within classes of entangled states gathering states that are stochastically equivalent through local operations assisted with classical communication. This function establishes a link between the amount of entanglement of a symmetric state and the distribution of its Majorana points on the Bloch sphere. We finally investigate the robustness of entanglement with respect to particle loss and provide a full description of all multiqubit states that are fragile for the loss of one of their qubits. For symmetric states, the fragility for the loss of one qubit is shown to be related to a particular symmetry of the Majorana points.
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.