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arXiv:2005.11607

Decomposition of symmetric separable states and ground state energy of bosonic systems

Stephan Weis

Published May 23, 2020• 0 citations
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Abstract

We prove that every symmetric separable state admits a convex decomposition into symmetric pure product states. While the result is not new in itself, here we focus on convex geometry. We discuss the decomposition in the context of numerical ranges and ground state problems of infinite bosonic systems.
DOI
10.48550/arxiv.2005.11607

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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.
Symmetry plays an important role in the field of quantum mechanics. We consider a subclass of symmetric quantum states in the multipartite system ${N}^{\ensuremath{\bigotimes}d}$, namely, the completely symmetric states, which are invariant under any index permutation. It was hypothesized by L. Qian and D. Chu (arXiv:1810.03125 [quant-ph]) that the completely symmetric states are separable if and only if it is a convex combination of symmetric pure product states. We prove that this conjecture is true for the both bipartite and multipartite cases. Further, we prove that the completely symmetric state $\ensuremath{\rho}$ is separable if its rank is at most 5 or N+1N+1N+1. For the states of rank 6 or N+2N+2N+2, they are separable if and only if their range contains a product vector. We apply our results to a few widely useful states in quantum information, such as symmetric states, edge states, extreme states, and non-negative states. We also study the relation of completely symmetric states to Hankel and Toeplitz matrices.
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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.
In their interesting paper [Phys. Rev. A 99, 032312 (2019)], Chen et al. consider the completely symmetric (CS) states, which are invariant under any index permutation. They proved that the CS states are separable if and only if they are a convex combination of real symmetric pure product states in Theorem 14, and claimed that any CS state in a two-qutrit system is separable in Theorem 20. We point out that their proof of Theorem 20 is flawed because of invalid assumptions. Furthermore, we present a sufficient and necessary condition for the separability of the CS states in two-qutrit systems.
A quantum state which can be written as a convex sum of product states is a separable state; such a convex sum allows for a local description. There are many different local descriptions for a given separable state, and some new results about the minimal number of product states in the convex sum wil be given in this paper. Any quantum state is either entangled or separable. However it is very difficult to tell whether a given state is separable or entangled. This difficulty has not been solved yet although a lot of separability criteria have already be proposed. The various separability criteria will also be discussed.