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arXiv:1812.04182 · Physical Review A

Separability of completely symmetric states in a multipartite system

Lin Chen, Delin Chu, Lilong Qian, Yi Shen

Published Mar 11, 2019• 18 citations
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Abstract

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 $N+1$. For the states of rank 6 or $N+2$, 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.

DOI
10.1103/physreva.99.032312

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Comment on “Separability of completely symmetric states in a multipartite system”

Kil-Chan Ha

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.

Physical Review A•Jul 6, 2021•2 citations

Decomposition of completely symmetric states states

Lilong Qian, Delin Chu

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 $|x,x\rangle\langle x,x|$, 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 $N$. 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 $4$ or $N+1$. 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.

arXiv:1810.03125•Oct 7, 2018•1 citation

On the decomposition of the supersymmetric state

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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 $|x,x\rangle\langle x,x|$, 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 $N$. 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 $4$ or $N+1$. 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.

•Oct 7, 2018•0 citations

Necessary and sufficient condition of separability for D -symmetric diagonal states

A. Rutkowski, Michał Banacki, Marcin Marciniak

For multipartite states, we consider a notion of $D$ symmetry. For a system of $N$ qubits, it coincides with the usual permutational symmetry. In the case of $N$ qudits ($d\ensuremath{\ge}3$), the $D$ symmetry is stronger than the permutational one. For the space of all $\mathrm{D}$-symmetric vectors in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, we define a basis composed of vectors ${|{R}_{N,d;k}\ensuremath{\rangle}:\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}k\ensuremath{\le}N(d\ensuremath{-}1)}$ which are analogs of Dicke states. The aim of this paper is to discuss the problem of separability of $D$-symmetric states which are diagonal in the basis ${|{R}_{N,d;k}\ensuremath{\rangle}}$. We show that if $N$ is even and $d\ensuremath{\ge}2$ is arbitrary then a positive partial transposition property is a necessary and sufficient condition of separability for $D$-invariant diagonal states. In this way, we generalize results obtained by Yu [Phys. Rev. A 94, 060101(R) (2016)] and Wolfe and Yelin [Phys. Rev. Lett. 112, 140402 (2014)]. Our strategy is to use some classical mathematical results on a moment problem.

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Reply to “Comment on ‘Separability of completely symmetric states in a multipartite system’ ”

Yi Shen, Lin Chen, Lilong Qian +1 more

The author of the Comment [Phys. Rev. A 104, 016401 (2021)] pointed out the missing part in the proof of Theorem 20 in our work [L. Qian et al., Phys. Rev. A 99, 032312 (2019)], and presented a sufficient and necessary condition for the separability of completely symmetric (CS) states in the two-qutrit system. While being technically correct, the proposed example is still a separable CS state according to Ha's sufficient and necessary condition. We provide another proof to show that every bipartite CS state of rank five is separable. This result bridges the gap appearing in our previous proof. It turns out that every two-qutrit CS state is separable, and thus making our conclusion (Theorem 20 of our work) still valid.

Physical Review A•Jul 6, 2021•0 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
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⨂d{N}^{\ensuremath{\bigotimes}d}N\ensuremath⨂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 \ensuremathρ\ensuremath{\rho}\ensuremathρ 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.
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.
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 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.
For multipartite states, we consider a notion of DDD symmetry. For a system of NNN qubits, it coincides with the usual permutational symmetry. In the case of NNN qudits ($d\ensuremath{\ge}3$), the DDD symmetry is stronger than the permutational one. For the space of all D\mathrm{D}D-symmetric vectors in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, we define a basis composed of vectors ${|{R}_{N,d;k}\ensuremath{\rangle}:\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}k\ensuremath{\le}N(d\ensuremath{-}1)}$ which are analogs of Dicke states. The aim of this paper is to discuss the problem of separability of DDD-symmetric states which are diagonal in the basis ${|{R}_{N,d;k}\ensuremath{\rangle}}$. We show that if NNN is even and $d\ensuremath{\ge}2$ is arbitrary then a positive partial transposition property is a necessary and sufficient condition of separability for DDD-invariant diagonal states. In this way, we generalize results obtained by Yu [Phys. Rev. A 94, 060101(R) (2016)] and Wolfe and Yelin [Phys. Rev. Lett. 112, 140402 (2014)]. Our strategy is to use some classical mathematical results on a moment problem.
The author of the Comment [Phys. Rev. A 104, 016401 (2021)] pointed out the missing part in the proof of Theorem 20 in our work [L. Qian et al., Phys. Rev. A 99, 032312 (2019)], and presented a sufficient and necessary condition for the separability of completely symmetric (CS) states in the two-qutrit system. While being technically correct, the proposed example is still a separable CS state according to Ha's sufficient and necessary condition. We provide another proof to show that every bipartite CS state of rank five is separable. This result bridges the gap appearing in our previous proof. It turns out that every two-qutrit CS state is separable, and thus making our conclusion (Theorem 20 of our work) still valid.
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.