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.

arXiv:2005.11607May 23, 20200 citations

arXiv:1810.03125

Decomposition of completely symmetric states states

Lilong Qian, Delin Chu

Published Oct 7, 2018• 1 citation

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Abstract

|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

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.

DOI: 10.48550/arxiv.1810.03125

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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.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 AJul 6, 20212 citations

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

Stephan Weis

arXiv:2005.11607May 23, 20200 citations

|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

N+1

|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

N+1

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.

We prove that every bipartite state

ρ

of a qubit coupled to an

n

-level system whose operator Schmidt rank equals three can be written as a mixture of

\text{rank}(ρ)

pure product states, and as a mixture of at most

n+1

mixed product states in general. The second result is tight in the sense that there exist states that cannot be decomposed with a smaller number of terms. Our proof gives constructive methods to obtain the decompositions. It involves eigendecompositions of suitable unitary dilations of an operator explicitly determined by the state. Our framework recovers previously known results for the length of decompositions into pure product states and is also able to deal with decompositions into mixed states. For pure product states, our decomposition is different from existing ones.