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arXiv · 1905.02575

The set of separable states has no finite semidefinite representation except in dimension $3\times 2$

Hamza Fawzi
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Abstract

Given integers n $\geq$ m, let Sep(n,m) be the set of separable states on the Hilbert space $\mathbb{C}^n \otimes \mathbb{C}^m$. It is well-known that for (n,m)=(3,2) the set of separable states has a simple description using semidefinite programming: it is given by the set of states that have a positive partial transpose. In this paper we show that for larger values of n and m the set Sep(n,m) has no semidefinite programming description of finite size. As Sep(n,m) is a semialgebraic set this provides a new counterexample to the Helton-Nie conjecture, which was recently disproved by Scheiderer in a breakthrough result. Compared to Scheiderer's approach, our proof is elementary and relies only on basic results about semialgebraic sets and functions.

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Hamza Fawzi. 2019-05-04. The set of separable states has no finite semidefinite representation except in dimension $3\times 2$. https://arxiv.org/abs/1905.02575

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