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The Set of Separable States has no Finite Semidefinite Representation Except in Dimension 3 × 2

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Peer-reviewed

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Article

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Abstract

Given integers n m, let Sep(n,m) be the set of separable states on the Hilbert space CnCm. 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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Keywords

quant-ph, quant-ph, math.OC

Journal Title

Communications in Mathematical Physics

Conference Name

Journal ISSN

0010-3616
1432-0916

Volume Title

386

Publisher

Springer Science and Business Media LLC