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

2019/05/04 by Hamza Fawzi, Fawzi, Hamza · 6 citations
Computer Science · #Complexity and Algorithms in Graphs #FOS: Mathematics #FOS: Physical sciences #Machine Learning and Algorithms #Optimization and Control (math.OC) #Quantum Physics (quant-ph) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1905.02575

openalex publication_date 2019/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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