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Given integers n $\geq$ m, let Sep(n,m) be the set of separable states on theHilbert 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 usingsemidefinite programming: it is given by the set of states that have a positivepartial transpose. In this paper we show that for larger values of n and m theset Sep(n,m) has no semidefinite programming description of finite size. AsSep(n,m) is a semialgebraic set this provides a new counterexample to theHelton-Nie conjecture, which was recently disproved by Scheiderer in abreakthrough result. Compared to Scheiderer's approach, our proof is elementaryand relies only on basic results about semialgebraic sets and functions.

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