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

Understand

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.