  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.
