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

Given integers $$n \ge m$$ , let $$\text {Sep}(n,m)$$ 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 $$\text {Sep}(n,m)$$ has no semidefinite programming description of finite size. As $$\text {Sep}(n,m)$$ 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.