AbstractGiven integers $$n \ge m$$
n
≥
m
, let $$\text {Sep}(n,m)$$
Sep
(
n
,
m
)
be the set of separable states on the Hilbert space $$\mathbb {C}^n \otimes \mathbb {C}^m$$
C
n
⊗
C
m
. It is well-known that for $$(n,m)=(3,2)$$
(
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 $$\text {Sep}(n,m)$$
Sep
(
n
,
m
)
has no semidefinite programming description of finite size. As $$\text {Sep}(n,m)$$
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.