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

## TL;DR

This paper proves that the set of separable quantum states cannot be described by finite semidefinite programming representations in most dimensions, providing a new counterexample to a longstanding conjecture.

## Contribution

It demonstrates that for larger dimensions, the set of separable states lacks finite semidefinite representations, offering an elementary proof and a new counterexample to the Helton-Nie conjecture.

## Key findings

- Sep(n,m) has no finite semidefinite representation for most (n,m)
- Provides an elementary proof contrasting Scheiderer's approach
- Offers a new counterexample to the Helton-Nie conjecture

## Abstract

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.