Abstract
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.
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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