Year-end Sale % OFF 
Abstract
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.
Highlights
Entanglement is a fundamental aspect of quantum mechanics
The theorem above can be used to recover the result of Scheiderer [Sch18, Corollary 4.25], that the cone Pn,2d of nonnegative forms in n variables of degree 2d is not semidefinite representable when it is distinct from n,2d, the cone of sums of squares
We did not find any reference for this equivalence, so we include a proof here
Summary
Entanglement is a fundamental aspect of quantum mechanics. The set of separable states (i.e., nonentangled states) on the Hilbert space Cn ⊗ Cm is defined as: Sep(n, m) = conv x x† ⊗ yy† : x ∈ Cn, |x| = 1, y ∈ Cm, |y| = 1. Combining this property with a simple observation regarding smooth sum of squares decompositions of homogeneous polynomials allows us to prove Theorem 2 already in the special case where p is a homogeneous polynomial This allows us to prove that Sep(n, m) is not semidefinite representable when (n, m) = (5, 3) or (4, 4). The theorem above can be used to recover the result of Scheiderer [Sch, Corollary 4.25], that the cone Pn,2d of nonnegative (real) forms in n variables of degree 2d is not semidefinite representable when it is distinct from n,2d , the cone of sums of squares It suffices to take p in Theorem 3 to be a dehomogenization of a nonnegative form that is not a sum of squares, and to use the well-known fact that a convex set has a semidefinite representation if and only if its dual has one.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
