An improved semidefinite programming hierarchy for testing entanglement

TL;DR

This paper introduces a refined semidefinite programming hierarchy for testing quantum entanglement that converges finitely and offers an efficient algorithm for separability testing, improving upon previous methods.

Contribution

It presents a stronger hierarchy that converges exactly at finite steps and provides an elementary, more efficient algorithm for entanglement testing.

Findings

Hierarchy converges finitely for fixed dimensions.

Algorithm is singly exponential in dimension.

Provides polylogarithmic accuracy in testing.

Abstract

We present a stronger version of the Doherty-Parrilo-Spedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing which is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.