Relaxations of separability in multipartite systems: Semidefinite programs, witnesses and volumes

TL;DR

This paper explores relaxations of separability in multipartite quantum systems using semidefinite programs, introduces an operational entanglement witness, and analyzes the volume of biseparable states, highlighting limitations of simple relaxations.

Contribution

It systematically characterizes multipartite entanglement via relaxations, constructs a universal entanglement witness, and provides bounds on the volume of biseparable states.

Findings

Relaxations can detect all entangled states.

Volume of biseparable states is bounded and small.

Simple relaxations are insufficient for accurate approximation.

Abstract

While entanglement is believed to be an important ingredient in understanding quantum many-body physics, the complexity of its characterization scales very unfavorably with the size of the system. Finding super-sets of the set of separable states that admit a simpler description has proven to be a fruitful approach in the bipartite setting. In this paper we discuss a systematic way of characterizing multiparticle entanglement via various relaxations. We furthermore describe an operational witness construction arising from such relaxations that is capable of detecting every entangled state. Finally, we also derive an analytic upper-bound on the volume of biseparable states and show that the volume of the states with a positive partial transpose for any split rapidly outgrows this volume. This proves that simple semi-definite relaxations in the multiparticle case cannot be an equally good approximation for any scenario.