A two-way algorithm for the entanglement problem

TL;DR

This work proposes an algorithm which proves a given bipartite quantum state to be separable in a finite number of steps through the search for a decomposition via a countable subset of product states, which is dense within all product states.

Abstract

We propose an algorithm which proves a given bipartite quantum state to be separable in a finite number of steps. Our approach is based on the search for a decomposition via a countable subset of product states, which is dense within all product states. Performing our algorithm simultaneously with the algorithm by Doherty, Parrilo and Spedalieri (which proves a quantum state to be entangled in a finite number of steps) leads to a two-way algorithm that terminates for any input state. Only for a set of arbitrary small measure near the border between separable and entangled states the result is inconclusive.

Figures (1)

• Figure 1: Constructing a "generalized diamond", i.e. a regular crosspolytope.