Towards efficient algorithm deciding separability of distributed quantum states

TL;DR

This paper presents a finite recursive algorithm for determining the separability of multi-partite quantum states, which is efficient for generic states and applicable to multipartite systems, despite potential exponential complexity in worst cases.

Contribution

It introduces a recursive algorithm that fully solves the quantum state separability problem with improved efficiency for generic cases and extends applicability to multipartite states.

Findings

Algorithm fully solves separability problem for multi-partite states.

For generic states, reduces to solving linear equations with few variables.

Applicable to multipartite states with comparable efficiency.

Abstract

It is pointed out that separability problem for arbitrary multi-partite states can be fully solved by a finite size, elementary recursive algorithm. In the worse case scenario, the underlying numerical procedure, may grow doubly exponentially with the state's rank. Nevertheless, we argue that for generic states, analysis of concurrence matrices essentially reduces the task of solving separability problem in $m \times n$ dimensions to solving a set of linear equations in about $\binom{mn+D-1}{D}$ variables, where $D$ decreases with $mn$ and for large $mn$ it should not exceed 4. Moreover, the same method is also applicable to multipartite states where it is at least equally efficient.