Bipartite separability and non-local quantum operations on graphs

TL;DR

This paper explores the separability of bipartite quantum states derived from graphs, introducing new classes of graphs and combinatorial methods to identify entanglement and separability properties.

Contribution

It introduces partially symmetric and degree symmetric graphs, providing a graph-theoretic framework for analyzing quantum state separability and entanglement.

Findings

Partially symmetric graphs correspond to separable states.

A combinatorial procedure can generate entanglement from partially symmetric graphs.

The degree criterion is extended via new graph classes.

Abstract

In this paper we consider the separability problem for bipartite quantum states arising from graphs. Earlier it was proved that the degree criterion is the graph-theoretic counterpart of the familiar positive partial transpose criterion for separability, although there are entangled states with positive partial transpose for which the degree criterion fails. Here we introduce the concept of partially symmetric graphs and degree symmetric graphs by using the well-known concept of partial transposition of a graph and degree criteria, respectively. Thus, we provide classes of bipartite separable states of dimension $m \times n$ arising from partially symmetric graphs. We identify partially asymmetric graphs that lack the property of partial symmetry. We develop a combinatorial procedure to create a partially asymmetric graph from a given partially symmetric graph. We show that this combinatorial operation can act as an entanglement generator for mixed states arising from partially symmetric graphs.