The tripartite separability of density matrices of graphs

TL;DR

This paper extends the study of entanglement properties of graph-based density matrices from bipartite to tripartite systems, establishing conditions for their separability.

Contribution

It generalizes the entanglement analysis of graph density matrices to tripartite states and proves the sufficiency and necessity of a degree condition for their separability.

Findings

Degree condition is necessary and sufficient for tripartite separability.

Generalization from bipartite to tripartite graph density matrices.

Characterization of entanglement in graph-based quantum states.

Abstract

The density matrix of a graph is the combinatorial laplacian matrix of a graph normalized to have unit trace. In this paper we generalize the entanglement properties of mixed density matrices from combinatorial laplacian matrices of graphs discussed in Braunstein {\it et al.} Annals of Combinatorics, {\bf 10}(2006)291 to tripartite states. Then we proved that the degree condition defined in Braunstein {\it et al.} Phys. Rev. A {\bf 73}, (2006)012320 is sufficient and necessary for the tripartite separability of the density matrix of a nearest point graph.