Combinatorial laplacians and positivity under partial transpose

TL;DR

This paper investigates the properties of density matrices derived from graph Laplacians, establishing the equivalence of certain criteria for separability, providing counterexamples, and deriving bounds on quantum entanglement measures.

Contribution

It proves the equivalence of the degree-criterion and PPT-criterion for graph density matrices, and establishes conditions under which these criteria are sufficient for separability.

Findings

Degree-criterion is equivalent to PPT-criterion for graph density matrices.

Counterexample shows the degree-criterion is not sufficient for separability.

Derived a rational upper bound on concurrence, exact for four-vertex graphs.

Abstract

Density matrices of graphs are combinatorial laplacians normalized to have trace one (Braunstein \emph{et al.} \emph{Phys. Rev. A,} \textbf{73}:1, 012320 (2006)). If the vertices of a graph are arranged as an array, then its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for separability of density matrices of graphs, is equivalent to the PPT-criterion. As such it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (for an alternative proof see Wu, \emph{Phys. Lett. A}\textbf{351} (2006), no. 1-2, 18--22). Finally we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.