Direct evaluation of pure graph state entanglement

TL;DR

This paper presents a method to evaluate multipartite entanglement in pure graph states by linking graph theory problems to quantum entanglement measures, enabling explicit calculations of various entanglement metrics.

Contribution

It introduces a novel approach connecting maximum independent set in graph theory to the evaluation of multiple entanglement measures in pure graph states.

Findings

Efficient evaluation of Schmidt measure for large classes of graph states.

Explicit construction of closest separable states for relative entropy of entanglement.

Analysis of noise effects on entanglement destruction.

Abstract

We address the question of quantifying entanglement in pure graph states. Evaluation of multipartite entanglement measures is extremely hard for most pure quantum states. In this paper we demonstrate how solving one problem in graph theory, namely the identification of maximum independent set, allows us to evaluate three multipartite entanglement measures for pure graph states. We construct the minimal linear decomposition into product states for a large group of pure graph states, allowing us to evaluate the Schmidt measure. Furthermore we show that computation of distance-like measures such as relative entropy of entanglement and geometric measure becomes tractable for these states by explicit construction of closest separable and closest product states respectively. We show how these separable states can be described using stabiliser formalism as well as PEPs-like construction. Finally we discuss the way in which introducing noise to the system can optimally destroy entanglement.