Shor-Laflamme distributions of graph states and noise robustness of entanglement

TL;DR

This paper develops a graph-theoretical approach to compute Shor-Laflamme distributions of graph states, revealing their typical binomial form, and introduces an entanglement criterion based on these distributions to assess noise robustness.

Contribution

It provides a novel graph-based method to derive SLDs for graph states, including closed forms for certain families, and introduces an easy-to-use entanglement criterion applicable to qubits and qudits.

Findings

SLD of graph states can be derived via graph theory.

SLD for cluster states resembles a binomial distribution.

New entanglement criterion yields noise thresholds for entanglement.

Abstract

The Shor-Laflamme distribution (SLD) of a quantum state is a collection of local unitary invariants that quantify $k$-body correlations. We show that the SLD of graph states can be derived by solving a graph-theoretical problem. In this way, the mean and variance of the SLD are obtained as simple functions of efficiently computable graph properties. Furthermore, this formulation enables us to derive closed expressions of SLDs for some graph state families. For cluster states, we observe that the SLD is very similar to a binomial distribution, and we argue that this property is typical for graph states in general. Finally, we derive an SLD-based entanglement criterion from the purity criterion and apply it to derive meaningful noise thresholds for entanglement. Our new entanglement criterion is easy to use and also applies to the case of higher-dimensional qudits. In the bigger picture, our results foster the understanding both of quantum error-correcting codes, where a closely related notion of Shor-Laflamme distributions plays an important role, and of the geometry of quantum states, where Shor-Laflamme distributions are known as sector length distributions.