Random graph states, maximal flow and Fuss-Catalan distributions

TL;DR

This paper introduces a method to analyze entanglement in graph-based random quantum states, revealing connections to Fuss-Catalan distributions and providing explicit constructions for these spectral properties.

Contribution

It develops a new technique using Weingarten calculus and flow problems to compute average entanglement entropy for graph states, and identifies conditions leading to Fuss-Catalan eigenvalue distributions.

Findings

Spectral properties of these states follow Marchenko-Pastur or Fuss-Catalan distributions.

Explicit graph constructions yield ensembles with Fuss-Catalan eigenvalue distributions.

The technique enables analysis of entanglement entropy in complex quantum networks.

Abstract

For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.