Generating random density matrices

TL;DR

This paper explores methods for generating ensembles of random density matrices, analyzing their statistical properties and entanglement distributions, including new ensembles related to maximally entangled states and the Fuss-Catalan law.

Contribution

It introduces structured ensembles of random pure states invariant under local unitaries and derives their entanglement properties, extending standard random state models.

Findings

Distribution of Schmidt coefficients for superpositions of maximally entangled states

Fuss-Catalan law describes entanglement distribution after selective measurements

Asymptotic behavior of structured ensembles relates to Marchenko-Pastur distribution

Abstract

We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi--partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N \to \infty, by the Marchenko-Pastur distribution.