Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace

TL;DR

This paper derives explicit formulas for the smallest eigenvalue distribution in Wishart-Laguerre ensembles with fixed trace, relevant for quantum entanglement, and demonstrates universality in the large-N limit.

Contribution

It provides new explicit expressions and proves universality for the smallest eigenvalue distribution in fixed-trace Wishart-Laguerre ensembles, extending previous results.

Findings

Explicit formulas for fixed-trace orthogonal WL ensemble eigenvalues.

Universality of distributions in large-N limit after rescaling.

Equivalence of hypergeometric function results to Pfaffian/determinant forms.

Abstract

The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt eigenvalues. For a bipartition of size M\geq N, these are distributed according to a Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a fixed-trace constraint. We first compute the distribution and moments of the smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary M\geq N. Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large-N limit with M-N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.