Extremal spacings between eigenphases of random unitary matrices and their tensor products

TL;DR

This paper investigates the extremal eigenphase spacings of random unitary matrices, deriving explicit distributions for small sizes and analyzing their behavior in tensor product ensembles as the system size grows, revealing deviations from Poisson statistics.

Contribution

It provides explicit probability distributions for minimal eigenphase spacings in small ensembles and analyzes the asymptotic behavior of extremal spacings in tensor product ensembles of random unitaries.

Findings

Explicit distributions for minimal spacings at N=4

Asymptotic Poissonian behavior of nearest neighbor distributions

Deviations from Poisson statistics in extremal spacings for large systems

Abstract

Extremal spacings between eigenvalues of random unitary matrices of size N pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for N = 4. We study ensembles of tensor product of k random unitary matrices of size n which describe independent evolution of a composite quantum system consisting of k subsystems. In the asymptotic case, as the total dimension N = n^k becomes large, the nearest neighbor distribution P(s) becomes Poissonian, but statistics of extreme spacings P(s_min) and P(s_max) reveal certain deviations from the Poissonian behavior.