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

Marek Smaczynski, Tomasz Tkocz, Marek Kus, Karol Zyczkowski

Published 2012-04-13, updated 2013-06-11Version 5

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.