Random unistochastic matrices

Abstract

An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U (N ) (or O(N ), respectively). An ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble. We study the distribution of complex eigenvalues of bistochastic, unistochastic and orthostochastic matrices in the complex plane. We compute averages (entropy, traces) over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices.

Time evolution of Wikipedia network ranking

“…The spectrum for all Wikipedia time slots is shown in Figure 10 for G and in Figure 11 pronounced star structure which can be recognized as a composition of triplet and quadruplet leaves (triangle and cross). This structure is very similar to those found in random unistochastic and orthostochastic matrices of size N = 3 and 4 [24] (see Fig. 4 therein).…”

Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4

“…4). An amusing side remark is that the boundary of the unistochastic set in this picture is a 3-hypocycloid [19]. It can be obtained by rolling a circle of radius 1/3 inside the unit circle.…”

“…For some of its properties consult S lomczyński [26] et al [19]. When N = 2 there are just two permutation matrices and B 2 is a line segment between these two points.…”

Quantum state discrimination: A geometric approach

“…Hence matrix B is by construction unistochastic [25] and thus bistochastic. It is convenient to introduce at this place two non-normalised vectors, |ψ := p ′ i |µ i , and |φ := q ′ j |µ j , where p ′ i = p s i and q ′ j = q t j are non-negative.…”