Generalized unistochastic matrices

TL;DR

This paper introduces a new class of bistochastic matrices called generalized unistochastic matrices, explores their geometric properties within the Birkhoff polytope, and studies their probabilistic measures derived from unitary groups.

Contribution

It generalizes unistochastic matrices by constructing a broader family from bipartite unitaries, characterizes their structure within the Birkhoff polytope, and analyzes associated probability measures.

Findings

The closure of generalized unistochastic matrices covers the entire Birkhoff polytope.

Different levels of the matrix family have a complex inclusion structure.

Probability measures interpolate between known measures on unistochastic matrices and the van der Waerden matrix.

Abstract

We study a class of bistochastic matrices generalizing unistochastic matrices. Given a complex bipartite unitary operator, we construct a bistochastic matrix having as entries the normalized squares of Frobenius norm of the blocks. We show that the closure of the set of generalized unistochastic matrices is the whole Birkhoff polytope. We characterize the points on the edges of the Birkhoff polytope that belong to a given level of our family of sets, proving that the different (non-convex) levels have a rich inclusion structure. We also study the corresponding generalization of orthostochastic matrices. Finally, we introduce and study the natural probability measures induced on our sets by the Haar measure of the unitary group. These probability measures interpolate between the natural measure on the set of unistochastic matrices and the Dirac measure supported on the van der Waerden matrix.