Generalized unistochastic matrices
- URL: http://arxiv.org/abs/2310.03436v2
- Date: Mon, 13 Nov 2023 20:30:07 GMT
- Title: Generalized unistochastic matrices
- Authors: Ion Nechita, Zikun Ouyang, Anna Szczepanek
- Abstract summary: We measure a class of bistochastic matrices generalizing unistochastic matrices.
We show that the generalized unistochastic matrices is the whole Birkhoff polytope.
- Score: 0.4604003661048266
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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