Algebraic and geometric structures inside the Birkhoff polytope

Abstract

The Birkhoff polytope ${\mathcal{B}}_d$ consisting of all bistochastic matrices of order $d$ assists researchers from many areas, including combinatorics, statistical physics and quantum information. Its subset ${\mathcal{U}}_d$ of unistochastic matrices, determined by squared moduli of unitary matrices, is of a particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantised. In order to investigate the problem of unistochasticity we introduce the set ${\mathcal{L}}_d$ of bracelet matrices that forms a subset of ${\mathcal{B}}_d$, but a superset of ${\mathcal{U}}_d$. We prove that for every dimension $d$ this set contains the set of factorisable bistochastic matrices ${\mathcal{F}}_d$ and is closed under matrix multiplication by elements of ${\mathcal{F}}_d$. Moreover, we prove that both ${\mathcal{L}}_d$ and ${\mathcal{F}}_d$ are star-shaped with respect to the flat matrix. We also analyse the set of $d\times d$ unistochastic matrices arising from circulant unitary matrices, and show that their spectra lie inside $d$-hypocycloids on the complex plane. Finally, applying our results to small dimensions, we fully characterise the set of circulant unistochastic matrices of order $d\le 4$, and prove that such matrices form a monoid for $d=3$.

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Cited by in corpus (1)

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