Algebraic and geometric structures inside the Birkhoff polytope

TL;DR

This paper explores the algebraic and geometric properties of the Birkhoff polytope and its subsets, introducing bracelet matrices and analyzing unistochastic matrices, especially circulant ones, with results on their structure and spectra.

Contribution

It introduces the set of bracelet matrices within the Birkhoff polytope, proves their properties, and fully characterizes circulant unistochastic matrices for small dimensions.

Findings

Bracelet matrices contain factorisable bistochastic matrices.

Both bracelet and factorisable matrices are star-shaped around the flat matrix.

Spectra of circulant unistochastic matrices lie inside d-hypocycloids.

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 $d\leq 4$, and prove that such matrices form a monoid for $d=3$.