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arXiv · 2101.11288

Algebraic and geometric structures inside the Birkhoff polytope

Grzegorz Rajchel-Mieldzioć·Kamil Korzekwa·Zbigniew Puchała·Karol Życzkowski
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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 $d=3$.

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Grzegorz Rajchel-Mieldzioć, Kamil Korzekwa, Zbigniew Puchała, Karol Życzkowski. 2021-01-27. Algebraic and geometric structures inside the Birkhoff polytope. https://doi.org/10.1063/5.0046581

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