Algebraic and geometric structures inside the Birkhoff polytope

• URL: http://arxiv.org/abs/2101.11288v1
• Date: Wed, 27 Jan 2021 09:51:24 GMT
• Title: Algebraic and geometric structures inside the Birkhoff polytope
• Authors: Grzegorz Rajchel-Mieldzio\'c, Kamil Korzekwa, Zbigniew Pucha{\l}a and Karol \.Zyczkowski
• Abstract summary: Birkhoff polytope $mathcalB_d$ consists of all bistochastic matrices of order $d$. We prove that $mathcalL_d$ and $mathcalF_d$ are star-shaped with respect to the flat matrix.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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