01608nas a2200157   4500000000100000000000100001008004100002100003200043700001900075700002200094700002200116245006800138856003800206490000700244520119900251       2022                            d1 aGrzegorz Rajchel-Mieldzioć1 aKamil Korzekwa1 aZbigniew Puchała1 aKarol Życzkowski00aAlgebraic and geometric structures inside the Birkhoff polytope  uhttps://doi.org/10.1063/5.00465810 v633 a<p>The Birkhoff polytope Bd consisting of all bistochastic matrices of order d assists researchers from many areas, including combinatorics, statistical physics and quantum information. Its subset Ud 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 Ld of bracelet matrices that forms a subset of Bd , but a superset of Ud . We prove that for every dimension d this set contains the set of factorisable bistochastic matrices Fd and is closed under matrix multiplication by elements of Fd. Moreover, we prove that both Ld and Fd are star-shaped with respect to the flat matrix. We also analyse the set of d × 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 ≤ 4, and prove that such matrices form a monoid for d = 3.</p>