Algebraic and geometric structures inside the Birkhoff polytope Available to Purchase

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 particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantized. 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 factorizable 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 analyze 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 characterize the set of circulant unistochastic matrices of order d ≤ 4 and prove that such matrices form a monoid for d = 3.