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arXiv:2101.11288 · Journal of Mathematical Physics

Algebraic and geometric structures inside the Birkhoff polytope

Grzegorz Rajchel-Mieldzioć, Kamil Korzekwa, Zbigniew Puchała, Karol Życzkowski

Published Jan 1, 2022• 2 citations
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Abstract

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.

DOI
10.1063/5.0046581

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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.
A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that Bij=|Uij|2 for i,j=1,…,N. The set U3 of all unistochastic matrices of order N=3 forms a proper subset of the Birkhoff polytope, which contains all bistochastic (doubly stochastic) matrices. We compute the volume of the set U3 with respect to the flat (Lebesgue) measure and analytically evaluate the mean entropy of an unistochastic matrix of this order. We also analyze the Jarlskog invariant J, defined for any unitary matrix of order three, and derive its probability distribution for the ensemble of matrices distributed with respect to the Haar measure on U(3) and for the ensemble which generates the flat measure on the set of unistochastic matrices. For both measures the probability of finding |J| smaller than the value observed for the Cabbibo–Kobayashi–Maskawa matrix, which describes the violation of the CP parity, is shown to be small. Similar statistical reasoning may also be applied to the Maki–Nakagawa–Sakata matrix, which plays role in describing the neutrino oscillations. Some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions.
We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a 2-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order n exists, if there exists a skew Hadamard matrix or a symmetric conference matrix of this size. This is the case for any even n≤20n\le 20n≤20 , and for these dimensions we demonstrate that a bistochastic matrix B located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix U, such that Bij=∣Uij∣2B_{ij}=|U_{ij}|^2Bij​=∣Uij​∣2 , is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order n×nn \times nn×n . Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis. In the case n=4n=4n=4 we study geometry of the set U4{\mathcal U}_4U4​ of unistochastic matrices, conjecture that this set is star-shaped and estimate its relative volume in the Birkhoff polytope B4{\mathcal B}_4B4​ .
In the present paper we show a link between bistochastic quantum channels and classical maps. The primary goal of this work is to analyse the multiplicative structure of the Birkhoff polytope of order 3 (the simplest nontrivial case). A suitable complex parametrization of the Birkhoff polytope is proposed, which reveals several its symmetries and characteristics, in particular: (i) the structure of Markov semigroups inside the Birkhoff polytope, (ii) the relation between the set of Markov time evolutions, the set of positive definite matrices and the set of divisible matrices. A condition for Markov time evolution of semigroups in the set of symmetric bistochastic matrices is derived, which leads to an universal conserved quantity for all Markov evolutions. Finally, the complex parametrization is extended to the Birkhoff polytope of order 4.
An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U ( N ) (or O ( N )). An ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble. We study the distribution of complex eigenvalues of bistochastic, unistochastic and orthostochastic matrices in the complex plane. We compute averages (entropy, traces) over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices.
Abstract We demonstrate that a complex equiangular tight frame composed of N vectors in dimension d , denoted ETF ( d , N ), exists if and only if a certain bistochastic matrix, univocally determined by N and d , belongs to a special class of unistochastic matrices. This connection allows us to find new complex ETFs in infinitely many dimensions and to derive a method to introduce non-trivial free parameters in ETFs. We present an explicit six-parametric family of complex ETF(6,16), which defines a family of symmetric POVMs. Minimal and maximal possible average entanglement of the vectors within this qubit–qutrit family are described. Furthermore, we propose an efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix, which we apply to find all existing classes of complex ETFs containing up to 20 vectors.
The H-unistochastic matrices are a special class of symmetric bistochastic matrices obtained by taking the square of the absolute value of each entry of a Hermitian unitary matrix. We examine the geometric relationship of the convex hull of the n by n H-unistochastic matrices relative to the larger convex set of n by n symmetric bistochastic matrices. We show that any line segment in the convex set of the n by n symmetric bistochastic matrices which passes through the centroid of this convex set must spend at least two-thirds of its length in the convex hull of the n by n H-unistochastic matrices when n is either three or four and we prove a partial result for higher n. A class of completely positive linear maps called the self-dual doubly stochastic maps is useful for studying this problem. Some results on self-dual doubly stochastic maps are given including a self-dual version of the Laudau-Streater theorem.