On orthostochastic, unistochastic and qustochastic matrices

TL;DR

This paper introduces quaternionic analogues of stochastic matrices, Hadamard matrices, and mutually unbiased bases, establishing bounds and classifications in quaternionic Hilbert spaces.

Contribution

It defines qustochastic matrices and quaternionic Hadamard matrices, and determines the maximum number of MUBs in quaternionic spaces, extending classical concepts to quaternionic settings.

Findings

Maximum of 2n+1 MUBs in n-dimensional quaternionic space

All quaternionic Hadamard matrices of size up to 4 classified

Introduction of quaternionic stochastic matrices as quaternionic unistochastic matrices

Abstract

We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by physicists. We also introduce quaternionic Hadamard matrices and quaternionic mutually unbiased bases (MUB). In particular we show that the number of MUB in an n-dimensional quaternionic Hilbert space is at most 2n+1. The bound is attained for n=2. We also determine all quaternionic Hadamard matrices of size at most 4.