Joint quasiprobability distribution on the measurement outcomes of MUB-driven operators

Abstract

We propose a method to define quasiprobability distributions for general spin-j systems of dimension n=2j+1, where n is a prime or power of prime. The method is based on a complete set of orthonormal commuting operators related to Mutually Unbiased Bases which enable (i) a parameterisation of the density matrix and (ii) construction of measurement operators that can be physically realised. As a result we geometrically characterise the set of states for which the quasiprobability distribution is non-negative, and can be viewed as a joint distribution of classical random variables assuming values in a finite set of outcomes. The set is an (n2−1)-dimensional convex polytope with n+1 vertices as the only pure states, nn+1 number of higher dimensional faces, and n3(n+1)/2 edges.