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 = 2 j + 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 ( n 2 − 1 ) -dimensional convex polytope with n + 1 vertices as the only pure states, n n + 1 number of higher dimensional faces, and n 3 ( n + 1 ) / 2 edges.