Operator system characterizations of SIC-POVMs and mutually unbiased bases

We show that a symmetric informationally-complete positive operator-valued measure exists in a given dimension $d$ if and only if there exists a $d^2$-dimensional operator system satisfying certain order-theoretic conditions. We also describe a method of constructing such an operator system and demonstrate that the first step of this construction can be carried out successfully. We obtain analogous results for the existence of $d+1$ mutually unbiased bases in a given dimension.