  A prepare-and-measure scenario is naturally described by a communication
matrix which collects all conditional outcome probabilities of the scenario
into a row-stochastic matrix. The set of all possible communication matrices is
partially ordered via the ultraweak matrix majorization relation. By
considering maximal elements in this preorder for a subset of matrices
implementable in a given theory, it becomes possible to identify communication
matrices of maximum utility, i.e., matrices that are not majorized by any other
matrices in the theory. The identity matrix of an appropriate size is the
greatest element in classical theories, while the maximal elements in quantum
theory have remain unknown. We completely characterize the maximal elements in
quantum theory. In particular, we show that the identity matrix is the only
maximal element in quantum theory but, as opposed to a classical theory, it is
not the greatest element. Quantum theory can hence be seen to be distinct from
classical theory by the existence of incompatible communication matrices.
