  Optimized, necessary and sufficient conditions for the identification of the
Schmidt number will be derived in terms of general Hermitian operators. These
conditions apply to arbitrary mixed quantum states. The optimization procedure
delivers equations similar to the eigenvalue problem of an operator. The
properties of the solution of these equations will be studied. We solve these
equations for classes of operators. The solutions will be applied to phase
randomized two-mode squeezed-vacuum states in continuous variable systems.
