Determination of the Schmidt number

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.