Determination of the Schmidt number

TL;DR

This paper develops optimized conditions for identifying the Schmidt number in quantum states using Hermitian operators, providing a mathematical framework applicable to mixed states and demonstrating solutions in continuous variable systems.

Contribution

It introduces necessary and sufficient conditions for Schmidt number identification using Hermitian operators, extending previous methods to mixed states and continuous variable systems.

Findings

Derived equations similar to eigenvalue problems for Schmidt number detection

Solved these equations for specific classes of operators

Applied solutions to phase randomized two-mode squeezed-vacuum states

Abstract

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.