Nonlinear squeezing of superpositions of quadrature eigenstates

We introduce a family of operators exploiting the symmetry of superpositions of quadrature eigenstates (SQE) and demonstrate how the associated nonlinear squeezing, quantified by the expectation value of such operators, serves both as a witness of non-Gaussianity and as an indicator of the quality of SQE approximations. To establish the usefulness of this measure, we connect it to quantum state fidelity and evaluate its implications in breeding protocols. Finally, we construct optimal approximations of SQE states in truncated Fock spaces.