  The well known metrological linear squeezing parameters (such as quadrature
or spin squeezing) efficiently quantify the sensitivity of Gaussian states.
Yet, these parameters are insufficient to characterize the much wider class of
highly sensitive non-Gaussian states. Here, we introduce a class of
metrological nonlinear squeezing parameters obtained by analytical optimization
of measurement observables among a given set of accessible (possibly nonlinear)
operators. This allows for the metrological characterization of non-Gaussian
quantum states of discrete and continuous variables. Our results lead to
optimized and experimentally-feasible recipes for high-precision moment-based
estimation of a phase parameter and can be used to systematically construct
multipartite entanglement and non-classicality witnesses for complex quantum
states.
