  Squeezed states of the harmonic oscillator are a common resource in
applications of quantum technology. If the noise is suppressed in a nonlinear
combination of quadrature operators below threshold for all possible
up-to-quadratic Hamiltonians, the quantum states are non-Gaussian and we refer
to the noise reduction as nonlinear squeezing. Non-Gaussian aspects of quantum
states are often more vulnerable to decoherence due to imperfections appearing
in realistic experimental implementations. Therefore, a stability of nonlinear
squeezing is essential. We analyze the behavior of quantum states with cubic
nonlinear squeezing under loss and dephasing. The properties of nonlinear
squeezed states depend on their initial parameters which can be optimized and
adjusted to achieve the maximal robustness for the potential applications.
