From the Wigner function to the $s$-ordered phase-space distribution via a Gaussian noise channel

Various phase-space distributions, from the celebrated Wigner function, to the Husimi $Q$ function and the Glauber–Sudarshan $P$ distribution, have played an interesting and important role in the phase-space formulation of quantum mechanics in general, and quantum optics in particular. A unified approach to all these distributions based on the notion of the $s$-ordered phase-space distribution was introduced by Cahill and Glauber. With the intention of illuminating the physical meaning of the parameter $s$, we interpret the $s$-ordered phase-space distribution as the Wigner function of a state under the Gaussian noise channel, and thus reveal an intrinsic connection between the $s$-ordered phase-space distribution and the Gaussian noise channel, which yields a physical insight into the $s$-ordered phase-space distribution. In this connection, the parameter $-s/2$ (rather than the original $s$) acquires the role of the noise occurring in the Gaussian noise channel. An alternative representation of the Gaussian noise channel as the scaling-measurement preparation in a coherent states is illuminated. Furthermore, by exploiting the freedom in the parameter $s$, we introduce a computable and experimentally testable quantifier for optical nonclassicality, reveal its basic properties, and illustrate it by typical examples. A simple and convenient criterion for optical nonclassicality in terms of the $s$-ordered phase-space distribution is derived.