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From the Wigner function to the $s$-ordered phase-space distribution via a Gaussian noise channel
Yue Zhangab,
Shunlong Luocd a Beijing Academy of Quantum Information Sciences, Beijing, China
b State Key Laboratory of Mesoscopic Physics, School of Physics, Frontiers Science Center
for Nano-optoelectronics, Peking University, Beijing, China
c Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
d School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing, China
Abstract:
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.
Keywords:
Wigner function, $s$-ordered phase-space distribution, Gaussian noise channel, noise parameter, nonclassicality.
PACS:
03.65.Ta,
03.67.-a,
Received: 20.08.2021
Revised: 26.10.2021
DOI:
10.4213/tmf10164
Fulltext:
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