Filter functions in quantum phase-space representations

Abstract

Representation of quantum states in terms of phase-space quasiprobability distributions provides practical tools for identifying features such as the nonclassicality of quantum states. In this setting, filter functions are commonly used to regularize or smooth the phase-space quasiprobability distributions, in particular, the Glauber-Sudarshan P -function. We show that the quantum map associated with a filter function is completely positive and trace-preserving if and only if the Fourier transform of the filter function is a probability density distribution. In this case, filtering the quasiprobability distributions of a quantum state can be viewed as applying a random displacement operation on the quantum state according to the Fourier transform of the filter function. We derive a lower bound on the fidelity between the input and output states of a quantum filtering map. We illustrate several examples of filter functions corresponding to physical and nonphysical maps, in particular, a class of positive but not completely positive maps. We also discuss interesting applications of our results in estimating the output state of unknown quantum channels and estimating the outcome probabilities of quantum measurements.