Filter functions for the Glauber-Sudarshan $P$-function regularization

TL;DR

This paper investigates filter functions used to regularize the Glauber-Sudarshan P function in quantum optics, establishing conditions for physical realizability and demonstrating that any quantum state can be approximated with a regularized P function.

Contribution

It provides a complete characterization of physically realizable filter functions and shows that any quantum state can be approximated by states with regularized P functions.

Findings

Quantum map associated with a filter function is completely positive and trace preserving if its Fourier transform is a probability density.

A lower bound on the fidelity between input and output states of a quantum filtering map is derived.

Any quantum state can be approximated arbitrarily closely by a state with a regularized P function.

Abstract

The phase-space quasi-probability distribution formalism for representing quantum states provides practical tools for various applications in quantum optics such as identifying the nonclassicality of quantum states. We study filter functions that are introduced to regularize the Glauber-Sudarshan $P$ function. We show that the quantum map associated with a filter function is completely positive and trace preserving and hence physically realizable if and only if the Fourier transform of this function is a probability density distribution. We also derive a lower bound on the fidelity between the input and output states of a physical quantum filtering map. Therefore, based on these results, we show that any quantum state can be approximated, to arbitrary accuracy, by a quantum state with a regular Glauber-Sudarshan $P$ function. We propose applications of our results for estimating the output state of an unknown quantum process and estimating the outcome probabilities of quantum measurements.