  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.
