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Paper   IPM / P / 17212
School of Physics
  Title:   Filter functions for the Glauber-Sudarshan $P$-function regularization
1.  M. Zartab
2.  E. Shojaee
3.  S. Rahimi-Keshari
  Status:   Published
  Journal: Phys. Rep.
  Vol.:  107
  Year:  2023
  Pages:   053706
  Supported by:  IPM
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.

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