Existence Criterion of Solutions to the Inverse Problem of Photocount Statistics Obtained by the Inverse Bernoulli Transform

TL;DR

This paper establishes the conditions under which the inverse Bernoulli transform method can reliably solve the inverse photocount statistics problem, especially for low quantum efficiencies and few-photon light.

Contribution

It introduces a general criterion for the applicability of the inverse Bernoulli transform in photocount statistics and identifies the limitations for quantum efficiency below 0.5.

Findings

Applicability depends on matrix associativity condition.

Critical quantum efficiency $ta_{cr}$ is identified for compound Poisson distribution.

Normalization alone is insufficient for correct solutions.

Abstract

It is shown that the applicability conditions for the inverse Bernoulli transform method for solving the inverse problem of photocount statistics are determined by the fulfillment of the associativity condition for multiplying the matrices included in this transformation. A general criterion for evaluating the photocount distributions $Q_{m}$ in the case of few-photon light, which makes it possible to establish whether the solution to the inverse problem of photocount statistics by inverse Bernoulli transform method is applicable for $\eta<0.5$, is found. As an example of application of the obtained criterion, the critical quantum efficiency $\eta_{cr}$ is found for compound Poisson distribution, below which the solution of the inverse problem of photocount statistics becomes incorrect. Additionally it is shown that the normalization of $Q_{m}$ is not sufficient to obtain a correct solution using the inverse Bernoulli transform.