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 $η<0.5$ , is found. As an example of application of the obtained criterion, the critical quantum efficiency $η_{cr}$ 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.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 $η<0.5$, is found. As an example of application of the obtained criterion, the critical quantum efficiency $η_{cr}$ 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.

arXiv:2110.04918Oct 10, 20210 citations

Statistical problems in quantum physics

Alexander Semenovich Holevo

Lecture notes in mathematicsJan 1, 1973105 citations

Inverse problem of photocount statistics: Applicability criterion for the inverse Bernoulli transform method

P. P. Gostev, Sergey A Magnitskiy, A. S. Chirkin

It is shown that the applicability conditions for the inverse Bernoulli transform method to solve 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 the inverse Bernoulli transform method is applicable for $\ensuremath{\eta}<0.5$, is found. As an example of the application of the obtained criterion, the critical quantum efficiency ${\ensuremath{\eta}}_{\text{cr}}$ 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.It is shown that the applicability conditions for the inverse Bernoulli transform method to solve 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 the inverse Bernoulli transform method is applicable for $\ensuremath{\eta}<0.5$, is found. As an example of the application of the obtained criterion, the critical quantum efficiency ${\ensuremath{\eta}}_{\text{cr}}$ 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.

Physical Review AApr 24, 20234 citations

Stability of solutions to the inverse problem of photocount statistics obtained by the inverse Bernoulli transform.

P. P. Gostev, Sergey A Magnitskiy, A. S. Chirkin

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 stable 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 unstable. It is also shown that the normalization of $Q_{m}$ is not sufficient to obtain a correct solution using the inverse Bernoulli transform.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 stable 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 unstable. It is also shown that the normalization of $Q_{m}$ is not sufficient to obtain a correct solution using the inverse Bernoulli transform.

Oct 10, 20210 citations

Problems in Quantum mechanics and quantum statistics

Vladimir V. Ulyanov

Jun 1, 19803 citations

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

η<0.5

, is found. As an example of application of the obtained criterion, the critical quantum efficiency

η_{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.

{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 the inverse Bernoulli transform method is applicable for $\ensuremath{\eta}<0.5$, is found. As an example of the application of the obtained criterion, the critical quantum efficiency ${\ensuremath{\eta}}_{\text{cr}}$ 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.

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 stable 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 unstable. It is also shown that the normalization of

Q_{m}

is not sufficient to obtain a correct solution using the inverse Bernoulli transform.