Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:physics/9712043 · Physical Review A

Maximum-likelihood estimation of photon-number distribution from homodyne statistics

Konrad Banaszek

Published Jun 1, 1998• 64 citations
arXivHTMLPDF

Abstract

We present a method for reconstructing the photon-number distribution from the homodyne statistics based on maximization of the likelihood function derived from the exact statistical description of a homodyne experiment. This method incorporates in a natural way the physical constraints on the reconstructed quantities, and the compensation for the nonunit detection efficiency.

DOI
10.1103/physreva.57.5013

Related papers

Reconstruction of photon distribution with positivity constraints

Konrad Banaszek

An iterative algorithm for reconstructing the photon distribution from the random phase homodyne statistics is discussed. This method, derived from the maximum-likelihood approach, yields a positive definite estimate for the photon distribution with bounded statistical errors even if numerical compensation for the detector imperfection is applied.

arXiv:quant-ph/9804050•Apr 22, 1998•2 citations

Quantum homodyne tomography with a priori constraints

Konrad Banaszek

I present an iterative algorithm for reconstructing the Wigner function from homodyne statistics. The proposed method, based on maximum-likelihood estimation, is capable of compensating for detection losses in a numerically stable way.

arXiv:quant-ph/9901064•Physical Review A•Jun 1, 1999•18 citations

Experimental Reconstruction of Photon Statistics without Photon Counting

Guido Zambra, Alessandra Andreoni, Maria Bondani +5 more

Experimental reconstructions of photon number distributions of both continuous-wave and pulsed light beams are reported. Our scheme is based on on/off avalanche photo-detection assisted by maximum-likelihood estimation and does not involve photon counting. Reconstructions of the distribution for both semiclassical and quantum states of light are reported for single-mode as well as for multi-mode beams.

arXiv:quant-ph/0502060•Physical Review Letters•Aug 5, 2005•167 citations

Characterizing photon number statistics using conjugate optical homodyne detection

Bing Qi, Pavel Lougovski, Brian P. Williams

We study the problem of determining the photon number statistics of an unknown quantum state using conjugate optical homodyne detection. We quantify the information gain in a single-shot measurement and show that the photon number statistics can be recovered in repeated measurements on an ensemble of identical input states without scanning the phase of the input state or randomizing the phase of the local oscillator used in homodyne detection. We demonstrate how the expectation maximization algorithm and Bayesian inference can be utilized to facilitate the reconstruction and illustrate our approach by conducting experiments to study the photon number distributions of a weak coherent state and a thermal state source.

arXiv:1702.02558•Optics Express•Jan 6, 2020•30 citations

On/off Detection Method for Reconstructing the Statistics of Quantum Optical States: An Overview

G. Brida, Marco Genovese, Marco Gramegna +3 more

We give a brief overview of the demonstration of a photon statistics reconstruction method based on Maximum Likelihood estimation and on/off detection. This method has been successfully applied to a large number of cases and, recently, extended to a scheme for a full reconstruction of the density matrix. Experimental results concerning the heralded single-photon state and the seeded parametric down-conversion are presented in some details.

International Conference on Quantum, Nano and Micro Technologies (ICQNM)•Feb 1, 2009•0 citations

Tutorial: Maximum likelihood estimation in the context of an optical measurement

Anthony Vella

The method of maximum likelihood estimation (MLE) is a widely used statistical approach for estimating the values of one or more unknown parameters of a probabilistic model based on observed data. In this tutorial, I briefly review the mathematical foundations of MLE, then reformulate the problem for the measurement of a spatially-varying optical intensity distribution. In this context, the detection of each individual photon is treated as a random event, the outcome being the photon's location. A typical measurement consists of many detected photons, which accumulate to form a spatial intensity profile. Here, I show a straightforward derivation for the likelihood function and Fisher information matrix (FIM) associated with a measurement of multiple photons incident on a detector comprised of a discrete array of pixels. An estimate for the parameter(s) of interest may then be obtained by maximizing the likelihood function, while the FIM determines the uncertainty of the estimate. To illustrate these concepts, several simple examples are presented for the one- and two-parameter cases, revealing many interesting properties of the MLE formalism, as well as some practical considerations for optical experiments. Throughout these examples, connections are also drawn to optical applications of quantum weak measurements, including off-null ellipsometry and scatterometry.

arXiv:1806.04503•Jun 8, 2018•4 citations
We present a method for reconstructing the photon-number distribution from the homodyne statistics based on maximization of the likelihood function derived from the exact statistical description of a homodyne experiment. This method incorporates in a natural way the physical constraints on the reconstructed quantities, and the compensation for the nonunit detection efficiency.
We study the problem of determining the photon number statistics of an unknown quantum state using conjugate optical homodyne detection. We quantify the information gain in a single-shot measurement and show that the photon number statistics can be recovered in repeated measurements on an ensemble of identical input states without scanning the phase of the input state or randomizing the phase of the local oscillator used in homodyne detection. We demonstrate how the expectation maximization algorithm and Bayesian inference can be utilized to facilitate the reconstruction and illustrate our approach by conducting experiments to study the photon number distributions of a weak coherent state and a thermal state source.
The method of maximum likelihood estimation (MLE) is a widely used statistical approach for estimating the values of one or more unknown parameters of a probabilistic model based on observed data. In this tutorial, I briefly review the mathematical foundations of MLE, then reformulate the problem for the measurement of a spatially-varying optical intensity distribution. In this context, the detection of each individual photon is treated as a random event, the outcome being the photon's location. A typical measurement consists of many detected photons, which accumulate to form a spatial intensity profile. Here, I show a straightforward derivation for the likelihood function and Fisher information matrix (FIM) associated with a measurement of multiple photons incident on a detector comprised of a discrete array of pixels. An estimate for the parameter(s) of interest may then be obtained by maximizing the likelihood function, while the FIM determines the uncertainty of the estimate. To illustrate these concepts, several simple examples are presented for the one- and two-parameter cases, revealing many interesting properties of the MLE formalism, as well as some practical considerations for optical experiments. Throughout these examples, connections are also drawn to optical applications of quantum weak measurements, including off-null ellipsometry and scatterometry.