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:quant-ph/9701010 · Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Systematic and statistical errors in homodyne measurements of the density matrix

Giacomo Mauro D’Ariano, Chiara Macchiavello, Nicoletta Sterpi

Published Dec 1, 1997• 20 citations
arXivHTMLPDF

Abstract

We study both systematic and statistical errors in radiation density matrix measurements. First we estimate the minimum number of scanning phases needed to reduce systematic errors to below a fixed threshold. Then, we calculate the statistical errors, intrinsic in the procedure that gives the density matrix. We present a detailed study of such errors versus the detectors' quantum efficiency and the matrix indices in the number representation, for different radiation states. For unit quantum efficiency, and for both coherent and squeezed states, the statistical errors of the diagonal matrix elements saturate for large n. In contrast, off-diagonal errors increase with the distance from the diagonal. For non-unit quantum efficiency the statistical errors along the diagonal do not saturate and increase dramatically versus both and the matrix indices.

DOI
10.1088/1355-5111/9/6/006

Related papers

Accuracy in Quantum Homodyne Tomography

Giacomo Mauro D’Ariano, Chiara Macchiavello, Nicoletta Sterpi

We study the statistical errors in homodyne tomography of radiation density matrix in the photon number representation. We give an asymptotic estimate for large matrix indexes at different values of the quantum efficiency η of homodyne detectors. We show that for fixed η the errors increase exponentially as functions of the matrix index.

Kluwer Academic Publishers eBooks•Dec 19, 2005•0 citations

Robustness of homodyne tomography to phase-insensitive noise

Giacomo Mauro D’Ariano, Nicoletta Sterpi

We study the effects of phase-insensitive noise on homodyne measurements of a radiation density matrix. We prove that this noise has an effect equivalent to a non-unit quantum efficiency at detectors. The overall effective quantum efficiency η∗ of the measurement is evaluated in terms of the quantum efficiency at detectors and of the average number of noise photons added to the radiation field. For pure Gaussian-displacement noise, we show that half a photon of noise is enough to prevent the homodyne measurement of the density matrix.

Journal of Modern Optics•Nov 1, 1997•7 citations

Homodyne detection of the density matrix of the radiation field

Giacomo Mauro D’Ariano, Ulf Leonhardt, Henryk Paul

We provide a simple analytic relation that connects the density operator \ensuremath{\rho}^ of the electromagnetic field with the tomographic homodyne probabilities for generic quantum efficiency \ensuremath{\eta} of detectors. The problem of experimentally ``sampling'' a general matrix element 〈\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\Phi}〉 is addressed in the statistically rigorous sense of the central-limit theorem. We show that experimental sampling is possible also for nonunit efficiency, provided that \ensuremath{\eta} satisfies a lower bound related to the ``resolutions'' of vectors \ensuremath{\Vert}\ensuremath{\psi}〉 and \ensuremath{\Vert}\ensuremath{\Phi}〉 in the quadrature representations. For coherent and number states the bound is \ensuremath{\eta}\ensuremath{\gtrsim}1/2. On the basis of computer-simulated experiments we show the feasibility of detecting delicate quantum probability oscillations, which otherwise would be smeared out by inefficient detection.

Physical Review A•Sep 1, 1995•189 citations

State estimation in quantum homodyne tomography with noisy data

Jean-Marie Aubry, Cristina Butucea, Katia Méziani

In the framework of noisy quantum homodyne tomography with efficiency parameter 0 < η ⩽ 1, we propose two estimators of a quantum state whose density matrix elements ρ m , n decrease as , for fixed known B > 0 and 0 < r ⩽ 2. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for 0 < η ⩽ 1/2), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in risk.

arXiv:0804.2434•Inverse Problems•Nov 13, 2008•13 citations

Tomographic measurement of the density matrix of the radiation field

Giacomo Mauro D’Ariano

Recently a novel homodine tomographic technique has been proposed which allows us to detect the density matrix experimentally in terms of averages on data. The method has been further extended to a general matrix element ( psi mod Q mod phi ) of the density operator Q, along with the possibility of using low-efficiency detectors. In this paper this technique is reconsidered as a new genuine quantum measurement. Numerical results for measured probabilities are given, and the mechanism underlying the generation of statistical errors in the measured matrix is illustrated. It is shown that the data processing of the novel technique can also be used as a new imaging algorithm for conventional tomography.

Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Aug 1, 1995•28 citations

Technique for photon statistics reconstruction by using on/off detectors.

Marco Gramegna, Marco Genovese, G. Brida +2 more

Report on first experimental applications of a measurement scheme with variable quantum efficiency single-photon ON/OFF detectors for evaluating the diagonal elements of the density matrix for various quantum optical states, without involving photon counting.

Quantum Electronics and Laser Science Conference•May 1, 2007•0 citations
We study both systematic and statistical errors in radiation density matrix measurements. First we estimate the minimum number of scanning phases needed to reduce systematic errors to below a fixed threshold. Then, we calculate the statistical errors, intrinsic in the procedure that gives the density matrix. We present a detailed study of such errors versus the detectors' quantum efficiency and the matrix indices in the number representation, for different radiation states. For unit quantum efficiency, and for both coherent and squeezed states, the statistical errors of the diagonal matrix elements saturate for large n. In contrast, off-diagonal errors increase with the distance from the diagonal. For non-unit quantum efficiency the statistical errors along the diagonal do not saturate and increase dramatically versus both and the matrix indices.
We study the effects of phase-insensitive noise on homodyne measurements of a radiation density matrix. We prove that this noise has an effect equivalent to a non-unit quantum efficiency at detectors. The overall effective quantum efficiency η∗ of the measurement is evaluated in terms of the quantum efficiency at detectors and of the average number of noise photons added to the radiation field. For pure Gaussian-displacement noise, we show that half a photon of noise is enough to prevent the homodyne measurement of the density matrix.
We provide a simple analytic relation that connects the density operator \ensuremath{\rho}^ of the electromagnetic field with the tomographic homodyne probabilities for generic quantum efficiency \ensuremath{\eta} of detectors. The problem of experimentally ``sampling'' a general matrix element 〈\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\Phi}〉 is addressed in the statistically rigorous sense of the central-limit theorem. We show that experimental sampling is possible also for nonunit efficiency, provided that \ensuremath{\eta} satisfies a lower bound related to the ``resolutions'' of vectors \ensuremath{\Vert}\ensuremath{\psi}〉 and \ensuremath{\Vert}\ensuremath{\Phi}〉 in the quadrature representations. For coherent and number states the bound is \ensuremath{\eta}\ensuremath{\gtrsim}1/2. On the basis of computer-simulated experiments we show the feasibility of detecting delicate quantum probability oscillations, which otherwise would be smeared out by inefficient detection.
In the framework of noisy quantum homodyne tomography with efficiency parameter 0 < η ⩽ 1, we propose two estimators of a quantum state whose density matrix elements ρ m , n decrease as , for fixed known B > 0 and 0 < r ⩽ 2. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for 0 < η ⩽ 1/2), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in risk.
Recently a novel homodine tomographic technique has been proposed which allows us to detect the density matrix experimentally in terms of averages on data. The method has been further extended to a general matrix element ( psi mod Q mod phi ) of the density operator Q, along with the possibility of using low-efficiency detectors. In this paper this technique is reconsidered as a new genuine quantum measurement. Numerical results for measured probabilities are given, and the mechanism underlying the generation of statistical errors in the measured matrix is illustrated. It is shown that the data processing of the novel technique can also be used as a new imaging algorithm for conventional tomography.