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

Reconstructing the density operator by using generalized field quadratures

Giacomo Mauro D’Ariano, Stefano Mancini, Paolo Tombesi

Published Jun 29, 1996• 93 citations
OpenAlex

Abstract

The Wigner function for one and two-mode quantum systems is explicitely expressed in terms of the marginal distribution for the generic linearly transformed quadratures. Then, also the density operator of those systems is written in terms of the marginal distribution of these quadratures. Some examples to apply this formalism, and a reduction to the usual optical homodyne tomography are considered.

Related papers

Reconstructing the density operator by using generalized field quadratures

G M D'Ariano, S Mancini, V I Man'ko +1 more

The Wigner function for one- and two-mode quantum systems is explicitly expressed in terms of the marginal distribution for the generic linearly transformed quadratures. Then, the density operator of those systems is also written in terms of the marginal distribution of these quadratures. Some applications and a reduction to the usual optical homodyne tomography are considered.

arXiv:quant-ph/9606034•Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Oct 1, 1996•106 citations

Measurement of the Wigner distribution and the density matrix of a light mode by using optical homodyne tomography: application to squeezed states and vacuum

D. T. Smithey, M. Beck, Adel Faridani +1 more

We report measurements of the Wigner distribution and the density matrix of an electromagnetic field mode, for both vacuum and quadrature-squeezed states. As proposed by Vogel and Risken,1 we obtained the Wigner distribution by tomographic inversion of a set of probability distributions of field-quadrature amplitudes measured by using balanced homodyne detection. A Fourier transform of the Wiener function yields the density matrix, which, according to the standard interpretation of quantum mechanics, contains all knowable information about a given quantum system.

Quantum Electronics and Laser Science Conference•May 2, 1993•1 citation

Wigner function and probability distribution for shifted and squeezed quadratures

Stefano Mancini, Vladimir I. Man’ko, P. Tombesi

The probability distribution for rotated, squeezed and shifted quadratures is shown to be expressed in terms of the Wigner function (as well as in terms of the Q-function and density operator in the coordinate representation). The inverse transformation generalizing the homodyne detection formula is obtained.

Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Aug 1, 1995•195 citations

On tomographic reconstruction of the quantum state of a two-mode light field

Th. Richter

We consider the state reconstruction of an optical two-mode light field from sum quadrature distributions measured with a single balanced homodyne detector. New explicit formulas for the pattern functions necessary to reconstruct the density matrix of the two-mode field in the photon-number basis are derived. Moreover, an expression of the measured quadature distribution in terms of the two-mode normally ordered moments is given and the determination of the moments from it is discussed.

Journal of Modern Optics•Nov 1, 1997•14 citations

Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum

D. T. Smithey, M. Beck, Michael G. Raymer +1 more

We have measured probability distributions of quadrature-field amplitude for both vacuum and quadrature-squeezed states of a mode of the electromagnetic field. From these measurements we demonstrate the technique of optical homodyne tomography to determine the Wigner distribution and the density matrix of the mode. This provides a complete quantum mechanical characterization of the measured mode.

Physical Review Letters•Mar 1, 1993•1,538 citations

Realistic pattern functions for optical homodyne tomography and determination of specific expectation values

Th. Richter

Within the framework of a unified approach, we derive several identical representations of the realistic (and ideal) pattern functions ${f}_{k,l}$ necessary to reconstruct the density-matrix element ${\ensuremath{\varrho}}_{\mathrm{kl}}$ in the Fock basis from realistic quadrature distributions measured with a nonideal balanced homodyne detector of overall efficiency $\ensuremath{\eta}.$ To this end we first establish a biorthogonality relation for the pattern functions from which the form of the Fourier-transformed pattern functions can be easily read off. Then a different handling of the inverse Fourier transform yields several identical representations for the desired pattern functions provided that $\ensuremath{\eta}>1/2.$ In particular, we derive a representation of the realistic pattern functions in terms of a finite weighted sum over scaled ideal ones. Compared to a previously given form, this expression has a nice analytical structure, reveals some general features of the realistic pattern functions, and is particularly well suited for their numerical computation. From this representation sum relations for the pattern functions follow. With their help we show that both a smoothed Wigner function and a generalized moment generating function can be directly sampled from realistic quadrature distributions. The corresponding sampling functions are basically shifted and/or scaled versions of the ideal pattern functions ${f}_{0,0}$ and ${f}_{0,k},$ respectively. As a by-product we derive the sampling functions needed to reconstruct the density matrix from the Fourier-transformed realistic quadrature distribution, i.e., from the s-ordered characteristic function with $s=\ensuremath{-}(1\ensuremath{-}\ensuremath{\eta})/\ensuremath{\eta}.$

Physical Review A•May 17, 2000•20 citations
The Wigner function for one and two-mode quantum systems is explicitely expressed in terms of the marginal distribution for the generic linearly transformed quadratures. Then, also the density operator of those systems is written in terms of the marginal distribution of these quadratures. Some examples to apply this formalism, and a reduction to the usual optical homodyne tomography are considered.
We report measurements of the Wigner distribution and the density matrix of an electromagnetic field mode, for both vacuum and quadrature-squeezed states. As proposed by Vogel and Risken,1 we obtained the Wigner distribution by tomographic inversion of a set of probability distributions of field-quadrature amplitudes measured by using balanced homodyne detection. A Fourier transform of the Wiener function yields the density matrix, which, according to the standard interpretation of quantum mechanics, contains all knowable information about a given quantum system.
Within the framework of a unified approach, we derive several identical representations of the realistic (and ideal) pattern functions fk,l{f}_{k,l}fk,l​ necessary to reconstruct the density-matrix element ${\ensuremath{\varrho}}_{\mathrm{kl}}$ in the Fock basis from realistic quadrature distributions measured with a nonideal balanced homodyne detector of overall efficiency $\ensuremath{\eta}.$ To this end we first establish a biorthogonality relation for the pattern functions from which the form of the Fourier-transformed pattern functions can be easily read off. Then a different handling of the inverse Fourier transform yields several identical representations for the desired pattern functions provided that $\ensuremath{\eta}>1/2.$ In particular, we derive a representation of the realistic pattern functions in terms of a finite weighted sum over scaled ideal ones. Compared to a previously given form, this expression has a nice analytical structure, reveals some general features of the realistic pattern functions, and is particularly well suited for their numerical computation. From this representation sum relations for the pattern functions follow. With their help we show that both a smoothed Wigner function and a generalized moment generating function can be directly sampled from realistic quadrature distributions. The corresponding sampling functions are basically shifted and/or scaled versions of the ideal pattern functions f0,0{f}_{0,0}f0,0​ and f0,k,{f}_{0,k},f0,k​, respectively. As a by-product we derive the sampling functions needed to reconstruct the density matrix from the Fourier-transformed realistic quadrature distribution, i.e., from the s-ordered characteristic function with $s=\ensuremath{-}(1\ensuremath{-}\ensuremath{\eta})/\ensuremath{\eta}.$