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

Journal of Modern Optics

Direct sampling of density matrix in displaced Fock-state basis from quadrature distributions and reconstruction of quasiprobability distributions

Th. Richter

Published Jul 1, 1999• 6 citations
Publisher

Abstract

The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions fmn used to reconstruct the density matrix elements Q mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f 00.

DOI
10.1080/09500349908231327

Related papers

Direct sampling of density matrix in displaced Fock-state basis from quadrature distributions and reconstruction of quasiprobability distributions

Th. Richter

The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions fmn used to reconstruct the density matrix elements Q mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f 00.

Journal of Modern Optics•Jul 10, 1999•3 citations

Letter Compensation of detection losses in quantum state reconstruction

Th. Richter

We present a new and quite simple representation of the realistic pattern functions needed to reconstruct the density matrix in the Fock basis from quadrature distributions measured with non-ideal detectors. We show that they can be expressed as a finite weighted sum over the ideal pattern functions of scaled argument. Moreover we propose an algorithm which takes the detection loss-affected density matrix elements and produces the Wigner function (s-parametrized quasiprobability) of the signal field.

Journal of Modern Optics•Jan 1, 2001•2 citations

Direct sampling of a smoothed Wigner function from quadrature distributions

Th. Richter

We propose the reconstruction of a smoothed Wigner function from quadrature distributions instead of the Wigner function itself to avoid the problems with the inverse Radon transformation. Starting from the well known relation between s -parametrized quasiprobability distributions W ( x , p , s ) and quadrature component distributions w ( x , ), we show that the sampling function needed to determine a smoothed Wigner function W ( x , p ,-| s |) is basically a shifted and scaled version of the well known pattern function necessary to reconstruct the probability of finding no photon in the signal field. An analogous result has been recently found in the literature. Here we obtain this result in a very concise and alternative way.

Journal of Optics B Quantum and Semiclassical Optics•Oct 29, 1999•11 citations

Determination ofs-ordered moments and moment generating function from quadrature distributions

T. Richter

The sampling functions needed to reconstruct from quadrature distributions both the s-ordered moments of the field and the generalized moment generating function are determined. The sampling function in the latter case turns out to be surprisingly simple. It is proportional to the well-known pattern function f 0k needed to reconstruct from homodyne data the density matrix element e 0k in the Fock basis. As a by-product a useful sum relation for the pattern functions f mn is found.

Journal of Modern Optics•Dec 1, 1999•2 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

Radon transform and pattern functions in quantum tomography

Alfred Wünsche

The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.

arXiv:quant-ph/9706024•Journal of Modern Optics•Nov 1, 1997•55 citations
The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions fmn used to reconstruct the density matrix elements Q mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f 00.
The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions fmn used to reconstruct the density matrix elements Q mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f 00.
We present a new and quite simple representation of the realistic pattern functions needed to reconstruct the density matrix in the Fock basis from quadrature distributions measured with non-ideal detectors. We show that they can be expressed as a finite weighted sum over the ideal pattern functions of scaled argument. Moreover we propose an algorithm which takes the detection loss-affected density matrix elements and produces the Wigner function (s-parametrized quasiprobability) of the signal field.
We propose the reconstruction of a smoothed Wigner function from quadrature distributions instead of the Wigner function itself to avoid the problems with the inverse Radon transformation. Starting from the well known relation between s -parametrized quasiprobability distributions W ( x , p , s ) and quadrature component distributions w ( x , ), we show that the sampling function needed to determine a smoothed Wigner function W ( x , p ,-| s |) is basically a shifted and scaled version of the well known pattern function necessary to reconstruct the probability of finding no photon in the signal field. An analogous result has been recently found in the literature. Here we obtain this result in a very concise and alternative way.
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 $s=\ensuremath{-}(1\ensuremath{-}\ensuremath{\eta})/\ensuremath{\eta}.$
The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.