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/0006023 · Physical Review A

Sampling functions for multimode homodyne tomography with a single local oscillator

Jaromı́r Fiurášek

Published Feb 8, 2001• 8 citations
arXivHTMLPDF

Abstract

We consider multimode homodyne tomography with a single local oscillator and derive sampling functions for multimode s-parametrized quasidistributions and s-ordered moments of creation and annihilation operators. In both cases, loss-compensating sampling functions, applicable to realistic measurement with imperfect detection, are provided. Results of Monte Carlo simulations for a squeezed three-mode state are reported and the feasibility of reconstruction of the three-mode Q function from ${10}^{7}$ sampled data points is demonstrated.

DOI
10.1103/physreva.63.033806

Related papers

Determination of Multimode Moments of Creation and Annihilation Operators via Multimode Balanced Homodyning with a Single Homodyne Detector

Jaromı́r Fiurášek

We investigate reconstruction of moments of creation and annihilation operators of multimode traveling light field from data recorded in multimode balanced homodyne detection with a single homodyne detector. We derive sampling functions whose average over the measured quadrature statistics provides the estimates of the moments. We show that detection losses may be compensated by a simple rescaling of the sampling functions. We also show that all moments of given order can be reconstructed from the statistics of a finite number of quadratures.

Fortschritte der Physik•Oct 1, 2001•1 citation

Universal homodyne tomography with a single local oscillator

Giacomo Mauro D’Ariano, Massimiliano Federico Sacchi, Prem Kumar

We propose a general method for measuring an arbitrary observable of a multimode electromagnetic field using homodyne detection with a single local oscillator. In this method the local oscillator scans over all possible linear combinations of the modes. The case of two modes is analyzed in detail and the feasibility of the measurement is studied on the basis of Monte Carlo simulations. We also provide an application of this method in tomographic testing of the Greenberger-Horne-Zeilinger state.

arXiv:quant-ph/9910085•Physical Review A•Dec 13, 1999•29 citations

Direct measurement of optical quasidistribution functions: Multimode theory and homodyne tests of Bell’s inequalities

Konrad Banaszek, Andrzej Dragan, Krzysztof Wódkiewicz +1 more

We develop a multimode theory of direct homodyne measurements of quantum optical quasidistribution functions. We demonstrate that unbalanced homodyning with appropriately shaped auxiliary coherent fields allows one to sample point-by-point different phase-space representations of the electromagnetic field. Our analysis includes practical factors that are likely to affect the outcome of a realistic experiment, such as nonunit detection efficiency, imperfect mode matching, and dark counts. We apply the developed theory to discuss feasibility of observing a loophole-free violation of Bell's inequalities by measuring joint two-mode quasidistribution functions by photon counting under locality conditions. We determine the range of parameters of the experimental setup that enable violation of Bell's inequalities for two states exhibiting entanglement in the Fock basis: a one-photon Fock state divided by a 50:50 beam splitter, and a two-mode squeezed vacuum state produced in the process of nondegenerate parametric down-conversion.

arXiv:quant-ph/0204103•Physical Review A•Oct 4, 2002•32 citations

Wigner function tomography via optical parametric amplification

Mahmoud Kalash, Maria V. Chekhova

Wigner function tomography is indispensable for characterizing quantum states, but its commonly used version, balanced homodyne detection, suffers from several weaknesses. First, it requires efficient detection, which is critical for measuring fragile non-Gaussian states, especially bright ones. Second, it needs a local oscillator, tailored to match the spatiotemporal properties of the state under test, and fails for multimode and broadband states. Here we propose Wigner function tomography based on optical parametric amplification followed by direct detection. The method is immune to detection inefficiency and loss, and suitable for broadband, spatially and temporally multimode quantum states. To prove the principle, we experimentally reconstruct the Wigner function of squeezed vacuum occupying a single mode of a strongly multimode state. We obtain a squeezing of −7.5±0.4dB and purity of 0.91 −0.08 +0.09 despite more than 97% loss caused mainly by filtering. Theoretically, we also consider the reconstruction of a squeezed single photon—a bright non-Gaussian state. Due to multimode parametric amplification, the method allows for simultaneous tomography of multiple modes. This makes it a powerful tool for optical quantum information processing.

arXiv:2207.10030•Optica•Jul 26, 2023•33 citations

Continuous-variable state moments from randomized homodyne and heterodyne measurements

Michael Tsesmelis, Moritz Straeter, Leong-Chuan Kwek

Continuous-variable (CV) quantum states are naturally characterized by their moments, defined as expectation values of products of single- or multimode ladder operators. Many CV Hamiltonians and quantum algorithms are formulated directly in terms of these moments, and therefore an efficient procedure to estimate moments with limited state measurements is necessary. In this paper, we present a protocol for shadow tomography of moment-generating functions (MGFs) of CV states based on randomized homodyne and heterodyne measurements. The resulting shadows enable an efficient and concurrent estimation of many multimode moments. Our complexity analysis shows that the number of measurements required to estimate a certain moment to a given precision grows exponentially in the order of the moments. Finally, we assess the precision of these moment estimators by applying them to two tasks: detecting entanglement in both Gaussian and non-Gaussian states via the Shchukin-Vogel protocol, and characterizing optical loss in a photonic chip. We demonstrate that both tasks can be accomplished with only a few thousand randomized measurements. The small number of required measurements, combined with efficient sample processing, makes our protocol applicable to a wide range of CV simulation tasks.

quant-pharXiv:2608.11811•Aug 13, 2026•0 citations

Direct sampling of exponential phase moments of smoothed Wigner functions

Jaromı́r Fiurášek

We investigate exponential phase moments of the s-parametrized quasidistributions (smoothed Wigner functions). We show that the knowledge of these moments as functions of s provides, together with photon-number statistics, a complete description of the quantum state. We demonstrate that the exponential phase moments can be directly sampled from the data recorded in balanced homodyne detection and we present simple expressions for the sampling kernels. The phase moments are Fourier coefficients of phase distributions obtained from the quasidistributions via integration over the radial variable in polar coordinates. We performed Monte Carlo simulations of the homodyne detection and we demonstrate the feasibility of direct sampling of the moments and subsequent reconstruction of the phase distribution.

arXiv:quant-ph/0005120•Physical Review A•Jun 16, 2000•2 citations
We consider multimode homodyne tomography with a single local oscillator and derive sampling functions for multimode s-parametrized quasidistributions and s-ordered moments of creation and annihilation operators. In both cases, loss-compensating sampling functions, applicable to realistic measurement with imperfect detection, are provided. Results of Monte Carlo simulations for a squeezed three-mode state are reported and the feasibility of reconstruction of the three-mode Q function from 107{10}^{7}107 sampled data points is demonstrated.
We investigate reconstruction of moments of creation and annihilation operators of multimode traveling light field from data recorded in multimode balanced homodyne detection with a single homodyne detector. We derive sampling functions whose average over the measured quadrature statistics provides the estimates of the moments. We show that detection losses may be compensated by a simple rescaling of the sampling functions. We also show that all moments of given order can be reconstructed from the statistics of a finite number of quadratures.
We develop a multimode theory of direct homodyne measurements of quantum optical quasidistribution functions. We demonstrate that unbalanced homodyning with appropriately shaped auxiliary coherent fields allows one to sample point-by-point different phase-space representations of the electromagnetic field. Our analysis includes practical factors that are likely to affect the outcome of a realistic experiment, such as nonunit detection efficiency, imperfect mode matching, and dark counts. We apply the developed theory to discuss feasibility of observing a loophole-free violation of Bell's inequalities by measuring joint two-mode quasidistribution functions by photon counting under locality conditions. We determine the range of parameters of the experimental setup that enable violation of Bell's inequalities for two states exhibiting entanglement in the Fock basis: a one-photon Fock state divided by a 50:50 beam splitter, and a two-mode squeezed vacuum state produced in the process of nondegenerate parametric down-conversion.
Wigner function tomography is indispensable for characterizing quantum states, but its commonly used version, balanced homodyne detection, suffers from several weaknesses. First, it requires efficient detection, which is critical for measuring fragile non-Gaussian states, especially bright ones. Second, it needs a local oscillator, tailored to match the spatiotemporal properties of the state under test, and fails for multimode and broadband states. Here we propose Wigner function tomography based on optical parametric amplification followed by direct detection. The method is immune to detection inefficiency and loss, and suitable for broadband, spatially and temporally multimode quantum states. To prove the principle, we experimentally reconstruct the Wigner function of squeezed vacuum occupying a single mode of a strongly multimode state. We obtain a squeezing of −7.5±0.4dB and purity of 0.91 −0.08 +0.09 despite more than 97% loss caused mainly by filtering. Theoretically, we also consider the reconstruction of a squeezed single photon—a bright non-Gaussian state. Due to multimode parametric amplification, the method allows for simultaneous tomography of multiple modes. This makes it a powerful tool for optical quantum information processing.
Continuous-variable (CV) quantum states are naturally characterized by their moments, defined as expectation values of products of single- or multimode ladder operators. Many CV Hamiltonians and quantum algorithms are formulated directly in terms of these moments, and therefore an efficient procedure to estimate moments with limited state measurements is necessary. In this paper, we present a protocol for shadow tomography of moment-generating functions (MGFs) of CV states based on randomized homodyne and heterodyne measurements. The resulting shadows enable an efficient and concurrent estimation of many multimode moments. Our complexity analysis shows that the number of measurements required to estimate a certain moment to a given precision grows exponentially in the order of the moments. Finally, we assess the precision of these moment estimators by applying them to two tasks: detecting entanglement in both Gaussian and non-Gaussian states via the Shchukin-Vogel protocol, and characterizing optical loss in a photonic chip. We demonstrate that both tasks can be accomplished with only a few thousand randomized measurements. The small number of required measurements, combined with efficient sample processing, makes our protocol applicable to a wide range of CV simulation tasks.
We investigate exponential phase moments of the s-parametrized quasidistributions (smoothed Wigner functions). We show that the knowledge of these moments as functions of s provides, together with photon-number statistics, a complete description of the quantum state. We demonstrate that the exponential phase moments can be directly sampled from the data recorded in balanced homodyne detection and we present simple expressions for the sampling kernels. The phase moments are Fourier coefficients of phase distributions obtained from the quasidistributions via integration over the radial variable in polar coordinates. We performed Monte Carlo simulations of the homodyne detection and we demonstrate the feasibility of direct sampling of the moments and subsequent reconstruction of the phase distribution.