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:1306.0508 · Physical Review A

Witnessing negativity of Wigner function by estimating fidelities of catlike states from homodyne measurements

Jaromı́r Fiurášek, Miroslav Ježek

Published Jun 21, 2013• 12 citations
arXivHTMLPDF

Abstract

We derive sampling functions for estimation of quantum-state fidelity with Schr\"odinger-cat-like states, which are defined as superpositions of two coherent states with opposite amplitudes. We also provide sampling functions for fidelity with squeezed Fock states that can approximate the catlike states and can be generated from Gaussian squeezed states by conditional photon subtraction. The fidelities can be determined by averaging the sampling functions over quadrature statistics measured by homodyne detection. The sampling functions are designed such that they can compensate for losses and inefficient homodyning provided that the overall efficiency exceeds a certain threshold. The fidelity with an odd coherent state and the fidelity with a squeezed odd Fock state provide convenient witnesses of negativity of the Wigner function of the measured state. The negativity of the Wigner function at the origin of phase space is certified if any of these fidelities exceeds $0.5$. Finally, we discuss the possibility of reducing the statistical uncertainty of the fidelity estimates by a suitable choice of the dependence of the number of quadrature samples on the relative phase shift between the local oscillator and the signal beam.

DOI
10.1103/physreva.87.062115

Related papers

Generation of Optical Schrödinger Cat States by Number-Resolved Squeezed Photon Subtraction

Thomas Gerrits, Scott Glancy, Tracy Sharp Clement +7 more

We have generated and measured an approximation of an optical Schrödinger cat state by photon subtraction from a squeezed state. Using single-photon avalanche photodiode detectors and photon-number-resolving transition edge sensors, we were able to extract Wigner distributions for one or two photons subtracted from the squeezed state, resulting in both an odd and even cat state. The one-photon-subtracted Wigner distribution shows measured negative features (without any correction), indicating quantum character. The Wigner distribution of the two-photon-subtracted state is negative after correcting for homodyne losses. A full statistical analysis is under way to quantify the magnitude and fidelity of the cat states generated.

Conference on Lasers and Electro-Optics•Jan 1, 2009•0 citations

Engineering and certification of multimode non Gaussian states of light

Niels Tripier-Mondancin, Ganaël Roeland, Yann Bouchereau +5 more

We use double homodyne detection to sample the Q-Husimi function of a photon-subtracted mutlimode squeezed vacuum state. Thereafter, we perform a fidelity estimation protocol to certify the Wigner negativity and stellar rank of the generated state.

Quantum 2.0 Conference and Exhibition•Jan 1, 2024•1 citation

Optical homodyne measurements and entangled coherent states

Barry C. Sanders, K. S. Lee, M. S. Kim

We show that optical homodyne measurements of coherent states, and of superpositions of coherent states, can be described using the joint photon-number distribution for entangled coherent states. The quadrature-phase distribution interference fringes for superpositions of macroscopically distinct coherent states (the so-called ``Schr\"odinger cat states'') are shown to arise from interference in the photon-number distribution for entangled coherent states. The entangled squeezed states are introduced here as squeezed superposition states which are optically mixed with an antisqueezed coherent local oscillator field (squeezing in the other quadrature) at a beam splitter, and we discuss the connection between entangled squeezed states and squeezed-state homodyne detection of squeezed light. Finally the relationship between interference in phase space and fringes in the joint photon-number distribution for the entangled squeezed state is explored.

Physical Review A•Jul 1, 1995•53 citations

Negativity of the Wigner function as an indicator of non-classicality

Anatole Kenfack, Karol yczkowski

A measure of non-classicality of quantum states based on the volume of the negative part of the Wigner function is proposed. We analyse this quantity for Fock states, squeezed displaced Fock states and cat-like states defined as coherent superposition of two Gaussian wavepackets.

arXiv:quant-ph/0406015•Journal of Optics B Quantum and Semiclassical Optics•Aug 24, 2004•778 citations

Sampling of the Wigner function adapted to time-multiplexed detection of photon statistics

Kaisa Laiho, Malte Avenhaus, K. N. Cassemiro +1 more

We investigate the capabilities of loss-tolerant quantum state characterization using a photon-number resolving, time-multiplexed detector (TMD). We employ the idea of probing the Wigner function point-by-point in phase space via photon parity measurements and displacement operations, replacing the conventional homodyne tomography. Our emphasis lies on reconstructing the Wigner function of non-Gaussian Fock states with highly negative values in a scheme that is based on a realistic experimental setup. In order to establish the concept of loss-tolerance for state characterization we show how losses can be decoupled from the impact of other experimental imperfections, i.e. the non-unity transmittance of the displacement beamsplitter and non-ideal mode overlap. We relate the experimentally accessible parameters to effective ones that are needed for an optimised state reconstruction. The feasibility of our approach is tested by Monte Carlo simulations, which provide bounds resulting from statistical errors that are due to limited data sets. Our results clearly show that high losses can be accepted for a defined parameter range, and moreover, that (in contrast to homodyne detection) mode mismatch results in a distinct signature, which can be evaluated by analysing the photon number oscillations of the displaced Fock states.

•Nov 3, 2008•1 citation

Effect of imperfect experimental condition on generation of Schrödinger cat state

Nana Zhang, Shujing Li, Yan Hong-Mei +2 more

Schrödinger cat state is an important non-classical state, and it can be used in quantum teleportation, quantum computation and quantum repeater. Schrödinger cat state is usually obtained experimentally by subtracting one photon from a squeezed-vacuum state. The fidelity between a photon-subtracted squeezed state and a cat state can be very high under suitable parameters. However, the quality of the generated state will be affected by the imperfect experimental conditions. In this paper, the effect of imperfect experimental conditions on the generation of cat state is theoretically calculated and analyzed. The input squeezed-vacuum field is represented by Weyl characteristic function, which contains the fluctuation variance of the squeezed and amplified noises. The characteristic function of generated state is obtained by using the transmission matrix of beam splitter and the measurement operator of single-photon detector. We acquire the expression of Wigner function of generated state by the Fourier transform of the Weyl characteristic function. The fidelity is calculated by using the formula F=1/π∫d2ζC1(ζ)C|cat->(ζ), where C1(ζ) and C|cat->(ζ) represent Weyl characteristic function of the generated state and the Schrodinger cat state, respectively. The imperfection of the input squeezed state, the imperfection of the single-photon detector and the loss of the balanced homodyne detection are included in our theoretical model. We calculate the Wigner function at the phase-space origin W(0) and the fidelity in terms of different experimental parameters. The results show that the fidelity and negativity of W(0) decrease with squeezing purity decreasing. A pure squeezed-vacuum state is composed of even photon number states. In the case of impure squeezing, some odd photon number states appear in the photon number distribution. After subtracting one photon from the impure squeezing state, the generated state consists of not only odd photon number state but also even photon states, which degrades the fidelity of the generated state. The lower squeezing purity is required to meet the demand for W(0)<0 under the condition of higher squeezing degree. There is an optimal squeezing degree to maximize the fidelity of generated state with impure squeezing. The use of inefficient on-ff single-photon detector and the loss of the balanced homodyne detection will further reduce the fidelity of the generated state. Under the practical experimental condition:squeezing degree s=-3 dB, the squeezing purity μ=99% and the quantum efficiency of balanced homodyne detection η=98%, the fidelity of generated state can reach 0.88 with using a commercially available on-off single-photon detector. This work can provide theoretical guidance for generating a high-quality Schrödinger cat state.

Acta Physica Sinica•Jan 1, 2018•3 citations
We derive sampling functions for estimation of quantum-state fidelity with Schr"odinger-cat-like states, which are defined as superpositions of two coherent states with opposite amplitudes. We also provide sampling functions for fidelity with squeezed Fock states that can approximate the catlike states and can be generated from Gaussian squeezed states by conditional photon subtraction. The fidelities can be determined by averaging the sampling functions over quadrature statistics measured by homodyne detection. The sampling functions are designed such that they can compensate for losses and inefficient homodyning provided that the overall efficiency exceeds a certain threshold. The fidelity with an odd coherent state and the fidelity with a squeezed odd Fock state provide convenient witnesses of negativity of the Wigner function of the measured state. The negativity of the Wigner function at the origin of phase space is certified if any of these fidelities exceeds 0.50.50.5. Finally, we discuss the possibility of reducing the statistical uncertainty of the fidelity estimates by a suitable choice of the dependence of the number of quadrature samples on the relative phase shift between the local oscillator and the signal beam.
We have generated and measured an approximation of an optical Schrödinger cat state by photon subtraction from a squeezed state. Using single-photon avalanche photodiode detectors and photon-number-resolving transition edge sensors, we were able to extract Wigner distributions for one or two photons subtracted from the squeezed state, resulting in both an odd and even cat state. The one-photon-subtracted Wigner distribution shows measured negative features (without any correction), indicating quantum character. The Wigner distribution of the two-photon-subtracted state is negative after correcting for homodyne losses. A full statistical analysis is under way to quantify the magnitude and fidelity of the cat states generated.
We show that optical homodyne measurements of coherent states, and of superpositions of coherent states, can be described using the joint photon-number distribution for entangled coherent states. The quadrature-phase distribution interference fringes for superpositions of macroscopically distinct coherent states (the so-called ``Schr\"odinger cat states'') are shown to arise from interference in the photon-number distribution for entangled coherent states. The entangled squeezed states are introduced here as squeezed superposition states which are optically mixed with an antisqueezed coherent local oscillator field (squeezing in the other quadrature) at a beam splitter, and we discuss the connection between entangled squeezed states and squeezed-state homodyne detection of squeezed light. Finally the relationship between interference in phase space and fringes in the joint photon-number distribution for the entangled squeezed state is explored.
We investigate the capabilities of loss-tolerant quantum state characterization using a photon-number resolving, time-multiplexed detector (TMD). We employ the idea of probing the Wigner function point-by-point in phase space via photon parity measurements and displacement operations, replacing the conventional homodyne tomography. Our emphasis lies on reconstructing the Wigner function of non-Gaussian Fock states with highly negative values in a scheme that is based on a realistic experimental setup. In order to establish the concept of loss-tolerance for state characterization we show how losses can be decoupled from the impact of other experimental imperfections, i.e. the non-unity transmittance of the displacement beamsplitter and non-ideal mode overlap. We relate the experimentally accessible parameters to effective ones that are needed for an optimised state reconstruction. The feasibility of our approach is tested by Monte Carlo simulations, which provide bounds resulting from statistical errors that are due to limited data sets. Our results clearly show that high losses can be accepted for a defined parameter range, and moreover, that (in contrast to homodyne detection) mode mismatch results in a distinct signature, which can be evaluated by analysing the photon number oscillations of the displaced Fock states.
Schrödinger cat state is an important non-classical state, and it can be used in quantum teleportation, quantum computation and quantum repeater. Schrödinger cat state is usually obtained experimentally by subtracting one photon from a squeezed-vacuum state. The fidelity between a photon-subtracted squeezed state and a cat state can be very high under suitable parameters. However, the quality of the generated state will be affected by the imperfect experimental conditions. In this paper, the effect of imperfect experimental conditions on the generation of cat state is theoretically calculated and analyzed. The input squeezed-vacuum field is represented by Weyl characteristic function, which contains the fluctuation variance of the squeezed and amplified noises. The characteristic function of generated state is obtained by using the transmission matrix of beam splitter and the measurement operator of single-photon detector. We acquire the expression of Wigner function of generated state by the Fourier transform of the Weyl characteristic function. The fidelity is calculated by using the formula F=1/π∫d2ζC1(ζ)C|cat->(ζ), where C1(ζ) and C|cat->(ζ) represent Weyl characteristic function of the generated state and the Schrodinger cat state, respectively. The imperfection of the input squeezed state, the imperfection of the single-photon detector and the loss of the balanced homodyne detection are included in our theoretical model. We calculate the Wigner function at the phase-space origin W(0) and the fidelity in terms of different experimental parameters. The results show that the fidelity and negativity of W(0) decrease with squeezing purity decreasing. A pure squeezed-vacuum state is composed of even photon number states. In the case of impure squeezing, some odd photon number states appear in the photon number distribution. After subtracting one photon from the impure squeezing state, the generated state consists of not only odd photon number state but also even photon states, which degrades the fidelity of the generated state. The lower squeezing purity is required to meet the demand for W(0)<0 under the condition of higher squeezing degree. There is an optimal squeezing degree to maximize the fidelity of generated state with impure squeezing. The use of inefficient on-ff single-photon detector and the loss of the balanced homodyne detection will further reduce the fidelity of the generated state. Under the practical experimental condition:squeezing degree s=-3 dB, the squeezing purity μ=99% and the quantum efficiency of balanced homodyne detection η=98%, the fidelity of generated state can reach 0.88 with using a commercially available on-off single-photon detector. This work can provide theoretical guidance for generating a high-quality Schrödinger cat state.