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arXiv:2109.13674 · Physical Review A

Estimation of the Wigner distribution of single-mode Gaussian states: A comparative study

Chandan Kumar, Arvind

Published Apr 12, 2022• 2 citations
arXivHTMLPDF

Abstract

In this work, we consider the estimation of single-mode Gaussian states using four different measurement schemes, namely, (1) homodyne measurement, (2) heterodyne measurement, (3) sequential measurement scheme, and (4) the Arthurs-Kelly measurement scheme, with a view to compare their relative performance. To this end, we work in the phase space formalism, specifically at the covariance matrix level, which provides an elegant and intuitive way to explicitly carry out the involved calculations. We show that the optimal performance of the Arthurs-Kelly and sequential measurement schemes is equal to the heterodyne measurement. While the heterodyne measurement outperforms the homodyne measurement in the mean estimation of squeezed state ensembles, the homodyne measurement outperforms the heterodyne measurement for variance estimation of squeezed state ensembles up to a certain range of the squeezing parameter. We then modify the Hamiltonian in the Arthurs-Kelly measurement scheme, such that the two meters can have correlations and show that the optimal performance is achieved when the meters are uncorrelated. We expect that these results will be useful in various quantum information and quantum communication protocols.

DOI
10.1103/physreva.105.042419

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In this work, we consider the estimation of single-mode Gaussian states using four different measurement schemes, namely, (1) homodyne measurement, (2) heterodyne measurement, (3) sequential measurement scheme, and (4) the Arthurs-Kelly measurement scheme, with a view to compare their relative performance. To this end, we work in the phase space formalism, specifically at the covariance matrix level, which provides an elegant and intuitive way to explicitly carry out the involved calculations. We show that the optimal performance of the Arthurs-Kelly and sequential measurement schemes is equal to the heterodyne measurement. While the heterodyne measurement outperforms the homodyne measurement in the mean estimation of squeezed state ensembles, the homodyne measurement outperforms the heterodyne measurement for variance estimation of squeezed state ensembles up to a certain range of the squeezing parameter. We then modify the Hamiltonian in the Arthurs-Kelly measurement scheme, such that the two meters can have correlations and show that the optimal performance is achieved when the meters are uncorrelated. We expect that these results will be useful in various quantum information and quantum communication protocols.
We examine the moment-reconstruction performance of both the homodyne and heterodyne (double-homodyne) measurement schemes for arbitrary quantum states and introduce moment estimators that optimize the respective schemes for any given data. In the large-data limit, these estimators are as efficient as the maximum-likelihood estimators. We then illustrate the superiority of the heterodyne measurement for the reconstruction of the first and second moments by analyzing Gaussian states and many other significant nonclassical states. Finally, we present an extension of our theories to two-mode sources, which can be straightforwardly generalized to all other multimode sources.
We reveal that quadrature squeezing can result in significantly better quantum-estimation performance with quantum heterodyne detection (of H. P. Yuen and J. H. Shapiro) as compared to quantum homodyne detection for Gaussian states, which touches an important aspect in the foundational understanding of these two schemes. Taking single-mode Gaussian states as examples, we show analytically that the competition between the errors incurred during tomogram processing in homodyne detection and the Arthurs-Kelly uncertainties arising from simultaneous incompatible quadrature measurements in heterodyne detection can often lead to the latter giving more accurate estimates. This observation is also partly a manifestation of a fundamental relationship between the respective data uncertainties for the two schemes. In this sense, quadrature squeezing can be used to overcome intrinsic quantum-measurement uncertainties in heterodyne detection.
Bayesian analysis is a framework for parameter estimation that applies even in uncertainty regimes where the commonly used local (frequentist) analysis based on the Cramer-Rao bound (CRB) is not well defined. In particular, it applies when no initial information about the parameter value is available, e.g., when few measurements are performed. Here, we consider three paradigmatic estimation schemes in continuous-variable (CV) quantum metrology (estimation of displacements, phases, and squeezing strengths) and analyse them from the Bayesian perspective. For each of these scenarios, we investigate the precision achievable with single-mode Gaussian states under homodyne and heterodyne detection. This allows us to identify Bayesian estimation strategies that combine good performance with the potential for straightforward experimental realization in terms of Gaussian states and measurements. Our results provide practical solutions for reaching uncertainties where local estimation techniques apply, thus bridging the gap to regimes where asymptotically optimal strategies can be employed.
Non-Gaussian quantum states are critical resources in photonic quantum information processing, rendering their generation and characterization of increasing importance in quantum optics. In this work, we theoretically and numerically analyze the relative efficiency of homodyne versus heterodyne measurements for reconstructing non-Gaussian states, a major outstanding question in continuous-variable tomography. Combining a Fisher information-based formalism with simulated experiments, we find homodyne tomography to outperform heterodyne measurements for all non-Gaussian states tested, although the separation between the two modalities proves significantly narrower than suggested by the asymptotic Cramer-Rao lower bound. Our results should find use for optimizing measurement strategies in practical continuous-variable quantum systems.
The quantum-to-classical transition of a quantum state is a topic of great interest in fundamental and practical aspects. A coarse-graining in quantum measurement has recently been suggested as its possible account in addition to the usual decoherence model. We here investigate the reconstruction of a Gaussian state (single mode and two modes) by coarse-grained homodyne measurements. To this aim, we employ two methods, the direct reconstruction of the covariance matrix and the maximum likelihood estimation (MLE), respectively, and examine the reconstructed state under each scheme compared to the state interacting with a Gaussian (squeezed thermal) reservoir. We clearly demonstrate that the coarse-graining model, though applied equally to all quadrature amplitudes, is not compatible with the decoherence model by a thermal (phase-insensitive) reservoir. Furthermore, we compare the performance of the direct reconstruction and the MLE methods by investigating the fidelity and the nonclassicality of the reconstructed states and show that the MLE method can generally yield a more reliable reconstruction, particularly without information on a reference frame (phase of input state).
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