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Pulsed homodyne Gaussian quantum tomography with low detection efficiency

Martina Esposito, Fabio Benatti, Roberto Floreanini B, Stefano Olivares, Francesco Randi, Kelvin Titimbo, Micol Pividori, Fabio Novelli, Federico Cilento, F. Parmigiani, Daniele Fausti

Published Jan 1, 2024• 20 citations
Publisher

Abstract

Pulsed homodyne quantum tomography usually requires a high detection efficiency, limiting its applicability in quantum optics. Here, it is shown that the presence of low detection efficiency (<50%) does not prevent the tomographic reconstruction of quantum states of light, specifically, of Gaussian states. This result is obtained by applying the so-called minimax adaptive reconstruction of the Wigner function to pulsed homodyne detection. In particular, we prove, by both numerical and real experiments, that an effective discrimination of different Gaussian quantum states can be achieved. Our finding paves the way to a more extensive use of quantum tomographic methods, even in physical situations in which high detection efficiency is unattainable.

DOI
10.25916/sut.26222240.v1

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Pulsed homodyne quantum tomography usually requires a high detection efficiency, limiting its applicability in quantum optics. Here, it is shown that the presence of low detection efficiency (<50%) does not prevent the tomographic reconstruction of quantum states of light, specifically, of Gaussian states. This result is obtained by applying the so-called minimax adaptive reconstruction of the Wigner function to pulsed homodyne detection. In particular, we prove, by both numerical and real experiments, that an effective discrimination of different Gaussian quantum states can be achieved. Our finding paves the way to a more extensive use of quantum tomographic methods, even in physical situations in which high detection efficiency is unattainable.
Pulsed homodyne quantum tomography usually requires a high detection efficiency, limiting its applicability in quantum optics. Here, it is shown that the presence of low detection efficiency ( ) does not prevent the tomographic reconstruction of quantum states of light, specifically, of Gaussian states. This result is obtained by applying the so-called 'minimax' adaptive reconstruction of the Wigner function to pulsed homodyne detection. In particular, we prove, by both numerical and real experiments, that an effective discrimination of different Gaussian quantum states can be achieved. Our finding paves the way to a more extensive use of quantum tomographic methods, even in physical situations in which high detection efficiency is unattainable.
Optical homodyne tomography consists in reconstructing the quantum state of an optical field from repeated measurements of its amplitude at different field phases (homodyne data). The experimental noise, which unavoidably affects the homodyne data, leads to a detection efficiency η<1\eta<1η<1 . The problem of reconstructing quantum states from noisy homodyne data sets prompted an intense scientific debate about the presence or absence of a lower homodyne efficiency bound ( η>0.5\eta> 0.5η>0.5 ) below which quantum features, like quantum interferences, cannot be retrieved. Here, by numerical experiments, we demonstrate that quantum interferences can be effectively reconstructed also for low homodyne detection efficiency. In particular, we address the challenging case of a Schrödinger cat state and test the minimax and adaptive Wigner function reconstruction technique by processing homodyne data distributed according to the chosen state but with an efficiency η<0.5\eta< 0.5η<0.5 . By numerically reproducing the Schrödinger’s cat interference pattern, we give evidence that quantum state reconstruction is actually possible in these conditions, and provide a guideline for handling optical tomography based on homodyne data collected by low efficiency detectors.
In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on R2\mathbb R^2R2 which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2<η≤11/2 < η\leq 11/2<η≤1, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the L∞\mathbb L_\inftyL∞​-risk over a class of infinitely differentiable. We compute also the lower bound for the L2\mathbb L_2L2​-risk. We construct adaptive estimator, i.e. which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness class functions. Finite sample behaviour of our adaptive procedure are explored through numerical experiments.
In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on R2\mathbb{R}^{2}R2, which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2<η≤11/2<\eta\leq11/2<η≤1, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the L∞\mathbb{L}_{\infty}L∞​-risk over a class of infinitely differentiable functions. We also compute the lower bound for the L2\mathbb{L}_{2}L2​-risk. We construct an adaptive estimator, that is, which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness of the class of functions up to a logarithmic factor in the sample size. Finite sample behaviour of our adaptive procedure is explored through numerical experiments.
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.