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The Annals of Statistics

Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography

Karim Lounici, Katia Méziani, Gabriel Peyré A

Published May 2, 2018• 2 citations
Publisher

Abstract

In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on $\mathbb{R}^{2}$, 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 $1/2<\eta\leq1$, 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 $\mathbb{L}_{\infty}$-risk over a class of infinitely differentiable functions. We also compute the lower bound for the $\mathbb{L}_{2}$-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.

DOI
10.1214/17-aos1586

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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.
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.
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.
Bosonic quantum systems provide a hardware-efficient platform for quantum information processing but remain challenging to characterise due to their large Hilbert space and the high measurement cost of state tomography. Existing approaches estimate the fidelity with respect to a single target state, making them unsuitable for applications in which physically equivalent states differ by phase space translations, rotations, or other transformations. Here, we introduce an adaptive reconstruction technique that estimates the fidelity with respect to a family of bosonic states while reconstructing the underlying Wigner function from a small number of measurements. The method combines a physics-informed parametric model with Bayesian inference, bootstrap, and active learning to iteratively select the most informative phase space sampling points. We implement the approach on a circuit quantum electrodynamics platform and benchmark it on Schrödinger cat states with amplitudes α∈[1,3]α\in[1,3]α∈[1,3]. The reconstruction yields reproducible fidelity estimates within a few minutes, remains robust to substantial displacements and rotations in phase space despite using a mismatched prior, and is sensitive to subtle state imperfections. We further compare the adaptive strategy with existing Wigner function sampling protocols experimentally, demonstrating the advantage of adaptive sampling for measurement-efficient fidelity estimation with respect to a family of cat states. Finally, we incorporate the reconstructed fidelity into the figure of merit used in a proof-of-principle closed-loop quantum optimal control experiment, demonstrating the applicability of the method to autonomous optimisation of bosonic quantum states.
In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2<η≤11/2 < η\leq 11/2<η≤1, we propose a novel estimator of a quantum state whose density matrix elements ρm,nρ_{m,n}ρm,n​ decrease like Ce−B(m+n)r/2Ce^{-B(m+n)^{r/ 2}}Ce−B(m+n)r/2, for fixed C≥1C\geq 1C≥1, B>0B>0B>0 and $0