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Frontiers in Optics

Estimation and Correction of Gaussian Random Displacement Error Using Simple Non-Gaussian States

Fumiya Hanamura, Warit Asavanant, Kosuke Fukui, Shunya Konno, Akira Furusawa

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We analyse estimation and error correction of Gaussian random displacement using experimentally feasible non-Gaussian states. We show that even simple states such as single-photon state is superior to Gaussian states in both settings.
DOI
10.1364/fio.2021.jw7a.48

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•May 22, 2026•0 citations
In continuous-variable quantum information processing, quantum error correction of Gaussian errors requires simultaneous estimation of both quadrature components of displacements in phase space. However, quadrature operators xxx and ppp are noncommutative conjugate observables, whose simultaneous measurement is prohibited by the uncertainty principle. Gottesman-Kitaev-Preskill (GKP) error correction deals with this problem using complex non-Gaussian states called GKP states. On the other hand, simultaneous estimation of displacement using experimentally feasible non-Gaussian states has not been well studied. In this paper, we consider a multiparameter estimation problem of displacements assuming an isotropic Gaussian prior distribution and allowing postselection of measurement outcomes. We derive a lower bound for the estimation error when only Gaussian operations are used and show that even simple non-Gaussian states such as single-photon states can beat this bound. Based on Ghosh's bound, we also obtain a lower bound for the estimation error when the maximum photon number of the input state is given. Our results reveal the role of non-Gaussianity in the estimation of displacements and pave the way toward the error correction of Gaussian errors using experimentally feasible non-Gaussian states.
Uncertainty principle prohibits the precise measurement of both components of displacement parameters in phase space. We have theoretically shown that this limit can be beaten using single-photon states, in a single-shot and single-mode setting [F. Hanamura et al., Estimation of gaussian random displacement using non-gaussian states, Phys. Rev. A 104, 062601 (2021).PLRAAN2469-992610.1103/PhysRevA.104.062601]. In this Letter, we validate this by experimentally beating the classical limit. In optics, this is the first experiment to estimate both parameters of displacement using non-Gaussian states. This result is related to many important applications, such as quantum error correction.
Abstract Multiple photon subtraction applied to a displaced phase-averaged coherent state, which is a non-Gaussian classical state, produces conditional states with a non trivial (positive) Glauber-Sudarshan Prepresentation. We theoretically and experimentally demonstrate that, despite its simplicity, this class of conditional states cannot be fully characterized by direct detection of photon numbers. In particular, the non-Gaussianity of the state is a characteristics that must be assessed by phase-sensitive measurements. We also show that the non-Gaussianity of conditional states can be manipulated by choosing suitable conditioning values and composition of phase-averaged states.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
Summary. A comparative, model-level study of four quantum algorithms ported to a common photonic continuous-variable (CV) model: control-free quantum phase estimation (QPE), randomized product formulas (qDRIFT), quantum Kolmogorov-Arnold networks (QKAN), and Regev's factoring algorithm. Algorithms are compared under a shared noise channel combining exact Fock-space photon loss and Gaussian displacement, with Gottesman-Kitaev-Preskill (GKP) error correction. Main findings. Control-free QPE is the most noise-robust because it needs no controlled unitaries and, for quadratic Hamiltonians, no non-Gaussian gates; the advantage survives for non-Gaussian Hamiltonians. GKP correction helps only once the physical noise exceeds the finite-squeezing floor (proved), and whether an algorithm benefits is set by a noise- versus systematic-limited dichotomy. The core GKP correction model is validated against an explicit Glancy-Knill error-correction Monte Carlo (agreement ≤0.1%). A discrete-variable surface-code baseline shows the CV advantage is structural (Gaussian operations are free), not threshold-based. Scope. This is a model-level numerical comparison, not an experiment; the GKP correction is a validated surrogate rather than a full Fock-space error-correction circuit, and instances are modest in size. Results, sensitivity analyses, and a two-dimensional noise phase diagram are included. Reproducibility. Standalone scripts and released JSON results reproduce every figure and table.