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arXiv:2202.09980

Quantum Multi-Parameter Adaptive Bayesian Estimation and Application to Super-Resolution Imaging

Kwan Kit Lee, Christos N. Gagatsos, Saikat Guha, Amit Ashok

Published Feb 21, 2022• 1 citation
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Abstract

In Bayesian estimation theory, the estimator ${\hat θ} = E[θ|l]$ attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest $θ$ from the observation of $l$ through a noisy channel $P_{l|θ}$, given a prior $P_θ$ on $θ$. In quantum sensing tasks, the user gets $ρ_θ$, the quantum state that encodes $θ$. They choose a measurement, a positive-operator valued measure (POVM) $Π_l$, which induces the channel $P_{l|θ} = {\rm Tr}(ρ_θΠ_l)$ to the measurement outcome $l$, on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM $Π_l$ that minimizes the MMSE over all possible measurements, and that MMSE. This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements, when $P_θ$ is unavailable. For multi-parameter estimation, i.e., when $θ$ is a vector, in Fisher quantum estimation theory, the inverse of the QFI matrix provides an operator lower bound to the covariance of an unbiased estimator. However, there has been little work on quantifying quantum limits and measurement designs, for multi-parameter quantum estimation in the {\em Bayesian} setting. In this paper, we build upon Personick's result to construct a Bayesian adaptive measurement scheme for multi-parameter estimation when $N$ copies of $ρ_θ$ are available. We illustrate an application to localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs far superior to quantum-noise-limited focal-plane direct imaging.

DOI
10.48550/arxiv.2202.09980

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Multiparameter estimation remains a fundamental challenge in quantum estimation theory because the optimal measurements associated with different parameters are generally incompatible. In this work, we develop a Bayesian framework for the simultaneous estimation of multiple parameters. Our approach builds on Personick's method, in which the estimation problem is reduced to a set of Lyapunov equations whose solutions define optimal observables for the individual parameters. Since these observables generally do not commute, we construct a convex combination of the solutions, parameterized by a set of variational parameters. The spectral decomposition of the resulting operator defines a parametrized projection-valued measure. The resulting projective measurement determines the likelihood function, from which the minimum mean-square error estimators and the corresponding Bayesian mean-square errors are obtained, following standard Bayesian procedures, as functions of the variational parameters. To determine their optimal values, we formulate a minimax optimization problem based on the normalized Bayesian mean-square errors, introducing a min-max normalization procedure that enables a meaningful comparison of estimation errors associated with different parameters. This optimization yields a variationally optimized projective measurement for simultaneous Bayesian estimation. Two qubit examples, involving phase estimation and the estimation of parameters defining convex combinations of unitary operations, demonstrate the construction of optimized projective measurements and the corresponding minimum mean-square error estimators within the proposed framework for finite data sets.
We consider a two-mode bosonic state with fixed photon number n∈Nn \in \mathbb {N}n∈N , whose upper and lower modes pick up a phase ϕ\phi ϕ and −ϕ-\phi −ϕ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating ϕ\phi ϕ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider ϕ\phi ϕ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider n=1n=1n=1 and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.
In quantum estimation for a ddd-parameter family of density operators on a finite-dimensional Hilbert space H\mathcal{H}H, an estimator is specified by a pair (M,θ^)\left(M,\hatθ\right)(M,θ^), where MMM is a POVM with a finite outcome set ΩΩΩ and θ^:Ω→Rd\hatθ:Ω\to\mathbb{R}^{d}θ^:Ω→Rd is a classical estimator map. Since the number of outcomes ∣Ω∣\left|Ω\right|∣Ω∣ is a priori unbounded, the space of admissible POVMs is vast, which makes the search for optimal estimators difficult. In this paper, for the minimization of the weighted trace of the mean squared error among locally unbiased estimators, we prove that it suffices to consider POVMs with at most (dim⁡H)2+d(d+1)/2−1\left(\dim\mathcal{H}\right)^{2}+d(d+1)/2-1(dimH)2+d(d+1)/2−1 outcomes, and that the optimization can be restricted to rank-one measurements. For Bayesian estimation with a general loss function, we show that rank-one POVMs with at most (dim⁡H)2(\dim\mathcal{H})^{2}(dimH)2 outcomes are sufficient for the infimum value of the Bayes risk. Furthermore, when the model admits a real sufficient subalgebra, we show that the (dim⁡H)2\left(\dim\mathcal{H}\right)^{2}(dimH)2 term in the above support-size bounds can be reduced in both the locally unbiased and Bayesian settings. These bounds substantially reduce the search space for optimal measurements and justify restricting numerical optimization to rank-one POVMs with finitely many outcomes.
For more than a century, the diffraction limit has defined the resolution achievable by passive optical imaging systems. Although some resolution improvement can be gained through classical data processing of the image, it is limited by the noise arising from quantum nature of light. Minimizing the effect of this noise requires quantum treatment of optical imaging. By reformulating imaging as a problem of quantum measurement and estimation, it becomes possible to identify optimal detection strategies that recover spatial information previously thought inaccessible. This review summarizes the theoretical framework that underpins this development, from the formulation of quantum Cramer-Rao bounds and Chernoff bounds to the construction of receivers that attain them, such as those based on spatial-mode demultiplexing. We show how these methods can beat conventional imaging in the classification, localization, and imaging of sub-Rayleigh incoherent sources. We then discuss extensions to multiparameter and partially coherent scenarios, and highlight the unifying connections between estimation and discrimination tasks. Finally, we survey recent experimental demonstrations that approach quantum-limited resolution and outline emerging applications in microscopy, astronomy, and optical sensing.