Fisher-information-based optimization of programmable optical front ends in situ generally requires a response model or parameter-labeled measurements. We show that natural parameter fluctuations can instead provide the training signal. We construct proxies for the Fisher information (FI) of one parameter and the prior-weighted FI spectrum of multiple parameters from measured covariances after subtracting conditional detection noise, assuming locally affine responses. We use simulated photon counts from two incoherent point sources to train a mode sorter that approaches the quantum limit for separation and the Nagaoka--Hayashi single-copy bound for joint centroid--separation estimation. We demonstrate count-only updates that require no instrument response model and achieve performance comparable to model-based gradients. Our results provide a route to adaptive optical measurements without controlled parameter scans.
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11 pages main text, 20 pages supplemental material
Fisher-information-based optimization of programmable optical front ends in situ generally requires a response model or parameter-labeled measurements. We show that natural parameter fluctuations can instead provide the training signal. We construct proxies for the Fisher information (FI) of one parameter and the prior-weighted FI spectrum of multiple parameters from measured covariances after subtracting conditional detection noise, assuming locally affine responses. We use simulated photon counts from two incoherent point sources to train a mode sorter that approaches the quantum limit for separation and the Nagaoka--Hayashi single-copy bound for joint centroid--separation estimation. We demonstrate count-only updates that require no instrument response model and achieve performance comparable to model-based gradients. Our results provide a route to adaptive optical measurements without controlled parameter scans.
In optics, when attempting to find the distance separations s between two incoherent emitters, it is standard practice to take a full image and extract the single parameter s from it. In such scenarios, the smallest precisely measurable separation is limited by the point-spread function on the image plane, with a vanishingly small amount of information available for smaller and smaller separations especially when photon shot noise is the dominant noise source (as in, for example, astronomical observations). This limit can be formalised by the Fisher information, which quantifies the amount of information accessible per photon detection and is directly associated with Cramér-Rao lower bound. For direct intensity imaging, the Fisher information drops to zero for object separations smaller than the spread of the optical field, sometimes referred to as the Rayleigh's curse, which limits the usefulness of photon counting for metrology.
Famously, the quantum Fisher information -- the maximum Fisher information over all physical measurements -- is additive for independent copies of a system and the optimal measurement acts locally. We are left to wonder: does the same hold when the set of accessible measurements is constrained? Such constraints are necessary to account for realistic experimental restrictions. Here, we consider a fully Gaussian scenario focusing on only Gaussian measurements. We prove that the optimal Gaussian measurement protocol remains local, if the information is encoded in either the displacement or the covariance matrix. However, when the information is imprinted on both, this no longer holds true: we construct a simple global Gaussian measurement where the Fisher information becomes super additive. These results can improve parameter estimation tasks via feasible tools. Namely, in quantum optical platforms our proposed global operation requires only passive global operations and single mode Gaussian measurements. We demonstrate this in two examples where we estimate squeezing and losses. While in the former case there is a significant gap between the Fisher information of the optimal Gaussian measurement and the quantum Fisher information for a single copy, this gap can be reduced with joint Gaussian measurements and closed in the asymptotic limit of many copies.
It is well known in Bayesian estimation theory that the conditional estimator attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest. In quantum, e.g., optical and atomic, imaging and sensing tasks the user has access to the quantum state that encodes the parameter. The choice of a measurement operator, i.e. a positive-operator valued measure (POVM), leads to a measurement outcome on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM that attains the MMSE over all possible physically allowable measurements and the resulting MMSE (Personick, 1997). 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 without considering any prior probability. For multi-parameter estimation, in quantum Fisher estimation theory the inverse of the QFI matrix provides an operator lower bound on the covariance of an unbiased estimator, and this bound is understood in the positive semidefinite sense. However, there has been little work on quantifying the quantum limits and measurement designs, for multi-parameter quantum estimation in aBayesiansetting. In this work, we build upon Personick's result to construct a Bayesian adaptive (greedy) measurement scheme for multi-parameter estimation. We illustrate our proposed measurement scheme with the application of 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 superior to quantum-noise-limited focal-plane direct imaging.
Resolving extended incoherent objects below the diffraction limit poses an application-rich imaging challenge whose solution may enable a new generation of observational instruments and capabilities. In this work, we invoke a practical model for general imaging by approximating an arbitrary extended incoherent object as a finite grid of thermal point emitters parameterized by their brightnesses. We derive the quantum Fisher information matrix (QFIM) for simultaneous brightness estimation and show that the symmetric logarithmic derivatives weakly commute, indicating that the Helstrom bound furnishes the ultimate quantum limit on the estimation error for incoherent imaging. Furthermore, for deeply sub-diffraction scenes, we find numerical evidence of a gap between the Nagaoka-Hayashi (NH) bound and the Helstrom bound. This gap reveals that separable measurements, though more experimentally accessible, are insufficient to reach the quantum limit, and points to the prospective advantage of joint measurements acting on multiple state copies. Additionally, we show that spatial mode-demultiplexing (SPADE) often saturates the NH bound solidifying its status as a near-optimal separable measurement strategy that significantly outperforms direct imaging. Finally, we articulate two joint detection receivers implemented with bona fide quantum resources that asymptotically achieve the Helstrom bound.
The optimal quantum measurements for estimating individual parameters might be incompatible with each other, so they cannot be jointly performed. The tradeoff between the estimation precisions for different parameters can be characterized by information regret: the difference between the Fisher information and its quantum limit. We show that the information-regret-tradeoff relation can give us not only an intuitive picture of the potential for improving the joint scheme for estimating the centroid and the separation but also some clues to the optimal measurements for the sequential scheme. In particular, we show that, for two incoherent point sources with a very small separation, the optimal measurement for the separation must extract little information about the centroid and vice versa.
Phase fluctuations between optical inputs limit programmable photonic processors because their output powers depend on coherent interference. We study the phase-drift penalty that arises when sine and cosine quadratures are measured sequentially rather than simultaneously. The analysis is motivated by measurements from an eight-mode programmable photonic processor, including 35 free-running recordings of 300 s acquired at approximately 125 samples per second per channel. These recordings provide an empirical route for estimating the phase-increment variance at a selected reconfiguration interval.
The estimate is defined at the time of the second measurement. For fixed quadrature order, perturbation of the atan2 reconstruction gives eC→S=−δτsin2φ0+O(δτ2) and eS→C=−δτcos2φ0+O(δτ2). Writing Qτ=Var(δτ), uniform phase averaging gives the first-order drift mean-square error 3Qτ/8. A phase-predicted ordering rule measures the locally less informative quadrature first and the more informative quadrature second. Its uniform first-order penalty is (3/8−1/π)Qτ, which is 84.9 percent below the fixed-order value.
We also derive an increment-aware estimator from a local state-space model. Marginalizing the unknown phase increment increases the variance of a stale phase observation by Qτ, reducing its Fisher information from I to I/(1+IQτ). For ideal balanced Poisson detection, the Fisher information of each quadrature equals its detected signal-photon number. This yields dimensionless architecture boundaries in spatial information and phase-increment variance. Nonlinear Monte Carlo simulations validate the perturbative laws, quantify robustness to prediction error, and compare simultaneous, fixed-order, increment-aware, and adaptive receivers under a common noise model.