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arXiv:2202.05647 · Physical Review A

Performance-tradeoff relation for locating two incoherent optical point sources

Jingjing Shao, Xiao-Ming Lu

Published Jun 9, 2022• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.105.062416

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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.
Abstract Simultaneously optimizing the estimation of the centroid and separation of two incoherent optical point sources is constrained by a tradeoff relation determined by an incompatibility coefficient. At the Rayleigh distance, the incompatibility coefficient vanishes and thus the tradeoff relation no longer restricts the simultaneous optimization of measurement for a joint estimation. We construct such a joint optimal measurement by an elaborated analysis on the operator algebra of the symmetric logarithmic derivative. Our work not only confirms the existence of a joint optimal measurement for this specific imaging model, but also gives a promising method to characterize the condition on measurement compatibility for general multiparameter estimation problems.
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A recent paper [Yu and Prasad Phys. Rev. Lett. 121, 180504 (2018)] considered the problem of quantum-limited estimation of the separation vector for a pair of incoherent point sources in all three dimensions. Here we extend our analysis to treat the problem of simultaneously estimating the location of the centroid and the separation of the source pair, which is equivalent to localizing both sources simultaneously. We first calculate the quantum Fisher information for simultaneous pair centroid-separation estimation and then discuss the fundamental, estimation-theoretic tradeoffs between the two tasks, which we confirm using simulations.
Joint or simultaneous measurements of non-commuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for defining what an "optimal" joint measurement is, with corresponding different tradeoff relations for the measurements. Understanding the limitations of such measurements is of fundamental interest and relevant for quantum technology. Here, we experimentally test a tradeoff relation for the sharpness of qubit measurements, a relation which refers directly to the form of the measurement operators, rather than to errors in estimates. We perform the first optical implementation of the simplest possible optimal joint measurement, requiring less quantum resources than have previously often been employed. Using a heralded single-photon source, we demonstrate quantum-limited performance of the scheme on single quanta.
We establish the multiparameter quantum Cram\'er-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cram\'er-Rao bound is independent of their separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost importance in realistic scenarios, such as observational astronomy.