Joint optimal measurement for locating two incoherent optical point sources near the Rayleigh distance

TL;DR

This paper demonstrates the existence of a joint optimal measurement for estimating the position and separation of two incoherent optical sources at the Rayleigh distance, overcoming traditional tradeoffs in quantum optical measurement.

Contribution

It constructs a joint optimal measurement for this specific imaging problem and offers a method to analyze measurement compatibility in multiparameter quantum estimation.

Findings

Joint optimal measurement exists at the Rayleigh distance.

The method characterizes measurement compatibility conditions.

Confirms the theoretical possibility of simultaneous optimal estimation.

Abstract

The simultaneous optimization of the centroid estimation and the separation estimation of two incoherent optical point sources is restricted by a tradeoff relation through 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.