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Optics Express

Quantum Fisher information for estimating N partially coherent point sources

Kevin Liang, S. A. Wadood, A. Nick Vamivakas

Published Dec 6, 2022• 12 citations
Publisher

Abstract

A partially coherent object's localization parameters are shown to be theoretically estimable with higher precision than those of an incoherent object, and the maximum number of independent parameters that have non-vanishing precision in the sub-Rayleigh regime is 3 (compared to 2 for an incoherent object). Normalization schemes, which are crucial in the proper interpretation of quantum Fisher information results in the presence of partial coherence, are introduced and detailed.

DOI
10.1364/oe.474036

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A partially coherent object's localization parameters are shown to be theoretically estimable with higher precision than those of an incoherent object, and the maximum number of independent parameters that have non-vanishing precision in the sub-Rayleigh regime is 3 (compared to 2 for an incoherent object). Normalization schemes, which are crucial in the proper interpretation of quantum Fisher information results in the presence of partial coherence, are introduced and detailed.
We consider imaging of two partially coherent sources and derive the ultimate quantum limits for estimating the separation, location, relative intensity, and coherence factor. We show that super-resolution in the separation is achievable both in the presence of nuisance parameters as well as when some of the parameters are assumed known. Nuisance parameters restrict super-resolution to balanced sources, whereas for known parameters super-resolution persists over a broader range of relative intensities and is lost only for perfectly correlated sources, i.e., γ=1γ=1γ=1. The achievable precision is governed primarily by interference-induced photon statistics and depends strongly on the degree of coherence. In the sub-Rayleigh regime, the imaging problem reduces to an effective two-dimensional Hilbert space description, provided a consistent reference position is used, with all parameters encoded in a Bloch vector representation. Finally, indirect estimation schemes based on the purity of the image-plane state are generally suboptimal for all non-zero coherence and become valid only in the incoherent limit.
In recent proposals for achieving optical super-resolution, variants of the Quantum Fisher Information (QFI) quantify the attainable precision. We find that claims about a strong enhancement of the resolution resulting from coherence effects are questionable because they refer to very small subsets of the data without proper normalization. When the QFI is normalized, accounting for the strength of the signal, there is no advantage of coherent sources over incoherent ones. Our findings have a bearing on further studies of the achievable precision of optical instruments.
In recent proposals for achieving optical super-resolution, variants of the Quantum Fisher Information (QFI) quantify the attainable precision. We find that claims about a strong enhancement of the resolution resulting from coherence effects are questionable because they refer to very small subsets of the data without proper normalization. When the QFI is normalized, accounting for the strength of the signal, there is no advantage of coherent sources over incoherent ones. Our findings have a bearing on further studies of the achievable precision of optical instruments.
Tsang et al. have shown that the Fisher information of the two incoherent point source separation, below the Rayleigh limit, is finite and achievable using optical modes measurements.1 However, recent claims regarding partial coherence of sources, no matter how small, leads to necessarily zero Fisher information as the source separation decreases below the Rayleigh limit approaching zero have proved to be controversial.2, 3 Thus, the impact of partial coherence on the photon counting optical modal measurements merits further exploration. In this work, we derive the mutual coherence function (image plane) of two partially coherent point sources and find the classical Fisher information of the source separation using both direct image plane and photon counting modal measurements. A classical Fisher information analysis of partially coherent source(s) leads to some rather surprising results for two-point source resolution as the source separation approaches zero. We find that the magnitude of the Fisher information strongly depends on the degree of (positive/negative) partial coherence, which can be understood using an intuitive semi-classical analysis of direct image plane and photon counting modal measurements. We also provide an error analysis of the maximum likelihood estimators for both measurements.
The quantum Fisher information (FI), when applied to the estimation of the separation of two point sources, has been shown to be non-zero in cases where the coherence between the sources are known. Although it has been claimed that ignorance of the coherence causes the quantum FI to vanish (a resurgence of Rayleigh's curse), a more complete analysis including both the magnitude and phase of the coherence parameter is given here. Partial ignorance of the coherence is shown to potentially break Rayleigh's curse, whereas complete ignorance guarantees its resurgence.