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arXiv:2103.07225

Exploring the ultimate limits: Super-resolution enhanced by partial coherence

Z. Hradil, Dominik Koutný, Jaroslav Řeháček

Published Jan 1, 2021• 0 citations
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Abstract

The resolution of separation of two elementary signals forming a partially coherent superposition, defined by Quantum Fisher Information and normalised with respect to detection probabilities, is always limited by the resolution of incoherent mixtures. However, when the partially coherent superpositions are prepared in a controlled way the precision can be enhanced by up to several orders of magnitude above this limit. Coherence also allows the sorting of information about various parameters into distinct channels as demonstrated by parameter of separation linked with the anti-phase superposition and the centroid position linked with the in-phase superposition.

DOI
10.48550/arxiv.2103.07225

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The resolution of separation of two elementary signals forming a partially coherent superposition, defined by Quantum Fisher Information and normalised with respect to detection probabilities, is always limited by the resolution of incoherent mixtures. However, when the partially coherent superpositions are prepared in a controlled way the precision can be enhanced by up to several orders of magnitude above this limit. Coherence also allows the sorting of information about various parameters into distinct channels as demonstrated by parameter of separation linked with the anti-phase superposition and the centroid position linked with the in-phase superposition.
The resolution of separation of two elementary signals forming a partially coherent superposition, defined by quantum Fisher information and normalized with respect to detection probabilities, is always limited by the resolution of incoherent mixtures. However, when the partially coherent superpositions are prepared in a controlled way, the precision can be enhanced by up to several orders of magnitude above this limit. Coherence also allows the sorting of information about various parameters into distinct channels as demonstrated by the parameter of separation linked with the anti-phase superposition and the centroid position linked with the in-phase superposition.
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.
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.
The diffraction, or Rayleigh, limit was long thought to determine the ultimate resolving power of optical imaging devices. Using techniques from quantum statistical inference, Tsang et al. [1] demonstrated that this limit can be circumvented, and derived an alternative measurement strategy capable of resolving two incoherent point sources no matter how arbitrarily small their separation is. However, these measurements require perfect knowledge of the sources centroid position, and are extremely sensitive to any source of misalignment about this position [2] , [3] . In this work, we study the problem of super-resolution from a more global point of view.
When two point sources are separated by less than the Abbe-Rayleigh limit, conventional resolution methods fail to distinguish whether there is one source or two. Consequently, it becomes nearly impossible to resolve important relative properties of the two sources, such as mutual coherence, relative phase, and brightness contrast. In this regime, the incoming composite quantum signal must be treated as a single entity described by an overall quantum wave function. In this study, we explore the superresolution of such two-point sources by examining a special wave property, i.e., entanglement between spatial degree of freedom (DoF) and remaining DoFs (both internal and external). Remarkably, we find that increasing this inter-DoF entanglement enhances the precision of superresolution, as quantified by the quantum Fisher information (QFI) in single-parameter estimation. Moreover, we show that this entanglement reliably ensures a nonvanishing QFI in two-parameter estimation, highlighting its role as a valuable resource for quantum superresolution.