Quantum scale estimation

TL;DR

This paper introduces a quantum framework for the most precise estimation of scale parameters, expanding quantum metrology beyond phase and location, with applications to thermometry and atomic lifetime measurement.

Contribution

It develops a Bayesian optimal measurement strategy for quantum scale parameters and generalizes the approach to multiple parameters, providing new tools for quantum metrology.

Findings

Optimal measurement rule for quantum scale estimation derived

Minimum mean logarithmic error identified

Framework applied to thermometry and atomic lifetime estimation

Abstract

Quantum scale estimation, as introduced and explored here, establishes the most precise framework for the estimation of scale parameters that is allowed by the laws of quantum mechanics. This addresses an important gap in quantum metrology, since current practice focuses almost exclusively on the estimation of phase and location parameters. For given prior probability and quantum state, and using Bayesian principles, a rule to construct the optimal probability-operator measurement is provided. Furthermore, the corresponding minimum mean logarithmic error is identified. This is then generalised as to accommodate the simultaneous estimation of multiple scale parameters, and a procedure to classify practical measurements into optimal, almost-optimal or sub-optimal is highlighted. As a means of illustration, the new framework is exploited to generalise scale-invariant global thermometry, as well as to address the estimation of the lifetime of an atomic state. On a more conceptual note, the optimal strategy is employed to construct an observable for scale parameters, an approach which may serve as a template for a more systematic search of quantum observables. Quantum scale estimation thus opens a new line of enquire - the precise measurement of scale parameters such as temperatures and rates - within the quantum information sciences.