Quantum Scale Estimation

Abstract

Quantum scale estimation, as introduced and explored here, establishes the most precise framework for the estimation of scale parameters which is allowed by the laws of quantum mechanics. This closes an important gap in quantum metrology, since current practice focuses almost exclusively on the estimation of phase and location parameters, using either periodic or square errors, and these do not necessarily apply when dealing with scale parameters, for which logarithmic errors are instead required. Using Bayesian principles, a general method to construct both the optimal estimator and the associated probability-operator measurement is first developed. An analytical expression for the true minimum mean logarithmic error is further provided, and a partial generalisation to accommodate the simultaneous estimation of multiple scale parameters is discussed. In addition, a procedure to identify whether a practical measurement is optimal, almost-optimal or sub-optimal is highlighted. 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 decay rates-within the quantum information sciences. 1 The square error can be used as an approximation to the sine error when the estimator θ(x) and the values for the hypothesis θ are close [3,46].

Optimal cold atom thermometry using adaptive Bayesian strategies

“…We have shown how quantum thermometry can deliver practical solutions leading to quantifiable precision enhancements, once the quantum-Fisher-information based 'local' paradigm [18] is abandoned in favour the global framework. While the former has proven useful when studying fundamental precision limits and maximising the responsiveness of temperature sensors, the more general Bayesian framework can, in addition to that [40,43], process finite experimental records optimally. Importantly, the exact same fundamental principles can be applied to other techniques in different temperature ranges and experimental platforms, which opens a new exciting avenue in quantum thermometry.…”

“…We start by outlining the Bayesian-global paradigm introduced by some of us in Refs. [35,40]. Even though we later focus on the specific release-recapture setup, this framework is fully applicable to any thermometry experiment in which temperature behaves as a 'scale parameter' [40,42].…”

“…As we shall see next, the energy scale in which our model p(n i |T, t i ) for release-recapture thermometry is applicable is determined by the unknown temperature itself. We thus say that temperature is a scale parameter [35,40,42,44], and adopt Jeffreys's prior…”

“…1, so that it is the unknown T itself which determines the energy scale in which our model for the fraction of recaptured atoms is applicable. Hence, we are dealing with a scale-estimation problem [40,42] for which Eqs. (2-5) are appropriate.…”

“…However, in the experimentally relevant scenario of finite measurement records, with just tens or hundreds of shots, F does not always capture-even qualitatively-the behaviour of optimal temperature estimates. In these cases, one must adopt the more general Bayesian framework [32][33][34], which is only starting to be explored in quantum thermometry [35][36][37][38][39][40][41].…”