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.

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