Abstract
PhD thesis, University of Sussex (2020) The number of times that we can access a system to extract information via
quantum metrology is always finite, and possibly small, and realistic amounts
of prior knowledge tend to be moderate. Thus theoretical consistency demands a
methodology that departs from asymptotic approximations and restricted
parameter locations, while practical convenience requires that it is also
flexible and easy to use in applications with limited data. We submit that this
methodology can and should be built on a Bayesian framework, and in this thesis
we propose, construct, explore and exploit a new non-asymptotic quantum
metrology. First we show the consistency of taking those solutions that are
optimal in the asymptotic regime of many trials as a guide to calculate a
Bayesian measure of uncertainty. This provides an approximate but useful way of
studying the non-asymptotic regime whenever an exact optimisation is
intractable, and it avoids the non-physical results that can arise when only
the asymptotic theory is used. Secondly, we construct a new non-asymptotic
Bayesian bound without relying on the previous approximation by first selecting
a single-shot optimal quantum strategy, and then simulating a sequence of
repetitions of this scheme. These methods are then applied to a
single-parameter Mach-Zehnder interferometer, and to multi-parameter qubit and
optical sensing networks. Our results provide a detailed characterisation of
how the interplay between prior information, correlations and a limited amount
of data affects the performance of quantum metrology protocols, which opens the
door to a vast set of unexplored possibilities to enhance non-asymptotic
schemes. Finally, we provide practical researchers with a numerical toolbox for
Bayesian metrology, while theoretical workers will benefit from the broader and
more fundamental perspective that arises from the unified character of our
methodology.