Non-asymptotic quantum metrology
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.