Ultimate precision bounds on quantum thermometry

This theoretical PhD project investigates the fundamental limits to the precision with which temperature can be estimated in quantum systems. The first part of the thesis presents a Bayesian formulation of temperature estimation theory, which is based on the thermodynamic length between thermal sample states. We show how the resulting temperature estimation theory can be mapped onto a Euclidean parameter estimation problem, and this insight makes it possible to apply precision bounds developed in this context. When mapping these bound from the Euclidean theory, back to the space of temperatures, it is found that the measure of precision resulting from a framework based on the thermodynamic length, has a natural interpretation as a generalized relative error.
The Bayesian approach to temperature estimation, in which a priori incomplete information is assumed, points to the need of effective adaptive strategies, with which the measurement strategy can be updated as temperature information is extracted. We propose two such adaptive strategies. In the case of equilibrium probe thermometry, we show that no non-adaptive measurement strategy can exhibit Heisenberg-like scaling with the probe dimension. Furthermore, it is illustrated that an adaptive strategy can restore Heisenberg-like scaling with the probe dimension. These results highlight the essential role of adaptation for thermometry.
The last part of this thesis investigates optimal thermometry under realistic constraints on the available measurements. In particular, we focus on the challenges associated with thermometry at very low temperatures. It is found that finite energy-resolution fundamentally constrains the attainable sensitivity of a measurement to temperature in the ultracold regime. The relevance of the derived bounds is illustrated by considering a numerically exact simulation of a probe-based thermometry protocol. Furthermore, we investigate fundamental limitations under constraints of measurements with only a finite number of distinguishable outcomes, and derive the associated precision bounds. It is shown that for the majority of many-body quantum systems, there exist coarse-grained measurements achieving a temperature sensitivity comparable to that of the many-body system itself.