Bayesian parameter estimation and thermodynamics in equilibrating quantum systems

The trend towards miniaturised devices has resulted in systems small enough to be governed by quantum mechanics. The goal of these systems is to perform thermodynamic tasks like refrigeration or work extraction, which requires a description of how they interact with their environment. This has spurred the development of quantum thermodynamics. However, achieving the desired experimental control requires precise knowledge of the system and highly accurate measurements. This thesis explores these two aspects of open quantum systems from a theoretical perspective. First, we explore how Bayesian techniques can be applied to the sensing of environmental parameters of a quantum system to find better estimation protocols, particularly in situations with little data, and where adaptive strategies are allowed. The advantages and drawbacks of various Bayesian estimation approaches are explored. These methods are specifically examined with respect to the role of prior distributions, estimators, and cost functions in achieving accurate estimates. The primary example involves qubit thermometry, where a two-level probe interacts with an environment. This example highlights the effectiveness of Bayesian estimation for small data sets and quantifies the scaling of the accuracy with number of measurements using Bayesian bounds. The sensitivity of probes based on environmental interactions is also analysed. The estimation of rate parameters is studied with particular emphasis on how the specification of the prior information can influence the entire estimation strategy. Next, another example of thermometry is considered using continuously monitored probes. An adaptive strategy is proposed which showcases the benefits of Bayesian estimation. We also study this scenario in the case when the measurement signal includes noise and finite detector bandwidth. Finally, we turn to our second aspect of open quantum systems and study work extraction from an open quantum system. Here, we investigate how collective effects arising from the interaction of permutationally invariant particles with their environment affect work extraction. The analysis includes various models of work extraction, including energy output in steady states, work done against dissipative loads, and power output when coupled to a driving field.