Characterisation and optimality of open quantum system dynamics

We develop tools to characterise the dynamics of open quantum systems. We start by introducing the concept of action quantum speed limits (QSLs). Unlike conventional geometric methods, these QSLs intricately depend on the instantaneous speed, offering bounds on the minimal time needed to connect states. The instantaneous speed along fixed trajectories is shown to be an important and optimisable degree of freedom, as exemplified through the thermalising qubit case. Beyond discussing the feasibility of geometric QSLs, we also critically examine their interpretation in terms of different metric choices. It is revealed that these open-system QSL times provide indications of optimality concerning geodesic paths, rather than being strict minimal time indicators.

Distinguishability, based on distance metrics employed in deriving QSLs (in particular the Fisher information), is also the key concept in the field of metrology. We consider the use of open system dynamics as a model to explore parameter estimation. We investigate how the presence of correlation between measurement results \new{affects} the Fisher information. These correlations are introduced through a sequential measurement scheme, where the same probe is measured multiple times in succession without allowing for equilibration. We prove that, for there to be any advantage in precision as a result of these correlations, there must be information encoded into the system-environment interaction term related to the parameter that we are trying to estimate. To emphasise this, we consider the specific case of temperature estimation where the thermalisation rate of the probe contains additional information about the temperature. The sequential scheme can be viewed as a form of collisional quantum thermometry, which further allows additional freedoms in the protocol, e.g. by introducing stochasticity in the waiting time between collisions. We establish that incorporating randomness in this manner leads to a significant expansion of the parameter range for achieving advantages over typical equilibrium approaches to thermometry. Intriguingly, we demonstrate that in certain settings optimal measurements can be performed locally, highlighting the limited role of genuine quantum correlations in this advantage.

Finally, we delve into the statistics of the work done on a quantum system via a two-point measurement scheme. The Shannon entropy of the work distribution is shown to possess a general upper bound tied to initial diagonal entropy and a distinct quantum term associated with the relative entropy of coherence. This approach is shown to capture signatures of underlying physics across diverse scenarios. In particular, through an in-depth exploration of the Aubry-André-Harper model, we illustrate how the entropy of the work distribution provides a useful tool for characterising the localisation transition. We further explore the use of the entropy of the work distribution as a tool for identifying the mobility edge in a generalised Aubry-André-Harper model.

Collectively, these results provide a toolbox to assess the optimality, either in terms of the dynamical paths taken, utility for metrological tasks, or ability to spotlight relevant physical properties of the model, for the dynamics of complex quantum systems.