Even though our microscopic physical theories, such as classical or quantum mechanics, are time-reversal invariant, time appears to march forward irreversibly. It is widely believed that effective entropy increase due to the second law of thermodynamics is at the origin of this emergent arrow of time. Any system that is out of thermal equilibrium and produces entropy can thus be used as a clock, be it a melting ice cube or a sophisticated atomic clock. The understanding of entropy in physics has significantly advanced with the advent of information theory in the last century. It has also led to the insight that the amount of entropy a clock produces is linked to how precisely the clock works, and thereby, how much information the clock reveals about the passage of time. In this thesis, we ask: what fundamental limits does the underlying physical theory impose on the precision of clocks within that theory? To rigorously answer this question, we devise a minimal clock model within quantum theory. By consistently modeling all clock constituents as quantum systems, we can ensure we account for all the resources used by the clock. This includes, in particular, the readout. Classically, one can obtain this for free, but quantum mechanically, this has a resource cost associated with it. To this end, we consolidate existing frameworks that describe clocks in both classical and quantum theory. Revising fluctuation theorems from quantum thermodynamics, we demonstrate how such theorems impose limits on the attainable precision of clocks. These precision bounds turn out to be more stringent for classical systems than for quantum ones. Quantum clocks can harness coherence and correlations to suppress fluctuations, allowing them to exponentially surpass precision bounds that hold for classical clocks. We show this separation between classical and quantum clocks by constructing an explicit example that uses quantum dynamics to saturate the quantum precision bounds. Finally, we close the thesis by exploring implications of these timekeeping bounds beyond fundamental science in quantum technologies. We show how quantum clocks can be used to quantify the energetic cost of quantum computation by linking computational fidelity to thermodynamic entropy production.
Florian Meier (Fri,) studied this question.