Time remains one of the least well-understood concepts in physics, most notably in quantum mechanics. A central goal is to find the fundamental limits of measuring time. One of the main obstacles is the fact that time is not an observable and thus has to be measured indirectly. Here, we explore these questions by introducing a model of time measurements that is complete and autonomous. Specifically, our autonomous quantum clock consists of a system out of thermal equilibrium—a prerequisite for any system to function as a clock—powered by minimal resources, namely, two thermal baths at different temperatures. Through a detailed analysis of this specific clock model, we find that the laws of thermodynamics dictate a trade-off between the amount of dissipated heat and the clock’s performance in terms of its accuracy and resolution. Our results furthermore imply that a fundamental entropy production is associated with the operation of any autonomous quantum clock, assuming that quantum machines cannot achieve perfect efficiency at finite power. More generally, autonomous clocks provide a natural framework for the exploration of fundamental questions about time in quantum theory and beyond.

%B Physical Review X %V 7 %P 031022 %U https://link.aps.org/doi/10.1103/PhysRevX.7.031022 %R 10.1103/PhysRevX.7.031022 %0 Journal Article %J Quantum %D 2017 %T Quantifying high dimensional entanglement with two mutually unbiased bases %A Erker, Paul %A Krenn, Mario %A Huber, Marcus %X Paul Erker, Mario Krenn, and Marcus Huber, Quantum 1, 22 (2017). https://doi.org/10.22331/q-2017-07-28-22 We derive a framework for quantifying entanglement in multipartite and high dimensional systems using only correlations in two unbiased bases. We furthermore develop such bounds in cases whe… %B Quantum %V 1 %P 22 %U https://quantum-journal.org/papers/q-2017-07-28-22/ %R 10.22331/q-2017-07-28-22 %0 Journal Article %J Physical Review A %D 2016 %T Heisenberg-{Weyl} {Observables}: {Bloch} vectors in phase space %A Asadian, Ali %A Erker, Paul %A Huber, Marcus %A Klockl, Claude %X We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anticommuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first nontrivial example beyond the dichotomic case. %B Physical Review A %V 94 %P 010301 %8 jul %U https://link.aps.org/doi/10.1103/PhysRevA.94.010301 %R 10.1103/PhysRevA.94.010301