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.

1 aErker, Paul1 aMitchison, Mark, T.1 aSilva, Ralph1 aWoods, Mischa, P.1 aBrunner, Nicolas1 aHuber, Marcus uhttps://link.aps.org/doi/10.1103/PhysRevX.7.03102200757nas a2200145 4500008003900000245007900039210006900118300000700187490000600194520030400200100001600504700001700520700001800537856005600555 2017 d00aQuantifying high dimensional entanglement with two mutually unbiased bases0 aQuantifying high dimensional entanglement with two mutually unbi a220 v13 aPaul 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…1 aErker, Paul1 aKrenn, Mario1 aHuber, Marcus uhttps://quantum-journal.org/papers/q-2017-07-28-22/01085nas a2200169 4500008003900000245006800039210006100107260000800168300001100176490000700187520059500194100001700789700001600806700001800822700001900840856005600859 2016 d00aHeisenberg-{Weyl} {Observables}: {Bloch} vectors in phase space0 aHeisenberg Weyl Observables Bloch vectors in phase space cjul a0103010 v943 aWe 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.1 aAsadian, Ali1 aErker, Paul1 aHuber, Marcus1 aKlockl, Claude uhttps://link.aps.org/doi/10.1103/PhysRevA.94.010301