Bounds on the entropy generated when timing information is extracted from microscopic systems

We consider Hamiltonian quantum systems with energy bandwidth \Delta E and show that each measurement that determines the time up to an error \Delta t generates at least the entropy (\hbar/(\Delta t \Delta E))^2/2. Our result describes quantitatively to what extent all timing information is quantum information in systems with limited energy. It provides a lower bound on the dissipated energy when timing information of microscopic systems is converted to classical information. This is relevant for low power computation since it shows the amount of heat generated whenever a band limited signal controls a classical bit switch. Our result provides a general bound on the information-disturbance trade-off for von-Neumann measurements that distinguish states on the orbits of continuous unitary one-parameter groups with bounded spectrum. In contrast, information gain without disturbance is possible for some completely positive semi-groups. This shows that readout of timing information can be possible without entropy generation if the autonomous dynamical evolution of the clock is dissipative itself.