A Bayesian view of single-qubit clocks, and an energy versus accuracy tradeoff

We bring a Bayesian viewpoint to the analysis of clocks. Using exponential distributions as priors for clocks, we analyze the case of a single precessing spin. We find that, at least with a single qubit, quantum mechanics does not allow exact timekeeping, in contrast to classical mechanics which does. We find the optimal ratio of angular velocity of precession to rate of the exponential distribution that leads to maximum accuracy. Further, we find an energy versus accuracy tradeoff - the energy cost is at least k
_B
T times the improvement in accuracy as measured by the entropy reduction in going from the prior distribution to the posterior distribution.