Lower Bounds for Quantum Parameter Estimation

The laws of quantum mechanics place fundamental limits on the accuracy of measurements and, therefore, on the estimation of unknown parameters of a quantum system. In this paper, we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success. Our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test. In group-covariant scenarios, we find that there is an ultimate bound for any estimation scheme, which depends only on the representation-theoretic data of the probe system, and we evaluate its asymptotics in the limit of many systems, establishing a general Heisenberg limit for region estimation. We apply our results to several examples, in particular, to phase estimation, where our bounds allow us to recover the well-known Heisenberg and shot-noise scaling.