Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information

A rigorous lower bound is obtained for the average resolution of any estimate of a shift parameter, such as an optical phase shift or a spatial translation. The bound has the asymptotic form k(I)/< 2 vertical bar G vertical bar > where G is the generator of the shift (with an arbitrary discrete or continuous spectrum), and hence establishes a universally applicable bound of the same form as the usual Heisenberg limit. The scaling constant k(I) depends on prior information about the shift parameter. For example, in phase sensing regimes, where the phase shift is confined to some small interval of length L, the relative resolution delta(Phi) over cap /L has the strict lower bound (2 pi e(3))(-1/2)/< 2m vertical bar G(1)vertical bar+1 >, where m is the number of probes, each with generator G(1), and entangling joint measurements are permitted. Generalizations using other resource measures and including noise are briefly discussed. The results rely on the derivation of general entropic uncertainty relations for continuous observables, which are of interest in their own right.