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A rigorous lower bound is obtained for the average resolution of any estimateof a shift parameter, such as an optical phase shift or a spatial translation.The bound has the asymptotic form k_I/<2|G|> where G is the generator of theshift (with an arbitrary discrete or continuous spectrum), and henceestablishes a universally applicable bound of the same form as the usualHeisenberg limit. The scaling constant k_I depends on prior information aboutthe shift parameter. For example, in phase sensing regimes, where the phaseshift is confined to some small interval of length L, the relative resolution\delta\hat{\Phi}/L has the strict lower bound (2\pi e^3)^{-1/2}/<2m| G_1 |+1>,where m is the number of probes, each with generator G_1, and entangling jointmeasurements are permitted. Generalisations using other resource measures andincluding noise are briefly discussed. The results rely on the derivation ofgeneral entropic uncertainty relations for continuous observables, which are ofinterest in their own right.

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