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The ultimate bound to the accuracy of phase estimates is often assumed to begiven by the Heisenberg limit. Recent work seemed to indicate that this boundcan be violated, yielding measurements with much higher accuracy than waspreviously expected. The Heisenberg limit can be restored as a rigorous boundto the accuracy provided one considers the accuracy averaged over the possiblevalues of the unknown phase, as we have recently shown [Phys. Rev. A 85,041802(R) (2012)]. Here we present an expanded proof of this result togetherwith a number of additional results, including the proof of a previouslyconjectured stronger bound in the asymptotic limit. Other measures of theaccuracy are examined, as well as other restrictions on the generator of thephase shifts. We provide expanded numerical results for the minimum error andasymptotic expansions. The significance of the results claiming violation ofthe Heisenberg limit is assessed, followed by a detailed discussion of thelimitations of the Cramer-Rao bound.

Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates