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

TL;DR

This paper establishes rigorous Heisenberg-style bounds for the average performance of arbitrary phase estimates, clarifying misconceptions about violations of the Heisenberg limit and analyzing various measures and restrictions.

Contribution

It provides an expanded proof of the Heisenberg limit for average phase estimation and introduces stronger bounds and numerical analyses.

Findings

Heisenberg limit holds for average phase accuracy

Stronger asymptotic bounds are proven

Numerical results support the theoretical bounds

Abstract

The ultimate bound to the accuracy of phase estimates is often assumed to be given by the Heisenberg limit. Recent work seemed to indicate that this bound can be violated, yielding measurements with much higher accuracy than was previously expected. The Heisenberg limit can be restored as a rigorous bound to the accuracy provided one considers the accuracy averaged over the possible values 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 together with a number of additional results, including the proof of a previously conjectured stronger bound in the asymptotic limit. Other measures of the accuracy are examined, as well as other restrictions on the generator of the phase shifts. We provide expanded numerical results for the minimum error and asymptotic expansions. The significance of the results claiming violation of the Heisenberg limit is assessed, followed by a detailed discussion of the limitations of the Cramer-Rao bound.