Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information
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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 kI/<2|G|> 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 kI 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 has the strict lower bound (2pe3)-1/2/<2m|G1| + 1>, where m is the number of probes, each with generator G1, 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.
New Journal of Physics
Copyright 2012 Institute of Physics Publishing. The attached file is reproduced here in accordance with the copyright policy of the publisher. Please refer to the journal's website for access to the definitive, published version.
Page numbers are not for citation purposes. Instead, this article has the unique article number of 033040.
Quantum Information, Computation and Communication