Michael J. W. Hall, Howard M. Wiseman
A number of authors have suggested that nonlinear interactions can enhance resolution of phase shifts beyond the usual Heisenberg scaling of 1/n, where n is a measure of resources such as the number of subsystems of the probe state or the mean photon number of the probe state. These suggestions are based on calculations of `local precision' for particular nonlinear schemes. However, we show that there is no simple connection between the local precision and the average estimation error for these schemes, leading to a scaling paradox. This is partially resolved by a careful analysis of iterative implementations of the suggested nonlinear schemes. However, it is shown that adaptive multi-pass linear schemes can achieve an average estimation error having an exponential scaling in n, whereas the suggested nonlinear schemes are limited to an exponential scaling in \sqrt{n}. The question of whether nonlinear schemes may have a scaling advantage in the presence of loss is left open. Our results are based on a new bound for average estimation error that depends on (i) an entropic measure of the degree to which the probe state can encode a reference phase value, called the G-asymmetry, and (ii) any prior information about the phase shift. This bound is asymptotically stronger than bounds based on the variance of the phase shift generator. The G-asymmetry is also shown to directly bound the average information gained per estimate. Our results generalise to estimates of any shift generated by an operator with discrete eigenvalues.
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http://arxiv.org/abs/1205.2405
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