Does nonlinear metrology offer improved resolution? Answers from quantum information theory

TL;DR

This paper critically examines the potential of nonlinear quantum metrology to improve phase resolution, revealing fundamental limits and clarifying misconceptions through a new entropic bound and analysis of iterative schemes.

Contribution

It introduces a new bound based on G-asymmetry for average estimation error and clarifies the scaling limits of nonlinear schemes in quantum metrology.

Findings

Nonlinear schemes are limited to exponential scaling in √n.

Local precision does not directly translate to average estimation error.

New entropic bound improves understanding of quantum phase estimation.

Abstract

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 puzzle. This puzzle is partially resolved by a careful analysis of iterative implementations of the suggested nonlinear schemes. However, it is shown that the suggested nonlinear schemes are still limited to an exponential scaling in \sqrt{n}. (This scaling may be compared to the exponential scaling in n which is achievable if multiple passes are allowed, even for linear schemes.) 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 hold for any prior distribution of the shift parameter, and generalise to estimates of any shift generated by an operator with discrete eigenvalues.