Sub-Heisenberg estimation strategies are ineffective

TL;DR

This paper demonstrates that sub-Heisenberg strategies do not outperform traditional methods unless significant prior information is available, establishing fundamental limits on their effectiveness across various parameter estimation scenarios.

Contribution

It proves that sub-Heisenberg strategies cannot surpass the Heisenberg limit without prior information, extending the bounds to general parameter estimation.

Findings

Sub-Heisenberg strategies offer limited gain without prior information.

Maximum gain over Heisenberg limit is approximately 1.73 without prior knowledge.

Effective error bounds are proportional to the inverse expectation of the generator.

Abstract

In interferometry, sub-Heisenberg strategies claim to achieve a phase estimation error smaller than the inverse of the mean number of photons employed (Heisenberg bound). Here we show that one can achieve a comparable precision without performing any measurement, just using the large prior information that sub-Heisenberg strategies require. For uniform prior (i.e. no prior information), we prove that these strategies cannot achieve more than a fixed gain of about 1.73 over Heisenberg-limited interferometry. Analogous results hold for arbitrary single-mode prior distributions. These results extend also beyond interferometry: the effective error in estimating any parameter is lower bounded by a quantity proportional to the inverse expectation value (above a ground state) of the generator of translations of the parameter.