Ultimate limits to quantum metrology and the meaning of the Heisenberg limit

TL;DR

This paper clarifies the fundamental limits of quantum measurement precision by redefining the Heisenberg limit using a universal resource count, resolving longstanding ambiguities in the field.

Contribution

It introduces a universal resource measure based on the generator of translations, establishing the ultimate Heisenberg limit for quantum metrology.

Findings

The new limit applies to generators with bounded spectra.

It unifies various definitions of resources in quantum metrology.

The formulation clarifies the fundamental capabilities of quantum measurement.

Abstract

For the last 20 years, the question of what are the fundamental capabilities of quantum precision measurements has sparked a lively debate throughout the scientific community. Typically, the ultimate limits in quantum metrology are associated with the notion of the Heisenberg limit expressed in terms of the physical resources used in the measurement procedure. Over the years, a variety of different physical resources were introduced, leading to a confusion about the meaning of the Heisenberg limit. Here, we review the mainstream definitions of the relevant resources and introduce the universal resource count, that is, the expectation value of the generator (above its ground state) of translations in the parameter we wish to estimate, that applies to all measurement strategies. This leads to the ultimate formulation of the Heisenberg limit for quantum metrology. We prove that the new limit holds for the generators of translations with an upper-bounded spectrum.