Not all pure entangled states are useful for sub shot-noise interferometry

TL;DR

This paper explores which pure entangled states can achieve sub shot-noise sensitivity in linear interferometry, revealing that not all such states are useful for this purpose when the particle number exceeds two.

Contribution

It demonstrates that for more than two particles, not all pure entangled states can be optimized for sub shot-noise interferometry, providing a classification of useful states.

Findings

All pure entangled states with two particles can be used for sub shot-noise interferometry.

Most multi-particle entangled states, including cluster states, are not useful for sub shot-noise sensitivity.

The usefulness of states is determined via the Fisher information and Cramer-Rao bound.

Abstract

We investigate the connection between the shot-noise limit in linear interferometers and particle entanglement. In particular, we ask whether or not sub shot-noise sensitivity can be reached with all pure entangled input states of $N$ particles if they can be optimized with local operations. Results on the optimal local transformations allow us to show that for $N=2$ all pure entangled states can be made useful for sub shot-noise interferometry while for $N>2$ this is not the case. We completely classify the useful entangled states available in a bosonic two-mode interferometer. We apply our results to several states, in particular to multi-particle singlet states and to cluster states. The latter turn out to be practically useless for sub shot-noise interferometry. Our results are based on the Cramer-Rao bound and the Fisher information.