Random bosonic states for robust quantum metrology

TL;DR

This paper investigates the metrological usefulness of random bosonic states, revealing that symmetric subspace states typically achieve Heisenberg scaling with simple measurements, even under noise, and can be efficiently generated.

Contribution

It demonstrates that symmetric subspace random states naturally attain optimal quantum metrology scaling without complex optimization, and introduces practical methods for their preparation.

Findings

Symmetric subspace states typically achieve Heisenberg scaling.

Random pure states from the full Hilbert space do not surpass classical limits.

Simple photon-counting measurements suffice for Heisenberg scaling.

Abstract

We study how useful random states are for quantum metrology, i.e., surpass the classical limits imposed on precision in the canonical phase estimation scenario. First, we prove that random pure states drawn from the Hilbert space of distinguishable particles typically do not lead to super-classical scaling of precision even when allowing for local unitary optimization. Conversely, we show that random states from the symmetric subspace typically achieve the optimal Heisenberg scaling without the need for local unitary optimization. Surprisingly, the Heisenberg scaling is observed for states of arbitrarily low purity and preserved under finite particle losses. Moreover, we prove that for such states a standard photon-counting interferometric measurement suffices to typically achieve the Heisenberg scaling of precision for all possible values of the phase at the same time. Finally, we demonstrate that metrologically useful states can be prepared with short random optical circuits generated from three types of beam-splitters and a non-linear (Kerr-like) transformation.