Towards the Quantum Limits of Phase Retrieval

TL;DR

This paper investigates the fundamental quantum limits of phase retrieval for single-mode electromagnetic fields, deriving optimal measurement strategies and analyzing the quantum Fisher information for various quantum states.

Contribution

It derives the quantum Fisher information matrix for a broad class of input states and constructs optimal measurements, advancing understanding of quantum-enhanced phase estimation.

Findings

QFIM is diagonal for the considered states

Existence of measurements saturating the QFIM confirmed

Optimal measurements constructed for specific quantum states

Abstract

We consider the problem of determining the spatial phase profile of a single-mode electromagnetic field. Our attention is on input states that are a statistical mixture of displaced and squeezed number states, a superset of Gaussian states. In particular, we derive the quantum Fisher information matrix (QFIM) for estimating the expansion coefficients of the wavefront in an orthonormal basis, finding that it is diagonal. Moreover, we show that a measurement saturating the QFIM always exists, and point to an adaptive strategy capable of implementing it. We then construct the optimal measurements for three particular states: mixtures of photon number, coherent, and single-mode squeezed vacuum states. Sensitivity of the measurements to nuisance parameters is explored.