# Quantum-Limited Estimation of Phase Gradient

## TL;DR

This paper investigates the fundamental quantum limits on measuring optical phase gradients, showing entanglement enhances precision but practical measurement schemes may reduce this advantage, with implications for quantum sensing.

## Contribution

It derives the quantum Cramér-Rao bounds for phase gradient estimation, compares single-photon and entangled two-photon states, and analyzes measurement strategies affecting the quantum advantage.

## Key findings

- Entangled two-photon states outperform single-photon states in ideal conditions.
- Structured measurements can reduce the quantum advantage for large beam widths.
- Simultaneous estimation of phase and phase gradient is feasible with optimal strategies.

## Abstract

We show that the quantum Cram\'er-Rao bound on the precision of measurements of the optical phase gradient, or the wavefront tilt, with a beam of finite width is consistent with the Heisenberg uncertainty principle for a single-photon state, and is a factor of 2 better for the two-photon state that is maximally entangled. This fundamental bound governs a trade-off between quantum sensitivity and spatial resolution. Precision bounds based on a structured configuration using binary projective measurements implemented by an image-inversion interferometer, are higher, and the two-photon factor of 2 advantage is lost for large beam width or large phase gradient. In all cases, estimation of the phase gradient is compatible with estimation of the phase, allowing for optimal joint estimation of both parameters simultaneously.