Optimal estimation of joint parameters in phase space

TL;DR

This paper demonstrates that entangled Gaussian probes enable joint estimation of phase space displacement parameters with precision surpassing the standard quantum limit, approaching optimality across various conditions.

Contribution

It introduces a measurement scheme using entangled Gaussian states and homodyne detection to achieve near-optimal joint parameter estimation in phase space.

Findings

Entanglement improves measurement precision below the standard quantum limit.

The proposed scheme approaches the theoretical Cramér-Rao bound.

The method relates to generalized uncertainty relations.

Abstract

We address the joint estimation of the two defining parameters of a displacement operation in phase space. In a measurement scheme based on a Gaussian probe field and two homodyne detectors, it is shown that both conjugated parameters can be measured below the standard quantum limit when the probe field is entangled. We derive the most informative Cram\'er-Rao bound, providing the theoretical benchmark on the estimation and observe that our scheme is nearly optimal for a wide parameter range characterizing the probe field. We discuss the role of the entanglement as well as the relation between our measurement strategy and the generalized uncertainty relations.