The Most Informative Cram\'er--Rao Bound for Quantum Two-Parameter Estimation with Pure State Probes

TL;DR

This paper derives a simplified expression for the ultimate precision limit in two-parameter quantum estimation with pure states, and identifies optimal measurements, advancing quantum metrology techniques.

Contribution

It introduces a new, simpler formula for the quantum Cramér-Rao bound in two-parameter pure state estimation and determines the optimal measurement strategies.

Findings

Derived a simpler expression for the quantum Cramér-Rao bound.

Identified optimal measurement strategies for two-parameter estimation.

Applied results to estimate displacements with grid states.

Abstract

Optimal measurements for quantum multiparameter estimation are complicated by the uncertainty principle. Generally, there is a trade-off between the precision with which different parameters can be simultaneously estimated. The task of determining the minimum achievable estimation error is a central task of multiparameter quantum metrology. For estimating parameters encoded in pure quantum states, the ultimate limit is known, but is given by the solution of a non-trivial minimisation problem. We present a new expression for the achievable bound for two-parameter estimation with pure states that is considerably simpler. We also determine the optimal measurements, completing the problem of two-parameter estimation with pure state probes. To demonstrate the utility of our result, we determine the precision limit for estimating displacements using grid states.