Classical Fisher Information in Quantum Metrology -- Interplay of Probe, Dynamics and Measurement

TL;DR

This paper introduces a Fisher operator to analyze quantum measurement processes, establishing bounds on classical Fisher Information and proposing practical optimal measurement schemes using linear optics and photon counting.

Contribution

It develops a formalism linking Fisher information with Hamiltonian semi-norm and constructs optimal measurement strategies for quantum metrology.

Findings

Fisher operator formalism bounds classical Fisher Information by Hamiltonian semi-norm.

Optimal measurement schemes can be implemented with linear optics and photon counting.

Achieves maximum Fisher Information with simple projective measurements on qubits.

Abstract

We introduce a positive Hermitian operator, the Fisher operator, and use it to examine a measurement process incorporating unitary dynamics and complete measurements. We develop the idea of information complement, the minimization of which establishes the optimal precision for a fixed input. The formalism demonstrates that, in general, the classical Fisher Information has the Hamiltonian semi-norm as an upper bound. This is achievable with a qubit probe and only projective measurements, and is independent of the true value of the estimated parameter. In an interferometry context, we show that an optimal measurement scheme can be constructed from linear optics and photon counting, without recourse to generalised measurements or exotic unitaries outside of SU(2).