Finite-error metrological bounds on the multi-parameter Hamiltonian estimation

TL;DR

This paper establishes finite-error bounds on the time required for multi-parameter Hamiltonian estimation, comparing adaptive feedback and entanglement strategies, and providing explicit procedures and bounds based on quantum Fisher information.

Contribution

It derives both upper and lower bounds on estimation time, demonstrating the equivalence of adaptive feedback and entanglement methods in resource requirements.

Findings

Lower bound based on quantum Fisher information and Cramér-Rao inequality.

Explicit estimation procedures matching the bounds.

Equivalence of adaptive feedback and entanglement strategies in resource use.

Abstract

Estimation of multiple parameters in an unknown Hamiltonian is investigated. We present upper and lower bounds on the time required to complete the estimation within a prescribed tolerance $\delta$. The lower bound is given on the basis of the Cram\'er-Rao inequality, where the quantum Fisher information is bounded by the squared evolution time. The upper bound is obtained by an explicit construction of estimation procedures. By comparing the cases with different numbers of Hamiltonian channels, we also find that the few-channel procedure with adaptive feedback and the many-channel procedure with entanglement are equivalent in that they require the same amount of time resource up to a constant factor.