Multiparameter function estimation for general Hamiltonians

TL;DR

This paper establishes the fundamental quantum limits and provides an optimal protocol for estimating functions of multiple parameters encoded in general Hamiltonians, extending quantum sensing capabilities.

Contribution

It derives the ultimate quantum limit and presents an estimation protocol for any function of parameters in general Hamiltonians, unifying previous approaches.

Findings

Derived the quantum limit for multi-parameter function estimation.

Proposed an optimal estimation protocol achieving the bound.

Unified single-parameter and multi-parameter estimation frameworks.

Abstract

Estimation of physical parameters encoded in a Hamiltonian is a central task in quantum sensing and learning. While the ultimate precision limit for estimating a single parameter coupled to a single generator is well established, the corresponding bound for estimating a function of multiple parameters-each coupled to distinct and possibly non-commuting generators-remains unknown in general. Here, we derive the ultimate quantum limit and present an estimation protocol for any function of parameters in a general Hamiltonian that attains this bound. We show that, although the task is fundamentally a multiparameter problem, our tight bound reduces to an optimized single-parameter quantum Cram\'er-Rao bound, even for arbitrary generator sets. Our result unifies and extends previous works, providing a general framework for optimal function estimation in quantum systems.