Unified and computable approach to optimal strategies for multiparameter estimation

TL;DR

This paper introduces a unified, computable framework for optimal multiparameter quantum estimation, leveraging quantum resources and the quantum tester formalism to achieve ultimate precision bounds in noisy and noise-free scenarios.

Contribution

It develops a systematic, semidefinite programming-based approach integrating quantum resources for multiparameter estimation, addressing the challenge of parameter incompatibility.

Findings

Achieves ultimate quantum precision bounds for multiparameter estimation.

Reveals hierarchy among different quantum strategies in magnetic-field sensing.

Applicable to noisy and noise-free metrological scenarios.

Abstract

Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for estimating physical parameters as precisely as possible. While substantial progress has been made in single-parameter quantum metrology, the multiparameter scenario remains significantly more challenging due to the issue of parameter incompatibility. In this work, we present a unified and computable approach for the simultaneous estimation of multiple parameters that attains the ultimate precision permitted by quantum mechanics. The core of our approach is to integrate the quantum tester formalism into the recently proposed tight Cram\'er-Rao type bound. This formulation enables us to figure out the highest achievable precision via upper and lower bounds that are computable via semidefinite programs. More importantly, within this formulation, diverse quantum resources, including entanglement, coherence, quantum control, and indefinite causal order, are treated on equal footing and systematically optimized for the purpose of achieving the ultimate precision in multiparameter estimation. As a result, our approach is applicable to various metrological strategies both in the presence and absence of noise. To demonstrate its utility, we revisit three-dimensional magnetic-field estimation, uncovering the strengths and limitations of existing analytical results and further establishing a strict hierarchy among different types of strategies.