Optimal Multiparameter Metrology: The Quantum Compass Solution
arxiv(2024)
摘要
We study optimal quantum sensing of multiple physical parameters using
repeated measurements. In this scenario, the Fisher information framework sets
the fundamental limits on sensing performance, yet the optimal states and
corresponding measurements that attain these limits remain to be discovered. To
address this, we extend the Fisher information approach with a second
optimality requirement for a sensor to provide unambiguous estimation of
unknown parameters. We propose a systematic method integrating Fisher
information and Bayesian approaches to quantum metrology to identify the
combination of input states and measurements that satisfies both optimality
criteria. Specifically, we frame the optimal sensing problem as an optimization
of an asymptotic Bayesian cost function that can be efficiently solved
numerically and, in many cases, analytically. We refer to the resulting optimal
sensor as a `quantum compass' solution, which serves as a direct multiparameter
counterpart to the Greenberger-Horne-Zeilinger state-based interferometer,
renowned for achieving the Heisenberg limit in single-parameter metrology. We
provide exact quantum compass solutions for paradigmatic multiparameter problem
of sensing two and three parameters using an SU(2) sensor. Our metrological
cost function opens avenues for quantum variational techniques to design
low-depth quantum circuits approaching the optimal sensing performance in the
many-repetition scenario. We demonstrate this by constructing simple quantum
circuits that achieve the Heisenberg limit for vector field and 3D rotations
estimation using a limited set of gates available on a trapped-ion platform.
Our work introduces and optimizes sensors for a practical notion of optimality,
keeping in mind the ultimate goal of quantum sensors to precisely estimate
unknown parameters.
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