Hybrid Cramér-Rao Bound for Quantum Bayes Point Estimation with Nuisance Parameters

Area of Science:

• Quantum Information Science
• Quantum Metrology
• Statistical Inference

Background:

• Quantum parameter estimation is crucial for advancing quantum technologies.
• Nuisance parameters often degrade estimation precision in quantum systems.
• Existing methods struggle to effectively incorporate prior knowledge about nuisance parameters.

Purpose of the Study:

• To develop a novel hybrid framework for quantum parameter estimation in the presence of nuisance parameters.
• To introduce a new metric, the hybrid partial quantum Fisher information matrix (hpQFIM), for assessing estimation performance.
• To establish a theoretical foundation for exploiting prior information in quantum metrology.

Main Methods:

• A hybrid approach treating parameters of interest as fixed and nuisance parameters as random variables.
• Introduction and mathematical definition of the hybrid partial quantum Fisher information matrix (hpQFIM).
• Derivation of a Cramér-Rao-type lower bound for the hybrid risk.

Main Results:

• The hpQFIM is defined by prior-averaging the nuisance block of the QFIM and taking a Schur complement.
• Structural properties and limiting behaviors of the hpQFIM under various prior conditions were established.
• The hybrid approach demonstrates improved precision over pure point estimation by optimizing measurements based on prior distributions.

Conclusions:

• The developed hybrid framework offers a robust method for quantum parameter estimation with nuisance parameters.
• The hpQFIM provides a tractable tool for analyzing and optimizing quantum metrology protocols.
• Systematic exploitation of partial prior information on nuisance variables enhances the efficiency of quantum measurements.