Incompatibility measures in multi-parameter quantum estimation under hierarchical quantum measurements

Abstract

The incompatibility of the measurements constraints the achievable precisions in multi-parameter quantum estimation. Understanding the tradeoff induced by such incompatibility is a central topic in quantum metrology. Here we provide an approach to study the incompatibility under general $p$-local measurements, which are the measurements that can be performed collectively on at most $p$ copies of quantum states. We demonstrate the power of the approach by presenting a hierarchy of analytical bounds on the tradeoff among the precision limits of different parameters. These bounds lead to a necessary condition for the saturation of the quantum Cramér-Rao bound under $p$-local measurements, which recovers the partial commutative condition at p=1 and the weak commutative condition at $p=\infty$. As a further demonstration of the power of the framework, we present another set of tradeoff relations with the right logarithmic operators(RLD).

34 pages, 5 figures, with improved bounds

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• Fully-Optimized Quantum Metrology: Framework, Tools, and Applications
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• Comparison of estimation limits for quantum two-parameter estimation
• Critical Quantum Sensing: a tutorial on parameter estimation near quantum phase transitions
• Randomized measurements for multi-parameter quantum metrology
• Joint optimal measurement for locating two incoherent optical point sources near the Rayleigh distance
• Quantum thermometry in diffraction-limited systems
• Simultaneous optical phase and loss estimation revisited: measurement and probe incompatibility
• Minimal Trade-off and Optimal Measurement for Multiparameter Quantum Estimation
• Experimental Joint Estimation of Phase and Phase Diffusion via Deterministic Bell Measurements
• Approaching the Multiparameter Quantum Cramér-Rao Bound via Classical Correlation and Entangling Measurements