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

TL;DR

This paper develops a framework to analyze measurement incompatibility in multi-parameter quantum estimation using p-local measurements, providing bounds on precision tradeoffs and conditions for saturating quantum bounds.

Contribution

It introduces a hierarchy of analytical bounds for measurement incompatibility under p-local measurements, advancing understanding of quantum estimation limits.

Findings

Derived bounds on precision tradeoffs for multi-parameter estimation

Identified conditions for saturating quantum Cramér-Rao bounds

Extended analysis to RLD-based tradeoff relations

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 Cram\'er-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).