Hierarchical incompatibility measures in multi-parameter quantum estimation

When collective measurements on an infinite number of copies of identical quantum states can be performed, the precision limit of multi-parameter quantum estimation is quantified by the Holevo bound. In practice, however, the collective measurements are always restricted to a finite number of quantum states, under which the precision limit is still poorly understood. Here we provide an approach to study the multi-parameter quantum estimation with general $p$-local measurement where the collective measurements are restricted to at most $p$ copies of quantum states. We demonstrate the power of the approach by providing a hierarchy of nontrivial tradeoff relations for multi-parameter quantum estimation which quantify the incompatibilities of general $p$-local measurement. These tradeoff relations also provide a necessary condition for the saturation of the quantum Cramer-Rao bound under $p$-local measurement, which is shown reducing to the weak commutative condition when $p\rightarrow \infty$. To further demonstrate the versatility of the approach, we also derive another set of tradeoff relations in terms of the right logarithmic operators(RLD).