Information Geometry under Hierarchical Quantum Measurement

In most quantum technologies, measurements need to be performed on the parametrized quantum states to transform the quantum information to classical information. The measurements, however, inevitably distort the information. The characterization of the discrepancy is an important subject in quantum information science, which plays a key role in understanding the difference between the structures of quantum and classical informations. Here we analyze the difference in terms of the Fisher information metric and present a framework that can provide analytical bounds on the discrepancy under hierarchical quantum measurements. Specifically, we present a set of analytical bounds on the difference between the quantum and classical Fisher information metric under hierarchical p-local quantum measurements, which are measurements that can be performed collectively on at most p copies of quantum states. The results can be directly transformed to the precision limit in multiparameter quantum metrology, which leads to characterizations of the trade-off among the precision of different parameters. The framework also provides a coherent picture for various existing results by including them as special cases.