Characterizing Fisher information of quantum measurement

TL;DR

This paper explores the relationship between quantum measurement and parameter estimation by analyzing Fisher information, revealing fundamental tradeoffs in quantum information extraction.

Contribution

It establishes a general link between informationally complete measurements and quantum parameter estimation using operator frame theory, providing bounds on Fisher information ratios.

Findings

Bound the ratio of classical to quantum Fisher information.

Connect bounds to optimal and least optimal parameter encoding directions.

Reveal fundamental tradeoffs in quantum information extraction.

Abstract

Informationally complete measurements form the foundation of universal quantum state reconstruction, while quantum parameter estimation is based on the local structure of the manifold of quantum states. Here we establish a general link between these two aspects, in the context of a single informationally complete measurement, by employing a suitably adapted operator frame theory. In particular, we bound the ratio between the classical and quantum Fisher information in terms of the spectral decomposition of the associated frame operator, and connect these bounds to the optimal and least optimal directions for parameter encoding. The geometric and operational characterization of information extraction thus obtained reveals the fundamental tradeoff imposed by informational completeness on local quantum parameter estimation.