Extension of the Watanabe-Sagawa-Ueda uncertainty relation for measurement errors to infinite-dimensional systems

Abstract

We extend the Watanabe--Sagawa--Ueda (WSU) uncertainty relations for measurement errors to infinite-dimensional systems. The original WSU formulation provided a definition of measurement errors with a clear physical interpretation based on quantum estimation theory, but was restricted to finite-dimensional systems, excluding important observables such as position and momentum. Using pseudo-inverse forms of positive-semidefinite forms, we develop a framework for classical and quantum estimation theory for models whose parameter space is the set of full-rank states on an infinite-dimensional Hilbert space, and derive classical and quantum Cram\'{e}r--Rao inequalities. We extend the WSU measurement errors to both bounded and unbounded operators, and derive corresponding error-error uncertainty relations. The resulting uncertainty relation inequalities are stronger than the original WSU bound due to an improved derivation method. Our results provide a theoretical framework for applying estimation-based uncertainty relations to observables with continuous values in infinite-dimensional systems.