Extension of the Watanabe–Sagawa–Ueda Uncertainty Relations 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 Cramér–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.