Universally valid uncertainty relations in general quantum systems

Abstract

We study universally valid uncertainty relations in general quantum systems described by general $\sigma$-finite von Neumann algebras to foster developing quantitative analysis in quantum systems with infinite degrees of freedom such as quantum fields. We obtain the most stringent measurement-disturbance relation ever, applicable to systems with infinite degrees of freedom, by refining the proofs given by Branciard and one of the authors (MO) for systems with finite degrees of freedom. In our proof the theory of the standard form of von Neumann algebras plays a crucial role, incorporating with measurement theory for local quantum systems recently developed by the authors.