Universally valid uncertainty relations in general quantum systems

TL;DR

This paper develops the most comprehensive uncertainty relations applicable to general quantum systems, including those with infinite degrees of freedom, by leveraging advanced von Neumann algebra theory and measurement frameworks.

Contribution

It introduces universally valid uncertainty relations for infinite-dimensional quantum systems, extending previous finite-dimensional results with new mathematical techniques.

Findings

Derived the strongest measurement-disturbance relation for infinite systems

Utilized von Neumann algebra theory to generalize uncertainty relations

Enhanced understanding of quantum measurement in complex 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.