An inequality related to uncertainty principle in von Neumann algebras

TL;DR

This paper extends a recently discovered matrix inequality related to the uncertainty principle, involving Wigner-Yanase-Dyson information, to the broader setting of von Neumann algebras and general operator monotone functions.

Contribution

The authors prove the inequality in von Neumann algebras and generalize it to arbitrary operator monotone functions, answering a question posed by Kosaki.

Findings

Inequality holds in von Neumann algebras.

Generalization to arbitrary operator monotone functions.

Extension of uncertainty principle to a broader mathematical setting.

Abstract

Recently Kosaki proved an inequality for matrices that can be seen as a kind of new uncertainty principle. Independently, the same result was proved by Yanagi, Furuichi and Kuriyama. The new bound is given in terms of Wigner-Yanase-Dyson informations. Kosaki himself asked if this inequality can be proved in the setting of von Neumann algebras. In this paper we provide a positive answer to that question and moreover we show how the inequality can be generalized to an arbitrary operator monotone function.