Aspects of universally valid Heisenberg uncertainty relation

TL;DR

This paper numerically illustrates a recently proposed universally valid Heisenberg uncertainty relation using spin-measurement data, showing its validity and analyzing why other related relations fail under certain assumptions.

Contribution

It demonstrates the validity of the universally valid Heisenberg uncertainty relation with experimental data and clarifies its relation to Robertson's relation and other uncertainty formulations.

Findings

The universally valid Heisenberg relation always holds in the tested experiments.

Modified Arthurs-Kelly and Ozawa's error-disturbance relations fail under certain measurement assumptions.

All universally valid uncertainty relations derive from Robertson's relation.

Abstract

A numerical illustration of a universally valid Heisenberg uncertainty relation, which was proposed recently, is presented by using the experimental data on spin-measurements by J. Erhart, et al.[ Nature Phys. {\bf 8}, 185 (2012)]. This uncertainty relation is closely related to a modified form of the Arthurs-Kelly uncertainty relation which is also tested by the spin-measurements. The universally valid Heisenberg uncertainty relation always holds, but both the modified Arthurs-Kelly uncertainty relation and Heisenberg's error-disturbance relation proposed by Ozawa, which was analyzed in the original experiment, fail in the present context of spin-measurements, and the cause of their failure is identified with the assumptions of unbiased measurement and disturbance. It is also shown that all the universally valid uncertainty relations are derived from Robertson's relation and thus the essence of the uncertainty relation is exhausted by Robertson's relation as is widely accepted.