Conditionally valid uncertainty relations

TL;DR

This paper demonstrates that traditional uncertainty relations become singular under precise measurements due to the failure of unbiased measurement assumptions, clarifying contradictions in recent quantum measurement literature.

Contribution

It clarifies the conditions under which uncertainty relations are valid, highlighting the singular behavior of certain relations in precise measurements and reconciling conflicting results.

Findings

Naive Heisenberg error-disturbance relation is formally preserved in quantum estimation theory.

Heisenberg error-error and Arthurs-Kelly relations remain valid as conditionally valid uncertainty relations.

Singular behavior occurs for precise measurements when unbiased measurement assumptions fail.

Abstract

It is shown that the well-defined unbiased measurement or disturbance of a dynamical variable is not maintained for the precise measurement of the conjugate variable, independently of uncertainty relations. The conditionally valid uncertainty relations on the basis of those additional assumptions, which include most of the familiar Heisenberg-type relations, thus become singular for the precise measurement. We clarify some contradicting conclusions in the literature concerning those conditionally valid uncertainty relations: The failure of a naive Heisenberg-type error-disturbance relation and the modified Arthurs-Kelly relation in the recent spin measurement is attributed to this singular behavior. The naive Heisenberg-type error-disturbance relation is formally preserved in quantum estimation theory, which is shown to be based on the strict unbiased measurement and disturbance, but it leads to unbounded disturbance for bounded operators such as spin variables. In contrast, the Heisenberg-type error-error uncertainty relation and the Arthurs-Kelly relation, as conditionally valid uncertainty relations, are consistently maintained.