A Unified and Strengthened Framework for the Uncertainty Relation

TL;DR

This paper introduces a unified framework that strengthens quantum uncertainty relations by incorporating an information operator, improving the accuracy of uncertainty bounds for incompatible observables, including non-Hermitian operators.

Contribution

It develops a unified uncertainty relation framework that enhances existing relations and offers new insights into non-Hermitian operator uncertainties.

Findings

Uncertainty bounds are improved by including the information operator.

The framework applies to both product and sum forms of uncertainty relations.

Provides a new interpretation for uncertainties of non-Hermitian operators.

Abstract

We provide a unified and strengthened framework for the product form and the sum form variance-based uncertainty relations by constructing a unified uncertainty relation. In the unified framework, we deduce that the uncertainties of the incompatible observables are bounded by not only the commutator of themselves, but also the quantities related with the other operator. This operator can provide information so that we can capture the uncertainty of the measurement result more accurately, and thus is named as the information operator. The introduction of the information operator can fix the deficiencies in both the product form and the sum form uncertainty relations, and provides a more accurate description of the quantum uncertainty relation. The unified framework also proposes a new interpretation of the uncertainty relation for non-Hermitian operators; i.e., the "observable" second-order origin moments of the non-Hermitian operators cannot be arbitrarily small at the same time when they are generalized-incompatible on the new definition of the generalized commutator.