Optimal uncertainty relations in a modified Heisenberg algebra

TL;DR

This paper proves the existence of a minimal observable length arising from modified Heisenberg algebras, introduces a method for optimal uncertainty relations, and explores entropic measures of uncertainty, including a minimal length of one bit.

Contribution

It establishes the existence of a minimal length in modified quantum frameworks and develops a general approach for optimal and entropic uncertainty relations.

Findings

Minimal length in terms of min-entropy is exactly one bit.

Proved the existence of a minimal observable length under certain assumptions.

Developed a method for state-independent uncertainty relations.

Abstract

Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations which are thought (but not proven) to imply the existence of a minimal observable length. Here we prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, we make use of entropies as a measure of uncertainty and provide uncertainty relations in terms of min- and Shannon entropies. We compute the corresponding entropic minimal lengths and find that the minimal length in terms of min-entropy is exactly one bit.