Near-Optimal Variance-Based Uncertainty Relations

TL;DR

This paper develops near-optimal bounds for variance-based uncertainty relations in quantum mechanics, creating an uncertainty interval that characterizes measurement limitations and linking different uncertainty formulations.

Contribution

It introduces new near-optimal bounds for variance-based uncertainty relations, forming an uncertainty interval and connecting variance and entropic uncertainty principles.

Findings

Derived near-optimal lower bounds for measurement uncertainties

Established an uncertainty interval combining bounds on incompatible observables

Linked variance-based and entropic uncertainty relations

Abstract

Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg's uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as `uncertainty relation', plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several `near optimal' bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an \emph{uncertainty interval}, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.