Continuous Krishna-Parthasarathy Entropic Uncertainty Principle

TL;DR

This paper extends the entropic uncertainty principle to continuous settings using operator-valued frames, providing a generalized framework applicable to families of operators indexed by measure spaces, with specific applications to compact groups.

Contribution

It introduces a continuous version of the entropic uncertainty principle utilizing continuous operator-valued frames, broadening the scope beyond discrete cases.

Findings

Derived a continuous entropic uncertainty principle for measure-indexed operator families.

Applied the framework to the case of compact groups.

Generalized previous discrete uncertainty principles to continuous settings.

Abstract

In 2002, Krishna and Parthasarathy [\textit{Sankhy\={a} Ser. A}] derived discrete quantum version of Maassen-Uffink [\textit{Phys. Rev. Lett., 1988}] entropic uncertainty principle. In this paper, using the notion of continuous operator-valued frames, we derive an entropic uncertainty principle for arbitrary family of operators indexed by measure spaces having finite measure. We give an application to the special case of compact groups.