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Modular Deutsch Entropic Uncertainty Principle

K. Mahesh Krishna

TL;DR

This work extends entropic uncertainty principles to Hilbert C*-modules over commutative unital C*-algebras by leveraging the modular Buzano inequality. It defines modular Shannon entropy with respect to Parseval frames and proves a modular Deutsch entropic inequality S_tau(x) + S_omega(x) >= -2 log((1 + sup_{j,k} ||<tau_j, omega_k>||)/2) for x in E_tau ∩ E_omega. The result generalizes Deutsch and Maassen-Uffink bounds to a module setting and connects to Ricaud-Torresani's frame results, while proposing a Modular Kraus Entropic Conjecture. The work advances entropic uncertainty in noncommutative geometry and suggests directions for operator-valued entropy with modular structures.

Abstract

Khosravi, Drnovšek and Moslehian [\textit{Filomat, 2012}] derived Buzano inequality for Hilbert C*-modules. Using this inequality we derive Deutsch entropic uncertainty principle for Hilbert C*-modules over commutative unital C*-algebras.

Modular Deutsch Entropic Uncertainty Principle

TL;DR

This work extends entropic uncertainty principles to Hilbert C*-modules over commutative unital C*-algebras by leveraging the modular Buzano inequality. It defines modular Shannon entropy with respect to Parseval frames and proves a modular Deutsch entropic inequality S_tau(x) + S_omega(x) >= -2 log((1 + sup_{j,k} ||<tau_j, omega_k>||)/2) for x in E_tau ∩ E_omega. The result generalizes Deutsch and Maassen-Uffink bounds to a module setting and connects to Ricaud-Torresani's frame results, while proposing a Modular Kraus Entropic Conjecture. The work advances entropic uncertainty in noncommutative geometry and suggests directions for operator-valued entropy with modular structures.

Abstract

Khosravi, Drnovšek and Moslehian [\textit{Filomat, 2012}] derived Buzano inequality for Hilbert C*-modules. Using this inequality we derive Deutsch entropic uncertainty principle for Hilbert C*-modules over commutative unital C*-algebras.
Paper Structure (3 sections, 5 theorems, 11 equations)

This paper contains 3 sections, 5 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Modular Deutsch Entropic Uncertainty Principle
  3. Acknowledgments

Key Result

Theorem 1.1

DEUTSCH (Deutsch Entropic Uncertainty Principle) Let $\{\tau_j\}_{j=1}^n$, $\{\omega_k\}_{k=1}^n$ be two orthonormal bases for a finite dimensional Hilbert space $\mathcal{H}$. Then

Theorems & Definitions (9)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Conjecture 2.5