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Continuous Krishna-Parthasarathy Entropic Uncertainty Principle

K. Mahesh Krishna

TL;DR

This work extends the Krishna-Parthasarathy entropic uncertainty principle from a discrete setting to a continuous, measure-indexed framework by employing continuous operator-valued Parseval frames. It defines the continuous entropy $S_A(h)$ and $S_B(h)$ for frames indexed by finite-measure spaces and establishes the entropic bound $\log(\mu(\Omega)\nu(\Delta)) \ge S_A(h) + S_B(h) \ge -2 \log\bigl(\sup_{\alpha,\beta} \|B_\beta A_\alpha^*\|\bigr)$ via Riesz–Thorin interpolation of an associated transform. The methodology generalizes entropic uncertainty to non-discrete operator families and includes an application to compact groups through the Peter–Weyl framework, yielding corollaries expressed in terms of matrix coefficients and group representations. The results provide a versatile continuous-frames perspective on quantum measurements and harmonic analysis with potential impact on quantum information and group-frame theory.

Abstract

In 2002, Krishna and Parthasarathy [\textit{Sankhyā 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.

Continuous Krishna-Parthasarathy Entropic Uncertainty Principle

TL;DR

This work extends the Krishna-Parthasarathy entropic uncertainty principle from a discrete setting to a continuous, measure-indexed framework by employing continuous operator-valued Parseval frames. It defines the continuous entropy and for frames indexed by finite-measure spaces and establishes the entropic bound via Riesz–Thorin interpolation of an associated transform. The methodology generalizes entropic uncertainty to non-discrete operator families and includes an application to compact groups through the Peter–Weyl framework, yielding corollaries expressed in terms of matrix coefficients and group representations. The results provide a versatile continuous-frames perspective on quantum measurements and harmonic analysis with potential impact on quantum information and group-frame theory.

Abstract

In 2002, Krishna and Parthasarathy [\textit{Sankhyā 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.
Paper Structure (3 sections, 6 theorems, 40 equations)

This paper contains 3 sections, 6 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Continuous Krishna-Parthasarathy Entropic Uncertainty Principle
  3. Acknowledgments

Key Result

Theorem 1.1

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

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6