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Nonlinear Maccone-Pati Uncertainty Principle

K. Mahesh Krishna

TL;DR

This work extends the Maccone-Pati uncertainty principle to nonlinear settings on Lebesgue spaces by leveraging Clarkson inequalities. It defines a nonlinear uncertainty for maps on subsets of $L^p$ via $\Delta_f(A,a)=\|Af-af\|_p$ and derives $p$-dependent lower bounds that survive under dual and Lipschitz functionals, as well as in weak parallelogram spaces and Type-p Banach spaces. The contributions unify parallelogram-law based arguments with nonlinear maps, broadening the uncertainty framework beyond Hilbert spaces and offering tools for analyzing uncertainty in broader Banach-space contexts.

Abstract

We show that one of the two important uncertainty principles derived by Maccone and Pati \textit{[Phys. Rev. Lett., 2014]} can be derived for arbitrary maps defined on subsets of $\mathcal{L}^p$ spaces for $1< p<\infty$. Our main tool is the Clarkson inequalities. We also derive a nonlinear uncertainty principle for weak parallelogram spaces and Type-p Banach spaces.

Nonlinear Maccone-Pati Uncertainty Principle

TL;DR

This work extends the Maccone-Pati uncertainty principle to nonlinear settings on Lebesgue spaces by leveraging Clarkson inequalities. It defines a nonlinear uncertainty for maps on subsets of via and derives -dependent lower bounds that survive under dual and Lipschitz functionals, as well as in weak parallelogram spaces and Type-p Banach spaces. The contributions unify parallelogram-law based arguments with nonlinear maps, broadening the uncertainty framework beyond Hilbert spaces and offering tools for analyzing uncertainty in broader Banach-space contexts.

Abstract

We show that one of the two important uncertainty principles derived by Maccone and Pati \textit{[Phys. Rev. Lett., 2014]} can be derived for arbitrary maps defined on subsets of spaces for . Our main tool is the Clarkson inequalities. We also derive a nonlinear uncertainty principle for weak parallelogram spaces and Type-p Banach spaces.
Paper Structure (2 sections, 14 theorems, 20 equations)

This paper contains 2 sections, 14 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Nonlinear Maccone-Pati Uncertainty Principle

Key Result

Theorem 1.1

ROBERTSONHEISENBERGVONNEUMANNBOOKDEBNATHMIKUSINSKI (Heisenberg-Robertson Uncertainty Principle) Let $A: \mathcal{D}(A)\to \mathcal{H}$ and $B: \mathcal{D}(B)\to \mathcal{H}$ be self-adjoint operators. Then for all $h \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\|h\|=1$, we have

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • Definition 2.5
  • ...and 8 more