Noncommutative Maccone-Pati Uncertainty Principles

K. MAHESH KRISHNA

Noncommutative Maccone-Pati Uncertainty Principles

Primarily human-written textHuman understanding: all partsmath.OA — Operator Algebras math.MP — Mathematical Physics

Contributed by K. MAHESH KRISHNA ↗

Version 1 / Oct 16, 2026 / CC BY 4.0

Abstract

Let

\mathcal{E}

be a Hilbert C*-module over a unital C*-algebra

\mathcal{A}

. Let

A: \mathcal{D}(A) \subseteq \mathcal{E} \to \mathcal{E}

and

B: \mathcal{D}(B)\subseteq \mathcal{E}\to \mathcal{E}

be possibly unbounded self-adjoint morphisms. Then for all

x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)

with

\langle x, x \rangle =1

, we show that \begin{align*} (1) \quad\quad\quad d_x(A)^2+d_x(B)^2&\geq \mp \langle \{A,B\}x, x \rangle +\langle (A\pm B)x, y \rangle\langle y, (A\pm B)x \rangle\pm \{\langle Ax, x \rangle, \langle Bx, x \rangle\}, \\ &\quad \forall y \in \mathcal{E} \text{ satisfying } \|y\|\leq 1 \text{ and } \langle x,y \rangle =0 \end{align*} \begin{align*} (2) \quad \quad \quad d_x(A)^2+d_x(B)^2&\geq \mp i\langle [A,B]x, x \rangle +\langle (A\pm iB)x, y \rangle\langle y, (A\pm iB)x \rangle \mp i[\langle Ax, x \rangle, \langle Bx, x \rangle], \\ &\quad \forall y \in \mathcal{E} \text{ satisfying } \|y\|\leq 1 \text{ and } \langle x,y \rangle =0. \end{align*} where \begin{align*} & d_x(A):= \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2},\quad [A,B] := AB-BA, \\ & [\langle Ax, x \rangle,\langle Bx, x \rangle]:= \langle Ax, x \rangle\langle Bx, x \rangle -\langle Bx, x \rangle\langle Ax, x \rangle. \end{align*} We call Inequalities (1) and (2) as noncommutative Maccone-Pati uncertainty principles. They reduce to uncertainty principles derived by Maccone and Pati [\textit{Phys. Rev. Lett., 2014}] whenever

\mathcal{A}=\mathbb{C}

.

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Version history

v1Submitted by K. MAHESH KRISHNAInitial depositCurrentOct 16, 2026