Noncommutative Maccone-Pati Uncertainty Principles

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 $\mathcal{A}=\mathbb{C}$.