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Nonlinear Maccone-Pati Uncertainty Principle

K. Mahesh Krishna

Published Jan 31, 2024• 1 citation
Publisher

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.

DOI
10.20944/preprints202401.2224.v1

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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 Lp\mathcal{L}^pLp spaces for 1<p<∞1< p<\infty1<p<∞. Our main tool is the Clarkson inequalities. We also derive a nonlinear uncertainty principle for weak parallelogram spaces and Type-p Banach spaces.
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 Lp\mathcal{L}^pLp spaces for 1<p<∞1< p<\infty1<p<∞. Our main tool is the Clarkson inequalities. We also derive a nonlinear uncertainty principle for weak parallelogram spaces and Type-p Banach spaces.
Recently, Maccone and Pati [Phys. Rev. Lett. {\bf 113}, 260401 (2014)] derived few inequalities among variances of incompatible operators which they called stronger uncertainty relations, stronger than Heisenberg-Robertson or Schrodinger uncertainty relations. Here we generalize their study to get infinite number of such inequalities and propose that only one of them may be the correct uncertainty relation between incompatible operators. We get back well known uncertainty relations of Heisenberg-Robertson and Schrodinger under certain limits. We also reexamine the conclusions of Maccone and Pati and comment on their work.
Let E\mathcal{E}E be a Hilbert C*-module over a unital C*-algebra A\mathcal{A}A. Let A:D(A)⊆E→EA: \mathcal{D}(A) \subseteq \mathcal{E} \to \mathcal{E}A:D(A)⊆E→E and B:D(B)⊆E→EB: \mathcal{D}(B)\subseteq \mathcal{E}\to \mathcal{E}B:D(B)⊆E→E be possibly unbounded self-adjoint morphisms. Then for all x∈D(AB)∩D(BA)x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)x∈D(AB)∩D(BA) with ⟨x,x⟩=1\langle x, x \rangle =1⟨x,x⟩=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)\coloneqq \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2},\quad [A,B] \coloneqq AB-BA, \\ & [\langle Ax, x \rangle,\langle Bx, x \rangle]\coloneqq \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 A=C\mathcal{A}=\mathbb{C}A=C.