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 \Delta _x(B)^2d_x(A)^2+\Delta _x(A)^2d_x(B)^2\geq \frac{(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2}{2} \end{align*} and \begin{align*} (2) \quad \Delta _x(A)\Delta _x(B)\geq \frac{\sqrt{\|(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2\|}}{2}, \end{align*} where \begin{align*} &\Delta _x(A)\coloneqq \|Ax-\langle Ax, x \rangle x \|, \quad d_x(A)\coloneqq \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2},\\ &[A,B] \coloneqq AB-BA, \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, \\ & [\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 Heisenberg-Robertson-Schrodinger uncertainty principles. They reduce to the Heisenberg-Robertson-Schrodinger uncertainty principle (derived by Schrodinger in 1930) whenever $\mathcal{A}=\mathbb{C}$ .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 \Delta _x(B)^2d_x(A)^2+\Delta _x(A)^2d_x(B)^2\geq \frac{(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2}{2} \end{align*} and \begin{align*} (2) \quad \Delta _x(A)\Delta _x(B)\geq \frac{\sqrt{\|(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2\|}}{2}, \end{align*} where \begin{align*} &\Delta _x(A)\coloneqq \|Ax-\langle Ax, x \rangle x \|, \quad d_x(A)\coloneqq \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2},\\ &[A,B] \coloneqq AB-BA, \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, \\ & [\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 Heisenberg-Robertson-Schrodinger uncertainty principles. They reduce to the Heisenberg-Robertson-Schrodinger uncertainty principle (derived by Schrodinger in 1930) whenever $\mathcal{A}=\mathbb{C}$.

arXiv:2502.05154Jan 2, 20250 citations

Quantum Uncertainty in Decision Theory

Vyacheslav I. Yukalov

Studies in systems, decision and controlJan 1, 20221 citation

Octonionic Heisenberg-Robertson-Schrodinger and Maccone-Pati Uncertainty Principles

K. MAHESH KRISHNA

Let $\mathcal{O}$ be an octonionic Hilbert space. Let $A: \mathcal{D}(A)\subseteq \mathcal{O}\to \mathcal{O}$ and $B:\mathcal{D}(B)\subseteq \mathcal{O}\to \mathcal{O}$ be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\|h\|=1$ , we show that \begin{align*} (1) \quad \quad \quad \Delta _h(A)\Delta _h(B)\geq \frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \Delta _h(A)^2+ \Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where $ \Delta _h(A):= \|Ah-\langle Ah, h \rangle h \|$. We call Inequality (1) as octonionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as octonionic Maccone-Pati uncertainty principle.Let $\mathcal{O}$ be an octonionic Hilbert space. Let $A: \mathcal{D}(A)\subseteq \mathcal{O}\to \mathcal{O}$ and $B:\mathcal{D}(B)\subseteq \mathcal{O}\to \mathcal{O}$ be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\|h\|=1$, we show that \begin{align*} (1) \quad \quad \quad \Delta _h(A)\Delta _h(B)\geq \frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \Delta _h(A)^2+ \Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where $ \Delta _h(A):= \|Ah-\langle Ah, h \rangle h \|$. We call Inequality (1) as octonionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as octonionic Maccone-Pati uncertainty principle.

Oct 19, 20260 citations

A group-theoretical approach to extending the Schrödinger-Robertson uncertainty inequality

Wannisa Sudprakhon, Keng Wiboonton

Mathematical Inequalities & ApplicationsJan 1, 20250 citations

Uncertainty relations: a small zoo of remarkable inequalities discovered since 1927

V. V. Dodonov

A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.

quant-pharXiv:2603.07293Revista Brasileira de Ensino de FísicaMar 10, 20261 citation

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 \Delta _x(B)^2d_x(A)^2+\Delta _x(A)^2d_x(B)^2\geq \frac{(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2}{2} \end{align*} and \begin{align*} (2) \quad \Delta _x(A)\Delta _x(B)\geq \frac{\sqrt{\|(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2\|}}{2}, \end{align*} where \begin{align*} &\Delta _x(A)\coloneqq \|Ax-\langle Ax, x \rangle x \|, \quad d_x(A)\coloneqq \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2},\\ &[A,B] \coloneqq AB-BA, \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, \\ & [\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 Heisenberg-Robertson-Schrodinger uncertainty principles. They reduce to the Heisenberg-Robertson-Schrodinger uncertainty principle (derived by Schrodinger in 1930) whenever

\mathcal{A}=\mathbb{C}

Let

\mathcal{O}

be an octonionic Hilbert space. Let

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

and

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

be possibly unbounded self-adjoint linear operators. Then for all

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

with

\|h\|=1

, we show that \begin{align*} (1) \quad \quad \quad \Delta _h(A)\Delta _h(B)\geq \frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \Delta _h(A)^2+ \Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where $ \Delta _h(A):= \|Ah-\langle Ah, h \rangle h \|$. We call Inequality (1) as octonionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as octonionic Maccone-Pati uncertainty principle.