Position--momentum uncertainty relation for the average confidence width

Abstract:We introduce the average confidence width $\Delta_{a} x=\int_0^1\Delta_{c} x (\theta)\,\mathrm{d}\theta$: the confidence width $\Delta_{c} x(\theta)$---the measure of the smallest region of position space carrying probability $\theta$---averaged over all confidence levels. It is the first moment of the decreasing rearrangement of the position density, an $L^1$ (mean-absolute-deviation) measure of localization, and $\Delta_{a} x\,\Delta_{a} p$ is dilation invariant. We study the uncertainty relation $\Delta_{a} x\,\Delta_{a} p\ge c^\ast\hbar$ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives $c^\ast\ge\pi/e$, while the ground state $\psi_0$ of the Fourier-invariant operator $|x|+|p|$, with $E_0=1.1040744$, gives $c^\ast\le E_0^2\approx1.2190<4/\pi$: Gaussian states, which saturate the Heisenberg--Kennard and entropic relations, are not optimal here. We prove that $\psi_0$ is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations $\Delta_{a} x \cdot 2 \langle |p-p_0| \rangle \ge E_0^2\hbar$ and $2\langle|x-x_0| \rangle \cdot\Delta_{a} p\ge E_0^2\hbar$ hold for all states, so $\Delta_{a} x\,\Delta_{a} p\ge E_0^2\hbar$ whenever either density is symmetric unimodal; the second variation of $\Delta_{a} x\,\Delta_{a} p$ at $\psi_0$ is positive definite apart from the symmetry directions. Extensive numerical minimization, including coherent states, never finds a product below $E_0^2\hbar$ and consistently returns $\psi_0$, supporting the conjecture $c^\ast=E_0^2$. In two dimensions the relation is sharp and saturated by Gaussians, identifying the dimension as the origin of the non-Gaussian optimizer in one dimension. Finally, we discuss other complementary pairs and the relation of the width to interferometric predictability and visibility.

arXiv:2606.31443 [quant-ph]

(or arXiv:2606.31443v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.31443 Focus to learn more arXiv-issued DOI via DataCite

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