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 ${\Delta }_{a}x{\Delta }_{a}p\ge c^{\ast }\hslash$ 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 $c^{\ast }\le E_0^2\approx 1.2190<4/\pi$: 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 ${\Delta }_{a}x\cdot 2⟨|p-p_0|⟩\ge E_0^2\hslash$ and $2⟨|x-x_0|⟩\cdot {\Delta }_{a}p\ge E_0^2\hslash$ hold for all states, so ${\Delta }_{a}x{\Delta }_{a}p\ge E_0^2\hslash$ 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 $E_0^2\hslash$ 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 comment15 pages, 5 figures