Position--momentum uncertainty relation for the average confidence width

Abstract

We introduce the average confidence width $Δ_{a} x=\int_0^1Δ_{c} x (θ)\,\mathrm{d}θ$: the confidence width $Δ_{c} x(θ)$---the measure of the smallest region of position space carrying probability $θ$---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 $Δ_{a} x\,Δ_{a} p$ is dilation invariant. We study the uncertainty relation $Δ_{a} x\,Δ_{a} p\ge c^\ast\hbar$ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives $c^\ast\geπ/e$, while the ground state $ψ_0$ of the Fourier-invariant operator $|x|+|p|$, with $E_0=1.1040744$, gives $c^\ast\le E_0^2\approx1.2190<4/π$: Gaussian states, which saturate the Heisenberg--Kennard and entropic relations, are not optimal here. We prove that $ψ_0$ is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations $Δ_{a} x \cdot 2 \langle |p-p_0| \rangle \ge E_0^2\hbar$ and $2\langle|x-x_0| \rangle \cdotΔ_{a} p\ge E_0^2\hbar$ hold for all states, so $Δ_{a} x\,Δ_{a} p\ge E_0^2\hbar$ whenever either density is symmetric unimodal; the second variation of $Δ_{a} x\,Δ_{a} p$ at $ψ_0$ 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 $ψ_0$, 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.