Confidence uncertainty: position and momentum can be jointly determined with a guaranteed probability

TL;DR

This paper introduces confidence-based measures of position and momentum uncertainty, providing new bounds that allow joint localization with high probability, challenging traditional uncertainty limits.

Contribution

It defines confidence uncertainty metrics and derives complementary bounds, including a sharp Landau–Pollak bound, supported by analytical and numerical analysis.

Findings

Position and momentum can be jointly localized with at least 50% probability.

Derived a lower bound on the product of confidence uncertainties for probabilities summing over 1.

Provided analytical bounds, asymptotics, and optimal states saturating the interval bound.

Abstract

Heisenberg's uncertainty principle states that the position and momentum of a particle cannot be sharply determined simultaneously. Standard-deviation and entropic formulations capture the spread of the probability distribution but say little about the probability itself contained in a small region. We introduce the "confidence uncertainty" $\Delta^{c}x(\theta_x)$ as the minimal Lebesgue measure of the support set in which the particle is found with probability at least $\theta_x$, and the companion "interval confidence uncertainty" $\Delta^{I}x(\theta_x)$ which restricts the support to a single interval. We prove two complementary uncertainty inequalities. (i) For $\theta_x+\theta_p\le 1$ both confidence uncertainties can be made arbitrarily small simultaneously, so that no nontrivial product bound holds; in particular, position and momentum can be jointly localised with probability at least~$50\%$. (ii) For $\theta_x+\theta_p>1$ a lower bound holds: combining Lenard's projection inequality with the Donoho--Stark operator-norm bound we obtain $\Delta^{c}x\,\Delta^{c}p\geq 2\pi\hbar\bigl(\sqrt{\theta_x\theta_p}-\sqrt{(1-\theta_x)(1-\theta_p)}\bigr)^{\!2}$, and for the interval version we obtain the sharp implicit Landau--Pollak bound $\Delta^{I}x\,\Delta^{I}p\geq 4\hbar\,\lambda_{0}^{-1}\!\bigl((\sqrt{\theta_x\theta_p}-\sqrt{(1-\theta_x)(1-\theta_p)})^{2}\bigr)$, where $\lambda_{0}(c)$ is the largest prolate-spheroidal eigenvalue. We support the analytical bounds with numerical evaluation of $\lambda_{0}(c)$, provide closed-form small-$c$ and large-$c$ asymptotics, compute the optimal Slepian-superposition states that saturate the interval bound, and compare the resulting product against the variance Heisenberg--Kennard, the Bia\l{}ynicki-Birula--Mycielski entropic, and the Donoho--Stark concentration bounds.