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

Abstract

Standard-deviation and entropic formulations of uncertainty principle 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. The unified picture provides a complete phase diagram on $(\theta_x,\theta_p)\in[0,1]^{2}$.