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 $Δ^{c}x(θ_x)$ as the minimal Lebesgue measure of the support set in which the particle is found with probability at least $θ_x$, and the companion interval confidence uncertainty $Δ^{I}x(θ_x)$ which restricts the support to a single interval. We prove two complementary uncertainty inequalities. (i) For $θ_x+θ_p\le 1$ 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 $θ_x+θ_p>1$ a lower bound holds: combining Lenard's projection inequality with the Donoho--Stark operator-norm bound we obtain $Δ^{c}x\,Δ^{c}p\geq 2π\hbar\bigl(\sqrt{θ_xθ_p}-\sqrt{(1-θ_x)(1-θ_p)}\bigr)^{\!2}$, and for the interval version we obtain the sharp implicit Landau--Pollak bound $Δ^{I}x\,Δ^{I}p\geq 4\hbar\,λ_{0}^{-1}\!\bigl((\sqrt{θ_xθ_p}-\sqrt{(1-θ_x)(1-θ_p)})^{2}\bigr)$, where $λ_{0}(c)$ is the largest prolate-spheroidal eigenvalue. We support the analytical bounds with numerical evaluation of $λ_{0}(c)$, 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 Białynicki-Birula--Mycielski entropic, and the Donoho--Stark concentration bounds. The unified picture provides a complete phase diagram on $(θ_x,θ_p)\in[0,1]^{2}$.