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

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" $Δ^{c}x(θ_x)$ as the minimal Lebesgue measure of the support set in which the particle is found with probability at least $θx$, 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 paper presents a novel framework of confidence uncertainty to quantify minimal localization intervals in position and momentum spaces at preset probability thresholds.
• It distinguishes between unrestricted and bounded regimes in the (θₓ, θₚ) space, applying sharp lower bounds using Slepian functions in the bounded regime.
• The results outperform traditional variance and entropic uncertainty relations, offering practical insights for quantum measurement and state preparation.

Confidence Uncertainty: Rigorous Joint Localisation in Quantum Position and Momentum

Introduction

The paper "Confidence uncertainty: position and momentum can be jointly determined with a guaranteed probability" (2605.04484) introduces a new framework for quantifying uncertainty in quantum systems, focusing on the operational probability of finding a quantum particle within constrained regions of position and momentum space. Moving beyond variance-based and entropic uncertainty relations, the authors define confidence uncertainties that provide sharp probabilistic bounds. This affords a precise characterization of the extent to which position and momentum may be jointly localized, given a chosen confidence threshold.

Definitions and Framework

The principal objects introduced are the confidence uncertainty, $\Delta^{c}x(\theta_x)$, and the interval confidence uncertainty, $\Delta^{I}x(\theta_x)$, which quantify the minimal Lebesgue measure or interval length in position space capturing at least probability $\theta_x$. Analogous quantities are defined in momentum space. This approach reframes the uncertainty principle in terms of guaranteed probability, moving away from the spread (variance) or entropy of a distribution toward a direct operational measure.

Formally,

$$\Delta^{c}x(\theta_x) = \inf\left\{ \mu(X) : \int_{X} |\psi(x)|^2 dx \geq \theta_x \right\},$$

where $X$ ranges over measurable sets.

The paper addresses the question: for given confidence levels $\theta_x$ (position) and $\theta_p$ (momentum), what are the minimal supports such that a quantum particle is found within both with at least their respective probabilities?

Two Regimes: Joint Localisation Phase Diagram

A key result is the dichotomy in the $(\theta_x, \theta_p)$ space:

• Unrestricted regime ($\theta_x + \theta_p \leq 1$): No nontrivial lower bound exists for the product $\Delta^{c}x\,\Delta^{c}p$. The authors construct explicit quantum states for which both uncertainties can be made arbitrarily small simultaneously, including the special case $\Delta^{I}x(\theta_x)$0 whereby position and momentum may both be confined to arbitrarily narrow intervals with at least $\Delta^{I}x(\theta_x)$1 probability.
• Bounded regime ($\Delta^{I}x(\theta_x)$2): A sharp lower bound applies to the product $\Delta^{I}x(\theta_x)$3:

$\Delta^{I}x(\theta_x)$4

For interval confidence, a tighter bound involves the largest prolate spheroidal eigenvalue, $\Delta^{I}x(\theta_x)$5, linking directly to the Slepian–Pollak–Landau theory from signal processing.

Slepian–Pollak Theory and Saturation

The interval confidence case is rigorously mapped to an eigenvalue problem in classical signal theory. The saturating wavefunctions are Slepian superpositions—prolate spheroidal functions—providing explicit constructions for states achieving the bound. The sharpness of these bounds is validated numerically, with asymptotic expansions for $\Delta^{I}x(\theta_x)$6 derived for small and large $\Delta^{I}x(\theta_x)$7:

• For high-confidence ($\Delta^{I}x(\theta_x)$8), the uncertainty product diverges logarithmically:

$\Delta^{I}x(\theta_x)$9

Numerical results supplied in the paper give explicit uncertainty products for diverse $\theta_x$0 values, delineating the operational limitations for joint localisation.

Comparison to Classical Uncertainty Relations

Variance-based (Heisenberg–Kennard): The confidence uncertainty product achieves results strictly stronger than the minimum-variance Gaussian states, especially notable at intermediate-to-high confidence levels ($\theta_x$1), where the Slepian states yield products an order of magnitude below the Heisenberg bound. Variance and confidence uncertainties are fundamentally distinct notions; the former is insensitive to sharply bimodal distributions, while the latter directly quantifies probability concentration.

Entropic (Białynicki-Birula–Mycielski): Entropic bounds control the Shannon or differential entropy, integrating over the entire distribution. Confidence uncertainty addresses the operational question of finding a particle within a specified region, revealing limitations of entropic bounds at high-confidence regimes.

Donoho–Stark and Landau–Pollak: The new bounds from this paper generalize and strictly tighten the Donoho–Stark concentration principle, especially for $\theta_x$2, with numerical examples showing up to $\theta_x$3 tighter uncertainty products at high confidence. The Landau–Pollak angular inequalities are shown to be saturated by the Slepian states, fulfilling the sharp bound for interval confidence uncertainty.

Practical and Theoretical Implications

• Experimental design: The $\theta_x$4 joint-localisation claim has practical implications, suggesting that quantum measurements can be tailored such that both position and momentum have high operational certainty. Realizing states saturating the interval bounds might be feasible via engineered quantum systems or digital quantum simulation.
• Foundations of quantum mechanics: This framework rigorously distinguishes the operational content of the uncertainty principle, clarifying the limits of joint localization in quantum phase space, and revealing a probabilistic phase diagram.
• Signal processing and information theory: The formalism directly connects quantum mechanical uncertainty relations to classical signal concentration, opening avenues for applications in quantum information, cryptography, and tomography where confidence-level constraints are paramount.
• Generalization: The theoretical apparatus extends naturally to mixed states, multidimensional systems, and other conjugate observable pairs (e.g., time-energy, number-phase), with Slepian functions providing a maximal framework.

Conclusion

The paper establishes confidence uncertainty as a foundational operational extension of the quantum uncertainty principle, rigorously mapping out the probability-constrained joint localisation landscape for position and momentum observables. The sharp lower bounds and their saturation by Slepian-superposition states furnish explicit, quantitative constraints that outperform traditional variance and entropic bounds in relevant regimes. These results offer a precise, probabilistic paradigm for quantum measurement and state preparation, with broad implications for theory and experiment.