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arxiv: 2605.04484 · v1 · submitted 2026-05-06 · 🪐 quant-ph

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

Jia-Yi Lin , Xin-Yu Li , Wei Wang , Shengjun Wu This is my paper

Pith reviewed 2026-05-08 17:57 UTC · model grok-4.3

classification 🪐 quant-ph
keywords confidence uncertaintyHeisenberg uncertainty principleposition-momentum localizationLandau-Pollak boundprolate spheroidal functionsDonoho-Stark boundquantum joint determination
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The pith

Position and momentum can be jointly localized to arbitrarily small regions with probability at least 50 percent when their target probabilities sum to one or less.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper redefines uncertainty for position and momentum using the size of the smallest region that contains a chosen probability mass θ rather than variance or entropy. It shows that when θ_x plus θ_p is at most 1, these minimal region sizes have no positive lower bound on their product and can both be driven toward zero by suitable quantum states. This means a particle can be found inside small intervals for both observables simultaneously with joint probability 50 percent or higher. When the sum of the target probabilities exceeds 1, explicit positive lower bounds on the product are derived from operator-norm inequalities. The work thereby separates the regimes where joint localization is possible from those where a tradeoff remains.

Core claim

We define the confidence uncertainty Δ^c x(θ_x) as the smallest Lebesgue measure of any measurable set whose probability content is at least θ_x, and the interval version Δ^I x(θ_x) as the shortest single interval with the same property. For θ_x + θ_p ≤ 1 we prove that both quantities can be made arbitrarily small simultaneously, so the product has infimum zero over all states. For θ_x + θ_p > 1 we obtain the lower bound Δ^c x Δ^c p ≥ 2πℏ (√(θ_x θ_p) − √((1−θ_x)(1−θ_p)) )² by combining Lenard’s projection inequality with the Donoho–Stark operator-norm bound, together with the sharp implicit Landau–Pollak bound for the interval version that involves the largest prolate-spheroidal eigenvalue.

What carries the argument

Confidence uncertainty, defined as the minimal Lebesgue measure of a set (or single interval) that contains probability at least θ.

Load-bearing premise

That the minimal Lebesgue measure of a high-probability set is the right way to quantify how well position and momentum can be jointly determined.

What would settle it

A concrete calculation or measurement showing that, for θ_x = θ_p = 0.5, the product of minimal position and momentum support sizes remains bounded below by a positive number for every quantum state.

Figures

Figures reproduced from arXiv: 2605.04484 by Jia-Yi Lin, Shengjun Wu, Wei Wang, Xin-Yu Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Phase diagram for the confidence-uncertainty bound view at source ↗
Figure 3
Figure 3. Figure 3: compares the Landau–Pollak bound of Theo￾rem 2 against the elementary bound (16). Both bounds agree on θx = 1, but the new one extends to all θx+θp > 1 and is consistently strictly tighter on the diagonal. Ta￾ble II samples the tight bound across the (θx, θp)-plane; view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Left: the largest Slepian eigenvalue view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical landscape of the tight bound on view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Interval-product comparison along view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Position (top) and momentum (bottom) proba view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces confidence uncertainty Δ^c x(θ_x) as the minimal Lebesgue measure of a set containing probability at least θ_x, and the interval-restricted version Δ^I x(θ_x). It proves two complementary results: for θ_x + θ_p ≤ 1 both can be made arbitrarily small simultaneously (no nontrivial product bound, allowing joint localization with probability ≥50%), while for θ_x + θ_p >1 it derives product lower bounds by combining Lenard's projection inequality with the Donoho-Stark operator-norm bound, yielding Δ^c x Δ^c p ≥ 2πℏ (√(θ_x θ_p) - √((1-θ_x)(1-θ_p)))^2, and a sharp implicit Landau-Pollak bound for the interval case involving the largest prolate-spheroidal eigenvalue λ_0(c). The analytical results are supported by numerical evaluation of λ_0(c), small-c and large-c asymptotics, explicit Slepian-superposition states that saturate the interval bound, and comparisons to the variance Heisenberg-Kennard, entropic Bia lynicki-Birula-Mycielski, and Donoho-Stark bounds.

Significance. If the results hold, the work offers a probability-centric reformulation of joint localization that complements spread-based and entropic uncertainty principles. The demonstration that Δ^c x and Δ^c p (and interval versions) can be simultaneously small for θ_x + θ_p ≤1 is a clear, falsifiable statement with direct implications for quantum mechanics interpretations and applications requiring guaranteed localization probabilities. Credit is given for the parameter-free derivations obtained by direct combination of three external inequalities (Lenard, Donoho-Stark, Landau-Pollak) without internal circularity, the explicit construction of saturating states, closed-form asymptotics, and the numerical evaluation of λ_0(c) with comparisons that situate the new measures relative to existing bounds.

minor comments (2)
  1. [Abstract] Abstract: the mention of 'numerical evaluation of λ_0(c) and comparison plots' would be strengthened by a brief note on whether error bars or full reproducibility data are provided in the main text or supplement.
  2. [Introduction] §1 (Introduction): the choice of Lebesgue measure for support size is presented as natural, but a short explicit comparison to alternative quantifiers (e.g., other norms or entropic supports) would address potential reader questions about the definition's uniqueness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough summary of the manuscript and for recommending acceptance. We appreciate the recognition of the parameter-free derivations, the explicit saturating states, and the comparisons to existing uncertainty principles.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins with explicit new definitions of the confidence uncertainties Δ^c x(θ_x) and Δ^I x(θ_x) as minimal measures of sets carrying probability at least θ_x. The two main inequalities are then obtained by direct application of three independent external theorems (Lenard projection inequality, Donoho–Stark operator-norm bound, and Landau–Pollak prolate-spheroidal eigenvalue bound) whose statements contain no reference to the new quantities. For the regime θ_x + θ_p ≤ 1 the lower-bound expression evaluates to a non-positive number, which immediately implies that no nontrivial product bound exists; this is a straightforward algebraic consequence rather than a self-referential step. No self-citations appear as load-bearing premises, no parameters are fitted and then relabeled as predictions, and no ansatz is imported via prior work by the same authors. The entire chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on the standard mathematical definitions of Lebesgue measure and L2 inner products, plus three external operator inequalities whose proofs are not reproduced here. No free parameters are fitted; the only new objects are the two uncertainty measures introduced by definition.

axioms (2)
  • domain assumption Lebesgue measure on R is the appropriate size for support sets of wave functions
    Used to define Δ^c x(θ_x) and Δ^I x(θ_x) as minimal measures
  • standard math Lenard's projection inequality, Donoho-Stark operator-norm bound, and Landau-Pollak inequality hold for the relevant operators
    Invoked to obtain the product lower bounds for θ_x + θ_p >1
invented entities (2)
  • confidence uncertainty Δ^c x(θ_x) no independent evidence
    purpose: Quantify the smallest set containing probability ≥θ_x
    New definition; no independent physical evidence supplied beyond the mathematical construction
  • interval confidence uncertainty Δ^I x(θ_x) no independent evidence
    purpose: Restrict the support to a single interval
    New definition; no independent physical evidence supplied

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