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arXiv:2605.04484 · Physics Letters A

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

Jia-Yi Lin, Xin-Yu Li, Wei Wang, Shengjun Wu

quant-phSubmitted May 6, 2026• Announced May 7, 2026• 0 citations• Revised Jul 2, 2026 (v2)
v1 · May 6, 2026v2 · Jul 2, 2026
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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 $(θ_x,θ_p)\in[0,1]^{2}$.

Journal reference
Physics Letters A 592 (2026) 131943
DOI
10.1016/j.physleta.2026.131943
Comments
8 pages, 6 figures

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We introduce the average confidence width Δax=∫01Δcx(θ) dθΔ_{a} x=\int_0^1Δ_{c} x (θ)\,\mathrm{d}θΔa​x=∫01​Δc​x(θ)dθ: the confidence width Δcx(θ)Δ_{c} x(θ)Δc​x(θ)---the measure of the smallest region of position space carrying probability θθθ---averaged over all confidence levels. It is the first moment of the decreasing rearrangement of the position density, an L1L^1L1 (mean-absolute-deviation) measure of localization, and Δax ΔapΔ_{a} x\,Δ_{a} pΔa​xΔa​p is dilation invariant. We study the uncertainty relation Δax Δap≥c∗ℏΔ_{a} x\,Δ_{a} p\ge c^\ast\hbarΔa​xΔa​p≥c∗ℏ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives c∗≥π/ec^\ast\geπ/ec∗≥π/e, while the ground state ψ0ψ_0ψ0​ of the Fourier-invariant operator ∣x∣+∣p∣|x|+|p|∣x∣+∣p∣, with E0=1.1040744E_0=1.1040744E0​=1.1040744, gives c∗≤E02≈1.2190<4/πc^\ast\le E_0^2\approx1.2190<4/πc∗≤E02​≈1.2190<4/π: Gaussian states, which saturate the Heisenberg--Kennard and entropic relations, are not optimal here. We prove that ψ0ψ_0ψ0​ is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations Δax⋅2⟨∣p−p0∣⟩≥E02ℏΔ_{a} x \cdot 2 \langle |p-p_0| \rangle \ge E_0^2\hbarΔa​x⋅2⟨∣p−p0​∣⟩≥E02​ℏ and 2⟨∣x−x0∣⟩⋅Δap≥E02ℏ2\langle|x-x_0| \rangle \cdotΔ_{a} p\ge E_0^2\hbar2⟨∣x−x0​∣⟩⋅Δa​p≥E02​ℏ hold for all states, so Δax Δap≥E02ℏΔ_{a} x\,Δ_{a} p\ge E_0^2\hbarΔa​xΔa​p≥E02​ℏ whenever either density is symmetric unimodal; the second variation of Δax ΔapΔ_{a} x\,Δ_{a} pΔa​xΔa​p at ψ0ψ_0ψ0​ is positive definite apart from the symmetry directions. Extensive numerical minimization, including coherent states, never finds a product below E02ℏE_0^2\hbarE02​ℏ and consistently returns ψ0ψ_0ψ0​, supporting the conjecture c∗=E02c^\ast=E_0^2c∗=E02​. In two dimensions the relation is sharp and saturated by Gaussians, identifying the dimension as the origin of the non-Gaussian optimizer in one dimension. Finally, we discuss other complementary pairs and the relation of the width to interferometric predictability and visibility.
The exact analytical values of the position and momentum information entropies are calculated for the single-slit and double-slit diffraction experiments. In both cases, the product of the exponentials of the entropies is strictly greater than the lower bound $\ensuremath{\pi}e\ensuremath{\Elzxh}$ given by the optimal entropic uncertainty relation for position and momentum, which implies that the single-slit and double-slit configurations are not minimum-uncertainty states. The results obtained show that the position-momentum entropic uncertainty relation provides a rigorous quantitative expression for the uncertainty principle in these experiments, unlike the Heisenberg inequality for standard deviations. However, it is also shown that, in the double-slit experiment, wave-particle duality cannot be derived from the entropic uncertainty relation.
We consider successive measurements of position and momentum of a single particle. Let P be the conditional probability to measure the momentum k with precision dk, given a previously successful position measurement q with precision dq. Several upper bounds for the probability P are determined. For arbitrary, but given precision dq, dk, these bounds refer to the variation of q, k and the state vector of the particle. A weak bound is given by the inequality P <= dkdq/h, where h is Planck's quantum of action. It is non-trivial for all measurements with dkdq < h. A sharper bound is obtained by applying the Hilbert-Schmidt-norm. As our main result the least upper bound of P is determined. All bounds are independent of the order with which the measuring of position and momentum is made.
Heisenberg’s uncertainty principle establishes a fundamental restriction on the simultaneous resolution of certain pairs of quantum observables. In its familiar position–momentum form, the standard deviations of position and momentum satisfy a lower bound determined by the reduced Planck constant. This physical restriction, however, is distinct from the computational problem of resolving an uncertainty state from incomplete, discrete, noisy, or progressively acquired observational data. This paper explores the latter problem through the lens of a Calculus of Uncertainty, or CoU. The CoU framework considered here begins with partial data and applies a recursive reconstruction procedure consisting of rectangular approximation, trapezoidal approximation, midpoint approximation, numerical refinement, bounded logistic formulation, validation, and recursive state resolution. Applied to the uncertainty principle, this procedure does not attempt to recover simultaneously exact values of incompatible observables, nor does it modify the quantum-mechanical lower bound. Instead, it reconstructs estimates of the statistical moments of position and momentum distributions from partial observations, derives corresponding estimates of their standard deviations, evaluates those estimates relative to the uncertainty relation, and recursively determines whether the resulting state is sufficiently supported to be accepted. A distinction is therefore maintained between quantum uncertainty, which is a property of the admissible statistical structure of a quantum state, and reconstruction uncertainty, which arises from partial sampling, numerical approximation, truncation, measurement limitations, and incomplete contextual information. The proposed formulation treats the uncertainty relation as a constraint within the CoU context and validation structure. A state may be accepted only when its numerical reconstruction has converged to a specified tolerance and its estimated uncertainty product remains consistent with the quantum-mechanical lower bound. The resulting approach provides a tolerance-aware, constraint-consistent method for representing how an inferred state develops as additional evidence is acquired, approximations are refined, and prior states are recursively revised or retained. A worked minimum-uncertainty Gaussian reconstruction demonstrates that quadrature may converge accurately on a truncated domain while the reconstructed uncertainty product remains inadmissible until sufficient distributional mass is captured. **Keywords:** Calculus of Uncertainty, Heisenberg uncertainty principle, quantum uncertainty, partial data, numerical reconstruction, midpoint rule, logistic state formulation, recursive state resolution
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.