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arXiv:2606.31443

Wave-particle duality as an uncertainty relation for the average confidence width

Shengjun Wu

quant-phmath-phSubmitted Jun 30, 2026• Announced Jul 1, 2026• 0 citations
v1 · Jun 30, 2026v2 · Sep 22, 2026
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Abstract

We introduce the average confidence width $Δ_a x=\int_0^1 Δ_c x (θ_x) d θ_x$: the confidence width $Δ_c x(θ_x)$ -- the smallest position interval carrying a fraction $θ_x$ of the probability -- averaged over all levels. It is the first moment of the decreasing rearrangement of $|ψ|^2$, an $L^1$ mean-absolute-deviation measure of localization, so the product $Δ_{a} x\,Δ_{a} p$ is dilation invariant and obeys $Δ_{a} x\,Δ_{a} p\ge c\,\hbar$. Reading $1/Δ_{a} x$ as a particle character and $1/Δ_{a} p$ as a wave character, this lower bound on combined spread is identically an upper bound on combined particle-and-wave character: uncertainty and wave-particle duality are two faces of one inequality. A mean-entropy argument with the Bialynicki-Birula-Mycielski relation gives the rigorous $c\geπ/e$, while the achievable constant $c^\ast$ is set by the ground state of the Fourier-invariant operator $|x|+|p|$, $c^\ast\le E_0^2\approx 1.217$. Hence $π/e\le c^\ast\le E_0^2<4/π$: the optimal state is sub-Gaussian, so the Gaussian -- optimal for the Heisenberg and entropic relations -- is not the duality optimum.

Comments
7 pages, 4 figures

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We introduce the average confidence width Δax=∫01Δcx(θx)dθxΔ_a x=\int_0^1 Δ_c x (θ_x) d θ_xΔa​x=∫01​Δc​x(θx​)dθx​: the confidence width Δcx(θx)Δ_c x(θ_x)Δc​x(θx​) -- the smallest position interval carrying a fraction θxθ_xθx​ of the probability -- averaged over all levels. It is the first moment of the decreasing rearrangement of ∣ψ∣2|ψ|^2∣ψ∣2, an L1L^1L1 mean-absolute-deviation measure of localization, so the product Δax ΔapΔ_{a} x\,Δ_{a} pΔa​xΔa​p is dilation invariant and obeys Δax Δap≥c ℏΔ_{a} x\,Δ_{a} p\ge c\,\hbarΔa​xΔa​p≥cℏ. Reading 1/Δax1/Δ_{a} x1/Δa​x as a particle character and 1/Δap1/Δ_{a} p1/Δa​p as a wave character, this lower bound on combined spread is identically an upper bound on combined particle-and-wave character: uncertainty and wave-particle duality are two faces of one inequality. A mean-entropy argument with the Bialynicki-Birula-Mycielski relation gives the rigorous c≥π/ec\geπ/ec≥π/e, while the achievable constant c∗c^\astc∗ is set by the ground state of the Fourier-invariant operator ∣x∣+∣p∣|x|+|p|∣x∣+∣p∣, c∗≤E02≈1.217c^\ast\le E_0^2\approx 1.217c∗≤E02​≈1.217. Hence π/e≤c∗≤E02<4/ππ/e\le c^\ast\le E_0^2<4/ππ/e≤c∗≤E02​<4/π: the optimal state is sub-Gaussian, so the Gaussian -- optimal for the Heisenberg and entropic relations -- is not the duality optimum.
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A Gaussian distribution of a quantum state with continuous spectra is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$, the quantum entropic uncertainty relation can take a suggestive form, where the standard deviations ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{p}$ are featured explicitly. From the construction of the entropic uncertainty relation, it follows in a transparent manner that (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modified form, ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}\ensuremath{\hbar}{e}^{\mathcal{N}}/2$; (ii) the additional factor N\mathcal{N}N quantifies the quantum non-Gaussianity of the probability distributions of two observables; and (iii) the lower bound of the entropic uncertainty relation for a non-Gaussian continuous-variable (CV) mixed state becomes stronger with purity. The optimality of specific non-Gaussian CV states for the refined uncertainty relation has been investigated and the existence of a new class of CV quantum state is identified.
Heisenberg’s uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a `true’ probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) n-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension n; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.
The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system but also the entropic uncertainty measures of Rényi type, which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg’s uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in 1/D in similar systems with a non-standard dimensionality D; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-D limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The D-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work, we rigorously determine the leading term of the Rényi entropies of the D-dimensional hydrogenic atom at the limit of large D. As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations.
ABSTRACT The dimensional analysis of the position and momentum variances‐based quantum mechanical Heisenberg uncertainty measures was carried out for generalized Yukawa‐type potential. The entropic information measures given by Shannon entropy sums and Fisher information products were obtained out for generalized Yukawa‐type potential. It is interesting to observe that the Shannon entropy in position space, , increases and then decreases with the quantum number , suggesting a growth in delocalization due to a rapid increase in fluctuations. Likewise, the Shannon entropy in momentum space, , decreases and then increases with the quantum number , and vice versa. This unusual phenomenon may be explained by the parity restrictions inherent in the system. The Bialynicki‐Birula and Mycielski (BBM) entropic uncertainty relation was verified to be saturated; however, the sum of the entropies increases with the quantum number up to a certain point and then decreases as continues to increase. We also found that the Fisher information accurately increases with the quantum number . The numerical results that support the validity of the scaling properties for Shannon entropy sum and Fisher information product were presented. The existence of a squeezed phenomenon in the ground and first excited state for some values of the potential screening parameter in momentum space was found. The Heisenberg uncertainty relation is met when the squeeze in one coordinate is compensated by an increase in the other coordinate. The results obtained exceed the theoretical lower bounds for all the information‐theoretic measures considered, confirming their physical validity and accuracy.