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arxiv: 2606.31443 · v1 · pith:4GT4H3EXnew · submitted 2026-06-30 · 🪐 quant-ph · math-ph· math.MP

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

Shengjun Wu This is my paper

Pith reviewed 2026-07-01 05:22 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords wave-particle dualityuncertainty relationsaverage confidence widthquantum mechanicsposition-momentumFourier-invariant operatorsub-Gaussian states
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The pith

Uncertainty and wave-particle duality are two faces of one inequality using the average confidence width.

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

The paper defines the average confidence width as the integral from zero to one of the smallest interval that carries a given probability fraction, yielding an L1-type localization measure whose product with the momentum version is dilation-invariant. This product obeys a lower bound c ħ, which the author reads as an upper bound on the product of reciprocals once those reciprocals are taken as measures of particle and wave character. A mean-entropy argument combined with the Bialynicki-Birula-Mycielski relation proves c is at least π/e, while the ground state of the operator |x| + |p| shows that the best constant is at most roughly 1.217 and that the minimizing state is sub-Gaussian. The result therefore presents the standard uncertainty principle and the wave-particle complementarity relation as logically identical statements expressed in the same spread measure.

Core claim

The central claim is that the lower bound on the product Δ_a x Δ_a p is identically an upper bound on combined particle-and-wave character when the reciprocals are so interpreted, making uncertainty and wave-particle duality two faces of one inequality. The constant satisfies π/e ≤ c* ≤ E_0² ≈ 1.217, with the achievable optimum fixed by the ground state of the Fourier-invariant operator |x| + |p| and the optimal state therefore sub-Gaussian rather than Gaussian.

What carries the argument

Average confidence width Δ_a x, the integral of the confidence width over all probability fractions, which equals the first moment of the decreasing rearrangement of |ψ|² and serves as the L1 mean-absolute-deviation localization measure.

If this is right

  • The product Δ_a x Δ_a p is bounded below by c ħ with c at least π/e.
  • The best possible constant c* cannot exceed the square of the ground-state energy of the operator |x| + |p|.
  • Gaussian states are not optimal for the combined duality-uncertainty bound.
  • The states that saturate the bound are sub-Gaussian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical diagonalization of |x| + |p| in a suitable basis would yield the exact optimal constant and the corresponding wavefunction.
  • The same construction could be applied to other conjugate pairs to produce analogous duality bounds.
  • Preparation of sub-Gaussian states in an experiment would allow a direct test of whether the measured product approaches the tighter constant.

Load-bearing premise

That the reciprocals of the average confidence widths can be interpreted as quantifying particle character and wave character respectively.

What would settle it

A quantum state whose product of average confidence widths lies below (π/e) ħ, or an explicit ground-state wavefunction of |x| + |p| whose energy squared lies below the stated upper limit on c*.

Figures

Figures reproduced from arXiv: 2606.31443 by Shengjun Wu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

We introduce the average confidence width $\Delta_a x=\int_0^1 \Delta_c x (\theta_x) d \theta_x$: the confidence width $\Delta_c x(\theta_x)$ -- the smallest position interval carrying a fraction $\theta_x$ of the probability -- averaged over all levels. It is the first moment of the decreasing rearrangement of $|\psi|^2$, an $L^1$ mean-absolute-deviation measure of localization, so the product $\Delta_{a} x\,\Delta_{a} p$ is dilation invariant and obeys $\Delta_{a} x\,\Delta_{a} p\ge c\,\hbar$. Reading $1/\Delta_{a} x$ as a particle character and $1/\Delta_{a} p$ 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 $c\ge\pi/e$, 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 $\pi/e\le c^\ast\le E_0^2<4/\pi$: the optimal state is sub-Gaussian, so the Gaussian -- optimal for the Heisenberg and entropic relations -- is not the duality optimum.

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

1 major / 2 minor

Summary. The manuscript defines the average confidence width Δ_a x as the integral over θ_x of the smallest interval containing fraction θ_x of the probability mass (the first moment of the decreasing rearrangement of |ψ|²). It shows that the product Δ_a x Δ_a p is dilation-invariant and obeys Δ_a x Δ_a p ≥ c ħ. A mean-entropy argument based on the Bialynicki-Birula-Mycielski relation supplies the rigorous lower bound c ≥ π/e; the achievable constant c* is bounded above by the square of the ground-state energy E_0 of the Fourier-invariant operator |x|+|p|, yielding π/e ≤ c* ≤ E_0² ≈ 1.217 < 4/π. The paper then reads 1/Δ_a x as particle character and 1/Δ_a p as wave character, asserting that the spread lower bound is identically an upper bound on combined character and that uncertainty and wave-particle duality are therefore two faces of one inequality.

Significance. If the interpretive mapping is accepted, the result supplies a concrete, dilation-invariant measure that unifies uncertainty and duality with explicit, non-Gaussian constants and identifies the |x|+|p| ground state as the duality optimum. The derivation re-uses the established BBM relation in a mean-entropy setting and introduces a technically natural upper-bound operator; these are genuine strengths. The sub-Gaussian character of the optimum distinguishes the result from both Heisenberg and entropic cases.

major comments (1)
  1. [Abstract] Abstract and opening paragraphs: the assertion that the inequality is 'identically' an upper bound on combined particle-and-wave character rests on the direct re-interpretation of 1/Δ_a x and 1/Δ_a p as particle and wave characters. The manuscript states this mapping immediately after the definition but supplies no independent operational definition of wave-particle duality nor a derivation showing that the re-interpretation follows from the mathematics of the spread bound alone; this step is load-bearing for the central duality claim.
minor comments (2)
  1. [Abstract] The numerical value E_0² ≈ 1.217 should be accompanied by additional digits, an explicit statement of the numerical method used to obtain it, or a reference to a prior computation of the |x|+|p| ground state.
  2. Notation: the symbol c* is introduced for the achievable constant; its precise definition (supremum over states of 1/(Δ_a x Δ_a p)) should be stated explicitly in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the assertion that the inequality is 'identically' an upper bound on combined particle-and-wave character rests on the direct re-interpretation of 1/Δ_a x and 1/Δ_a p as particle and wave characters. The manuscript states this mapping immediately after the definition but supplies no independent operational definition of wave-particle duality nor a derivation showing that the re-interpretation follows from the mathematics of the spread bound alone; this step is load-bearing for the central duality claim.

    Authors: We agree that the mapping of 1/Δ_a x to particle character and 1/Δ_a p to wave character is presented as an interpretive step rather than a mathematical derivation internal to the spread bound. The motivation rests on the conventional association, in discussions of wave-particle duality, between position localization (small Δ_a x) and particle-like behavior versus momentum localization (small Δ_a p) and wave-like behavior. This is consistent with prior quantitative approaches to duality in the literature, though the manuscript does not supply an independent operational definition or experimental protocol that would derive the mapping from the inequality alone. To make this load-bearing interpretive step more transparent and better supported, we will expand the introduction with additional context and references to existing operational measures of wave-particle character. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on independent external relations

full rationale

The paper first defines the average confidence width Δ_a x as the first moment of the decreasing rearrangement of |ψ|^2 and derives the product bound Δ_a x Δ_a p ≥ c ħ via a mean-entropy argument from the established Bialynicki-Birula-Mycielski relation together with the ground-state energy of the separate Fourier-invariant operator |x|+|p|. These inputs are external and not reduced to the target duality statement. The subsequent claim that reading the inverses as particle/wave characters makes the inequality 'identically' a duality bound is an interpretive overlay stated after the mathematical result, not a self-definitional or fitted-input reduction. No self-citations, ansatz smuggling, or renaming of known results occur in the derivation chain. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the new definition of average confidence width, the interpretive mapping of inverses to particle/wave character, the Bialynicki-Birula-Mycielski entropic relation, and the existence of a ground state for the operator |x|+|p|.

axioms (2)
  • standard math Bialynicki-Birula-Mycielski entropic uncertainty relation
    Invoked to obtain the rigorous lower bound c ≥ π/e
  • standard math Existence and computability of the ground-state energy of the Fourier-invariant operator |x| + |p|
    Used to establish the achievable upper bound c* ≤ E0² ≈ 1.217

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Reference graph

Works this paper leans on

38 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    duality from entropy

    Settingℓ= 1,ℏ= 1 leads to ˆO=|x|+|p|for simplicity. Because ˆOcommutes with the Fourier transform,ψ 0 is self-dual ( ˜ψ0 =ψ 0), even, and unimodal, so the reduction is exact atψ 0 (Appendix C). Diagonalizing|x|+|p|on a grid (Appendix D) givesE 0 ≈1.1032 and π e|{z} ≈1.156 ℏ≤c ∗ℏ≤E 2 0|{z} ≈1.217 ℏ< 4 π|{z} ≈1.273 ℏ.(14) The strict last inequality is the h...

  2. [2]

    The bound is saturated by the mini- mizerψ 0, which is even, nodeless, and—because ˆOcom- mutes withF—self-dual ( ˜ψ0 =ψ 0); when its densi- ties are symmetric unimodal, (5) holds with equality and ∆ ax∆ ap(ψ0) = 4⟨|x|⟩⟨|p|⟩=E 2 0 ℏ, matching the bound. Appendix D: Numerical methods All computations useℏ= 1 on a uniform grid ofN points spanning [−L/2, L/2...

  3. [3]

    Heisenberg

    W. Heisenberg. ¨Uber den anschaulichen Inhalt der quan- tentheoretischen Kinematik und Mechanik.Z. Phys., 43:172–198, 1927

    1927

  4. [4]

    E. H. Kennard. Zur Quantenmechanik einfacher Bewe- gungstypen.Z. Phys., 44:326–352, 1927

    1927

  5. [5]

    W. K. Wootters and W. H. Zurek. Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle.Phys. Rev. D, 19:473–484, 1979

    1979

  6. [6]

    D. M. Greenberger and A. Yasin. Simultaneous wave and particle knowledge in a neutron interferometer.Phys. Lett. A, 128:391–394, 1988

    1988

  7. [7]

    B.-G. Englert. Fringe visibility and which-way informa- tion: an inequality.Phys. Rev. Lett., 77:2154–2157, 1996

    1996

  8. [8]

    Jaeger, A

    G. Jaeger, A. Shimony, and L. Vaidman. Two interfero- metric complementarities.Phys. Rev. A, 51:54–67, 1995

    1995

  9. [9]

    S. D¨ urr. Quantitative wave-particle duality in multibeam interferometers.Phys. Rev. A, 64:042113, 2001

    2001

  10. [10]

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

    J.-Y. Lin, X.-Y. Li, W. Wang, and S. Wu. Confidence uncertainty: position and momentum can be jointly de- termined with a guaranteed probability.arXiv preprint arXiv:2605.04484, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  11. [11]

    P. J. Coles, J. Kaniewski, and S. Wehner. Equivalence of wave–particle duality to entropic uncertainty.Nat. Commun., 5:5814, 2014

    2014

  12. [12]

    D. L. Donoho and Philip B. Stark. Uncertainty principles and signal recovery.SIAM J. Appl. Math., 49:906–931, 1989

    1989

  13. [13]

    Bia lynicki-Birula and J

    I. Bia lynicki-Birula and J. Mycielski. Uncertainty rela- tions for information entropy in wave mechanics.Com- mun. Math. Phys., 44:129–132, 1975

    1975

  14. [14]

    W. Beckner. Inequalities in Fourier analysis.Ann. of Math., 102:159–182, 1975

    1975

  15. [15]

    I. I. Hirschman. A note on entropy.Amer. J. Math., 79:152–156, 1957

    1957

  16. [16]

    H. P. Robertson. The uncertainty principle.Phys. Rev., 34:163–164, 1929

    1929

  17. [17]

    V. V. Dodonov. Variance-based uncertainty relations: a concise review of inequalities discovered since 1927. Quantum Reports, 7(3):34, 2025

    1927

  18. [18]

    Maassen and J

    H. Maassen and J. B. M. Uffink. Generalized entropic uncertainty relations.Phys. Rev. Lett., 60:1103–1106, 1988

    1988

  19. [19]

    S. Wu, S. Yu, and K. Mølmer. Entropic uncertainty relation for mutually unbiased bases.Phys. Rev. A 79:022104, 2009

    2009

  20. [20]

    P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner. Entropic uncertainty relations and their applications. Rev. Mod. Phys., 89:015002, 2017

    2017

  21. [21]

    H. Wang, Z. Ma, S. Wu, W. Zheng, Z. Cao, Z. Chen, Z. Li, S.-M. Fei, X. Peng, V. Vedral, and J. Du. Uncertainty equality with quantum memory and its experimental ver- ification.npj Quantum Information, 5:39, 2019

    2019

  22. [22]

    Huang, Z.-B

    S. Huang, Z.-B. Chen, and S. Wu. Entropic uncertainty relations for general symmetric informationally complete positive operator-valued measures and mutually unbiased measurements.Phys. Rev. A, 103:042205, 2021

    2021

  23. [23]

    H. J. Landau and H. O. Pollak. Prolate spheroidal wave functions, Fourier analysis and uncertainty—II.Bell Syst. Tech. J., 40:65–84, 1961

    1961

  24. [24]

    G. B. Folland and A. Sitaram. The uncertainty principle: a mathematical survey.J. Fourier Anal. Appl., 3:207– 7 238, 1997

    1997

  25. [25]

    Benedicks

    M. Benedicks. On Fourier transforms of functions sup- ported on sets of finite Lebesgue measure.J. Math. Anal. Appl., 106:180–183, 1985

    1985

  26. [26]

    W. O. Amrein and A. M. Berthier. On support properties ofL p-functions and their Fourier transforms.J. Funct. Anal., 24:258–267, 1977

    1977

  27. [27]

    Hossein Partovi

    M. Hossein Partovi. Majorization formulation of uncer- tainty in quantum mechanics.Phys. Rev. A, 84:052117, 2011

    2011

  28. [28]

    Friedland, V

    S. Friedland, V. Gheorghiu, and G. Gour. Universal un- certainty relations.Phys. Rev. Lett., 111:230401, 2013. Erratum: Phys. Rev. Lett.112, 119905 (2014)

    2013

  29. [29]

    Pucha la, L

    Z. Pucha la, L. Rudnicki, and K. ˙Zyczkowski. Majoriza- tion entropic uncertainty relations.J. Phys. A: Math. Theor., 46:272002, 2013

    2013

  30. [30]

    A. W. Marshall, I. Olkin, and B. C. Arnold.Inequalities: Theory of Majorization and Its Applications. Springer, 2nd edition, 2011

    2011

  31. [31]

    Bagan, J

    E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery. Relations between coherence and path information.Phys. Rev. Lett., 116:160406, 2016

    2016

  32. [32]

    D¨ urr and G

    S. D¨ urr and G. Rempe. Can wave–particle duality be based on the uncertainty relation?Am. J. Phys., 68:1021–1024, 2000

    2000

  33. [33]

    Spegel-Lexne, S

    D. Spegel-Lexne, S. G´ omez, J. Argillander, M. Paw lowski, P. R. Dieguez, A. Alarc´ on, and G. B. Xavier. Experimental demonstration of the equivalence of en- tropic uncertainty with wave–particle duality.Sci. Adv., 10:eadr2007, 2024

    2024

  34. [34]

    T. M. Cover and J. A. Thomas.Elements of Information Theory. Wiley, 2nd edition, 2006

    2006

  35. [35]

    A. N. Kolmogorov and V. M. Tikhomirov.ε-entropy and ε-capacity of sets in function spaces.Uspekhi Mat. Nauk, 14(2):3–86, 1959

    1959

  36. [36]

    J. L. Massey. Guessing and entropy. InProc. 1994 IEEE Int. Symp. Information Theory (ISIT), page 204, 1994

    1994

  37. [37]

    E. Arikan. An inequality on guessing and its application to sequential decoding.IEEE Trans. Inf. Theory, 42:99– 105, 1996

    1996

  38. [38]

    E. H. Lieb and M. Loss.Analysis, volume 14 ofGraduate Studies in Mathematics. American Mathematical Society, 2nd edition, 2001

    2001