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arxiv: 2606.13525 · v1 · pith:L6AFLO5Lnew · submitted 2026-06-11 · 🪐 quant-ph

Generalized Exact Fractional Quantum Information Model with Memory Effects

Abdelmalek Bouzenada , Allan R. P. Moreira This is my paper

Pith reviewed 2026-06-27 06:32 UTC · model grok-4.3

classification 🪐 quant-ph
keywords fractional quantum mechanicsShannon entropyFisher informationRiemann-Liouville derivativequantum harmonic oscillatornonlocal dynamicsmemory effectsquantum information measures
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The pith

The fractional parameter controls deviations in Shannon entropy and Fisher information from standard quantum forms.

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

This paper reconsiders the definitions of Shannon entropy and Fisher information and extends them to fractional quantum systems through the Riemann-Liouville derivative. It builds the corresponding fractional expressions and applies the formalism to the quantum harmonic oscillator to obtain explicit analytical results that depend on the fractional parameter. The work shows that these derivatives change the localization of probability densities and produce clear shifts in information content and sensitivity. The fractional parameter is identified as the quantity that sets the size of the departures from ordinary quantum information measures. The overall result is a consistent description of information properties for quantum systems that obey nonlocal dynamics.

Core claim

Within this generalized formulation, fractional expressions of Shannon entropy and Fisher information are constructed and their mathematical structures examined thoroughly. Also, the formalism is then applied to the quantum harmonic oscillator, yielding explicit analytical expressions derived as functions of the fractional parameter therein. The obtained results demonstrate that fractional derivatives alter the localization properties of probability densities and generate nontrivial variations in information content and sensitivity across system behavior. In this context, the fractional parameter plays a central role in controlling deviations from the standard quantum information measures fr

What carries the argument

Fractional extensions of Shannon entropy and Fisher information constructed with the Riemann-Liouville derivative and evaluated on the quantum harmonic oscillator.

If this is right

  • The fractional parameter sets the magnitude of departures from ordinary entropy and Fisher information values.
  • Nonlocal dynamics produce probability densities whose localization differs from the integer-order case.
  • Explicit formulas for the harmonic oscillator allow immediate computation of the altered information quantities.
  • A single consistent description covers information properties for any system obeying the nonlocal dynamics.

Where Pith is reading between the lines

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

  • The approach may supply quantitative tools for quantum systems whose evolution includes memory, such as certain open-system models.
  • It could be used to track how non-Markovian effects change the amount of extractable information in a state.
  • Comparison of the derived expressions against numerical solutions of the fractional Schrödinger equation would test the extension.

Load-bearing premise

Standard definitions of Shannon entropy and Fisher information can be extended directly to fractional quantum systems described by nonlocal operators without creating inconsistencies in their information-theoretic meaning.

What would settle it

Numerical evaluation of the probability density for the fractional harmonic oscillator at a chosen fractional order, followed by direct integration to obtain entropy or Fisher information and comparison against the paper's closed-form expressions.

read the original abstract

In this paper, we analyze quantum information measures in fractional quantum mechanics using the Riemann-Liouville derivative formalism adopted here. In this case, we initially reconsider the conventional definitions of Shannon entropy and Fisher information, subsequently extending them to fractional quantum systems described by nonlocal differential operator frameworks adopted. Within this generalized formulation, fractional expressions of Shannon entropy and Fisher information are constructed and their mathematical structures examined thoroughly. Also, the formalism is then applied to the quantum harmonic oscillator, yielding explicit analytical expressions derived as functions of the fractional parameter therein. The obtained results demonstrate that fractional derivatives alter the localization properties of probability densities and generate nontrivial variations in information content and sensitivity across system behavior. In this context, the fractional parameter plays a central role in controlling deviations from the standard quantum information measures framework. Also, the study establishes a consistent framework for describing information-theoretic properties of quantum systems governed by nonlocal dynamics.

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

2 major / 2 minor

Summary. The manuscript develops a generalized framework for quantum information measures in fractional quantum mechanics by extending the conventional definitions of Shannon entropy and Fisher information via Riemann-Liouville fractional derivatives. It constructs fractional expressions for these quantities, applies the formalism to the quantum harmonic oscillator to obtain explicit analytical expressions depending on the fractional order parameter, and concludes that the fractional parameter alters localization properties of probability densities while generating nontrivial variations in information content and sensitivity.

Significance. If the derivations and consistency checks hold, the work supplies concrete analytical expressions for a standard model system and a consistent framework for information-theoretic analysis under nonlocal dynamics. The explicit dependence on the fractional parameter and the treatment of memory effects via the chosen operator could be useful for applications in fractional quantum mechanics, provided the extensions preserve key information-theoretic properties such as positivity and normalization.

major comments (2)
  1. [Abstract, §3] Abstract and §3 (presumed derivation section): the central claim that the fractional extensions produce nontrivial, non-tautological variations in information content requires explicit verification that the constructed Shannon and Fisher expressions do not reduce by definition to functions of the fractional order alone; the abstract provides no such check or comparison to the integer-order limit.
  2. [§4] Application to harmonic oscillator (presumed §4): without the explicit forms of the fractional probability densities or the error analysis for the Riemann-Liouville operator applied to the oscillator wavefunctions, it is impossible to confirm that the reported alterations in localization are physically meaningful rather than artifacts of the nonlocal operator.
minor comments (2)
  1. [Abstract] The abstract states that 'explicit analytical expressions' are derived but does not display them; including at least the leading-order forms in the main text would improve readability.
  2. Clarify whether the fractional Shannon entropy remains non-negative and satisfies the usual inequalities when the fractional order deviates from 1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the major comments point by point below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (presumed derivation section): the central claim that the fractional extensions produce nontrivial, non-tautological variations in information content requires explicit verification that the constructed Shannon and Fisher expressions do not reduce by definition to functions of the fractional order alone; the abstract provides no such check or comparison to the integer-order limit.

    Authors: We agree that an explicit check strengthens the central claim. The fractional Shannon entropy and Fisher information are obtained by replacing the ordinary derivative with the Riemann-Liouville operator inside the standard integral definitions; the resulting expressions therefore contain the fractional order α both through the operator and through the eigenfunctions of the fractional Schrödinger equation. In the revised manuscript we will add, in §3, the direct substitution α → 1, recovering the conventional expressions, together with a short numerical comparison for the harmonic oscillator that shows the α-dependence is non-trivial and not a definitional artifact. revision: yes

  2. Referee: [§4] Application to harmonic oscillator (presumed §4): without the explicit forms of the fractional probability densities or the error analysis for the Riemann-Liouville operator applied to the oscillator wavefunctions, it is impossible to confirm that the reported alterations in localization are physically meaningful rather than artifacts of the nonlocal operator.

    Authors: The manuscript derives closed-form expressions for the fractional information measures as functions of α, but does not display the underlying fractional wavefunctions or perform a separate error analysis of the Riemann-Liouville operator on them. We will therefore include, in the revised §4, the explicit fractional probability densities obtained from the known solutions of the fractional harmonic-oscillator problem and a brief verification that the operator is applied consistently within the domain of the eigenfunctions. This addition will allow readers to assess the physical origin of the reported localization changes. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper reconsiders conventional Shannon entropy and Fisher information definitions, extends them via Riemann-Liouville fractional derivatives to nonlocal quantum systems, constructs the fractional expressions, and evaluates them analytically on the harmonic oscillator to obtain explicit dependence on the fractional order. No load-bearing steps reduce by construction to inputs: the fractional measures are newly defined rather than fitted or renamed from prior results, no predictions are made from subsets of the same data, and no self-citation chains or uniqueness theorems are invoked to force the framework. The central claim (nontrivial variation with fractional order) follows directly from applying the extended operators and is externally falsifiable via the resulting formulas. This is a standard definitional extension with independent content.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that Riemann-Liouville fractional derivatives provide a physically meaningful description of quantum dynamics with memory effects, plus the free choice of the fractional order as the key variable; no new entities are postulated.

free parameters (1)
  • fractional order parameter
    The order of the fractional derivative is treated as a free tunable parameter that controls deviations from standard results.
axioms (1)
  • domain assumption Riemann-Liouville derivative formalism is a valid nonlocal operator framework for quantum mechanics
    The paper adopts this formalism to extend the information measures.

pith-pipeline@v0.9.1-grok · 5675 in / 1269 out tokens · 18578 ms · 2026-06-27T06:32:01.300340+00:00 · methodology

discussion (0)

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Reference graph

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