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arxiv: 2512.14155 · v2 · submitted 2025-12-16 · 🪐 quant-ph

Fisher Information Measures under Lattice Combined Paul Trap

Precious Ogbonda Amadi , Paphon Pewkhom , Pruet Kalasuwan , Norshamsuri Ali , Syed Alwee Aljunid , Rosdisham Endut This is my paper

Pith reviewed 2026-05-16 22:13 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Fisher informationShannon entropyFisher-Shannon complexityPaul trapoptical latticeeffective frequencymotional statesion trap
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The pith

Fisher-Shannon complexity stays invariant under lattice control in Paul traps.

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

The paper studies the informational properties of a single ion in a Paul trap modified by an optical lattice, focusing on ground and first excited motional states. It finds that Fisher information and Shannon entropy change with the effective frequency set by the lattice strength, but their product—the Fisher-Shannon complexity—remains constant. This invariance shows that adjusting the lattice only changes how localized the ion is without affecting the underlying harmonic character of its motion. A reader cares because this gives a reliable starting point for applying information measures to design or analyze lattice-assisted ion traps in quantum technologies.

Core claim

The Fisher-Shannon complexity measure remains invariant under effective frequency control. The invariance demonstrates that optical modulation of kappa rescales localization, without altering the harmonic structure of the motional states. Beyond the harmonic limit, quartic corrections introduce non-Gaussian features that break this invariance, with stronger effects for excited states.

What carries the argument

The Fisher-Shannon complexity, defined as the product of Fisher information and Shannon entropy, which compensates changes from the effective frequency omega_eff = omega sqrt(1-kappa) to keep the product fixed in the harmonic regime.

If this is right

  • Fisher information and Shannon entropy redistribute between position and momentum spaces following the effective frequency.
  • The invariance holds specifically in the harmonic small-oscillation limit.
  • Retaining quartic lattice terms breaks the invariance by mixing higher states and creating non-Gaussian wavefunctions.
  • Departures from the invariant value grow with increasing kappa and are larger for the first excited state.

Where Pith is reading between the lines

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

  • This invariance could serve as a diagnostic for whether an ion trap operates in the pure harmonic regime.
  • Similar calculations might apply to other combined traps, such as those with static fields.
  • Experimental measurement of complexity in real ion systems could map the boundary where quartic effects appear.
  • Applications in quantum information might use lattice tuning to control localization while preserving complexity metrics.

Load-bearing premise

The ion motion is confined to the small-oscillation harmonic regime where quartic corrections from the lattice can be neglected.

What would settle it

Calculate the Fisher-Shannon complexity including the quartic lattice term in the potential and check whether it deviates from the constant value predicted in the harmonic approximation.

Figures

Figures reproduced from arXiv: 2512.14155 by Norshamsuri Ali, Paphon Pewkhom, Precious Ogbonda Amadi, Pruet Kalasuwan, Rosdisham Endut, Syed Alwee Aljunid.

Figure 1
Figure 1. Figure 1: Harmonic Paul trap lattice potential model with tunable parameters [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantized energy spectrum and wavefunction of the harmonic potential with the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Energy spectrum of the harmonic Paul trap lattice potential. The black solid 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Probability density in position space for the ground state where [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Probability density in momentum space for the ground state, where [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Fisher information in (a) Position space and (b) momentum space in ground [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Shannon Entropy in (a) Position space and (b) momentum space in ground state [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Fisher- Shannon complexity measures in (a) position space and (b) momentum [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
read the original abstract

We examine how the informational properties of a confined single ion response in a Paul trap modified by optical-lattice. We focus on the ground and first excited motional states and show that Fisher information, Shannon entropy, and Fisher-Shannon complexity track the effective frequency $\omega_{\mathrm{eff}}=\omega\sqrt{1-\kappa}$ of the potential. We show that the Fisher information and Shannon entropy reflect an effective frequency-driven redistribution of information between conjugate spaces. Our results show that the Fisher-Shannon complexity measure remains invariant under effective frequency control. The invariance demonstrates that optical modulation of $\kappa$ rescales localization, without altering the harmonic structure of the motional states. These results establish a controlled information-theoretic baseline for lattice-assisted Paul traps. Beyond the harmonic limit, retaining the quartic lattice correction introduces non-Gaussian wavefunction features through state-dependent mixing of higher eigenstates, which breaks the mutual compensation between Fisher information and Shannon entropy that sustains the invariant. The departure of $P'$ from its harmonic reference value intensifies with $\kappa$ and is stronger for the excited state, which confirms that the Fisher-Shannon complexity invariance is a distinctive property of the small-oscillation harmonic regime.

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 examines Fisher information, Shannon entropy, and Fisher-Shannon complexity for the ground and first excited motional states of a single ion in a Paul trap combined with an optical lattice. In the harmonic (small-oscillation) approximation, these quantities track the effective frequency ω_eff = ω √(1 − κ); the Fisher-Shannon complexity I_x exp(2 S_x) remains invariant under modulation of κ because the ω_eff scaling of I_x and the logarithmic scaling of S_x cancel exactly. The paper explicitly states that retention of the quartic lattice term produces non-Gaussian mixing and breaks the invariance, thereby scoping the result to the pure-harmonic regime.

Significance. If the algebraic invariance holds under the stated approximation, the work supplies a clean information-theoretic baseline for lattice-assisted Paul traps, showing that optical modulation of κ rescales localization without changing the underlying harmonic structure of the motional states. The explicit acknowledgment that the quartic term destroys the invariance is a strength, as is the focus on both ground and excited states. The result is modest in scope but could serve as a reference point for quantum-control protocols that rely on effective-frequency tuning.

major comments (2)
  1. [Abstract and main derivation section] The central invariance claim is presented as following directly from the known scaling properties of harmonic-oscillator Fisher information once ω_eff is substituted, yet the manuscript supplies no explicit derivation of I_x(ω_eff) or S_x(ω_eff) for the effective potential (see abstract and the paragraph following Eq. (presumably the definition of ω_eff)). Without these steps the cancellation I_x exp(2 S_x) = constant cannot be verified from the text alone.
  2. [Discussion of quartic term] The paper states that the quartic correction breaks the invariance via state-dependent mixing, but provides no quantitative estimate (e.g., overlap integrals or perturbative correction to P(x)) of how large the departure becomes as a function of κ for the ground versus excited state. This omission weakens the contrast drawn between the harmonic and anharmonic regimes.
minor comments (2)
  1. [Introduction / Methods] Notation for the effective frequency is introduced as ω_eff = ω √(1 − κ) without an accompanying equation number; subsequent references to “the effective frequency” would be clearer if the defining relation were labeled.
  2. [Abstract] The abstract refers to “P′” as the departure from the harmonic reference; this symbol is not defined in the provided text and should be introduced explicitly when first used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The comments are helpful for improving the clarity of the invariance result and the contrast with the anharmonic regime. We address each point below and will revise the manuscript as indicated.

read point-by-point responses

  1. Referee: [Abstract and main derivation section] The central invariance claim is presented as following directly from the known scaling properties of harmonic-oscillator Fisher information once ω_eff is substituted, yet the manuscript supplies no explicit derivation of I_x(ω_eff) or S_x(ω_eff) for the effective potential (see abstract and the paragraph following Eq. (presumably the definition of ω_eff)). Without these steps the cancellation I_x exp(2 S_x) = constant cannot be verified from the text alone.

    Authors: We agree that an explicit derivation strengthens the presentation. In the revised manuscript we will insert a short derivation immediately after the definition of ω_eff. For the ground state the rescaled Gaussian wavefunction yields I_x = 2mω_eff/ℏ and S_x = ½ + ½ log(2πℏ/(mω_eff)), so that I_x exp(2S_x) is exactly independent of ω_eff (and thus of κ). The same scaling holds for the first excited state by direct computation of its Hermite-Gaussian moments. This step-by-step verification will be added to the main text. revision: yes

  2. Referee: [Discussion of quartic term] The paper states that the quartic correction breaks the invariance via state-dependent mixing, but provides no quantitative estimate (e.g., overlap integrals or perturbative correction to P(x)) of how large the departure becomes as a function of κ for the ground versus excited state. This omission weakens the contrast drawn between the harmonic and anharmonic regimes.

    Authors: We acknowledge that a quantitative estimate would better illustrate the breakdown. Because the paper’s focus is the exact harmonic invariance, a full numerical diagonalization of the quartic problem is outside scope. We will add a first-order perturbative remark: the leading correction to P(x) arises from matrix elements of the quartic term, which are larger for the excited state owing to its broader support and nodal structure. The relative departure in I_x then grows linearly with κ at small κ, with a steeper coefficient for the excited state. This qualitative estimate will be inserted in the discussion section as a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in harmonic limit

full rationale

The paper defines the effective frequency ω_eff = ω √(1−κ) from the quadratic approximation to the combined Paul-lattice potential and notes that the motional states are then exact harmonic-oscillator eigenfunctions. The claimed invariance of the Fisher-Shannon complexity follows directly from the known scaling properties of position-space Fisher information (∝ ω_eff) and Shannon entropy (∝ −½ log ω_eff) for Gaussian wave functions; this algebraic identity is a standard feature of the harmonic oscillator and is not introduced by redefinition or self-citation within the paper. The abstract explicitly contrasts the result with the non-harmonic case where quartic terms break the invariance, confirming that the central claim is scoped to and exhausted by the stated approximation without reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Central claim rests on the definition of effective frequency from the modified potential and on the exact scaling properties of Gaussian wavefunctions under frequency rescaling.

free parameters (1)
  • κ
    Lattice modulation depth that defines the effective frequency ω_eff = ω√(1−κ); treated as a controllable parameter.
axioms (1)
  • domain assumption Motional states remain in the harmonic-oscillator ground or first-excited eigenstates of the effective potential.
    Invariance holds only inside the small-oscillation regime; quartic corrections are stated to break it.

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