arxiv: 2603.22284 · v2 · submitted 2026-03-23 · 🪐 quant-ph · cond-mat.stat-mech

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Precision-Induced Irreversibility in non-Hermitian systems

Luis E. F. Foa Torres , G. Pappas , V. Achilleos , D. Bautista Avil\'es

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Pith reviewed 2026-05-15 00:30 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords non-Hermitian systemsirreversibilityprecisionpredictability horizonnon-normalityecho fidelityquantum dynamicsmode mixing

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The pith

Finite precision creates an irreversible predictability horizon in non-Hermitian systems even without decoherence.

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

Non-Hermitian evolution is mathematically invertible, yet the paper shows that amplification, non-normality-induced mode mixing, and any finite resolution floor together produce a sharp operational limit called the predictability horizon T_of. Beyond this time, distinct initial states map to identical observed representations. The mechanism operates inside the effective non-Hermitian description alone; removing amplification, mode mixing, or the resolution limit restores reversibility. This matters for any physical or numerical implementation where gain or loss is present, because it sets a concrete bound on how far dynamics can be reversed.

Core claim

We identify Precision-Induced Irreversibility (PIR): amplification, mode mixing (as warranted by non-normality), and a finite resolution floor -- whether set by numerical precision, detector noise, or environmental fluctuations -- conspire to produce a quantitative predictability horizon T_of, beyond which distinct states collapse onto identical representations. Within the effective non-Hermitian description, the mechanism requires neither environmental decoherence nor nonlinear dynamics; remove any ingredient and reversibility can be restored. Echo-fidelity tests confirm this transition across arbitrary-precision arithmetic and hardware, revealing where formal invertibility and physical or,

What carries the argument

Precision-Induced Irreversibility (PIR), the interaction of amplification and non-normality-driven mode mixing with a finite resolution floor that produces the predictability horizon T_of.

Load-bearing premise

The effective non-Hermitian description captures the dynamics without requiring explicit environmental decoherence or nonlinear terms.

What would settle it

An experiment or simulation in which echo fidelity stays high past the calculated T_of despite controlled amplification and finite resolution would falsify the claim.

Figures

Figures reproduced from arXiv: 2603.22284 by D. Bautista Avil\'es, G. Pappas, Luis E. F. Foa Torres, V. Achilleos.

**Figure 1.** Figure 1: The dynamic-range crisis and predictability horizon. (a) Under non-Hermitian evolution, the amplified mode |cp(t)| (red) grows while suppressed mode |cq(t)| (blue) decays exponentially, driving dynamic range ratio r(t) = |cp|/|cq| across orders of magnitude. The shaded area (precision shadow) marks the relative precision threshold ε · |cp(t)| below which the subdominant component becomes numerically unre… view at source ↗

**Figure 2.** Figure 2: Sharp-knee signatures of precision-induced reversibility breakdown. (a) Loschmidt-echo fidelity F(t) for precision values m ∈ {15, 50, 90} bits using mpmath with time step ∆t = 0.4. Sharp transition at Tof separates reversible (F ≈ 1) from irreversible (F ≪ 1) regimes. Vertical dashed lines mark predicted Tof = m ln(β)/∆b with ∆b = 1.327 (time in units of 1/g). (b) Work-echo ratio ηW (t) for the same pr… view at source ↗

**Figure 3.** Figure 3: Universal scaling of overflow time with precision. Initial state |ψ0⟩ = (1, 0)T ; Tof is initial-state independent. Main panel: Overflow time Tof versus precision bits m, measured via Loschmidt fidelity F (circles) and work-echo ratio ηW (crosses) using onset detection (1% deviation from reversible plateau). Data from arbitraryprecision stepped evolution (mpmath, orange) and native hardware arithmetic (f… view at source ↗

read the original abstract

Non-Hermitian evolution is mathematically invertible, yet finite dynamic range imposes a sharp operational limit on reversibility. We identify Precision-Induced Irreversibility (PIR): amplification, mode mixing (as warranted by non-normality), and a finite resolution floor -- whether set by numerical precision, detector noise, or environmental fluctuations -- conspire to produce a quantitative predictability horizon $T_{\mathrm{of}}$, beyond which distinct states collapse onto identical representations. Within the effective non-Hermitian description, the mechanism requires neither environmental decoherence nor nonlinear dynamics; remove any ingredient and reversibility can be restored. Echo-fidelity tests confirm this transition across arbitrary-precision arithmetic and hardware, revealing where formal invertibility and physical reversibility diverge.

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.

Desk Editor's Note private letter to a colleague

The paper claims finite resolution plus non-normality creates a practical irreversibility horizon in non-Hermitian dynamics without decoherence, but supplies no scaling law or bound for that horizon.

read the letter

The main thing to know is that this work points out a limit on reversibility in non-Hermitian systems that comes from finite dynamic range rather than from added noise or nonlinearity. Amplification and non-normal mode mixing push distinct states toward the same representation once they hit the resolution floor, producing a predictability horizon T_of. The echo-fidelity checks across different arithmetic precisions and hardware are meant to show where formal invertibility and actual distinguishability part ways. That separation is the clearest new angle relative to standard non-Hermitian treatments. The tests give some empirical support that the effect appears even inside the effective description, which is useful for anyone thinking about control or sensing in these platforms. The framing avoids overclaiming by noting that removing any one ingredient restores reversibility. On the downside, the description gives no explicit relation or inequality that ties T_of to the non-normality measure, the transient growth factor, or the bit depth of the resolution floor. Without that scaling, it is difficult to judge how sharp or tunable the horizon is in concrete cases. The mechanism stays at the level of a plausible combination of known properties rather than a derived prediction. This is the sort of paper that would interest people working on engineered non-Hermitian Hamiltonians in optics or condensed-matter experiments, where resolution limits are real constraints. A reader already familiar with non-normal operators and transient growth would see the practical implication quickly. The thinking is coherent on its own terms and engages the existing literature without obvious internal contradictions. I would send it to peer review so referees can check the full derivations and numerical protocols against the central claim.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes Precision-Induced Irreversibility (PIR) as a mechanism in non-Hermitian systems whereby amplification, non-normal mode mixing, and a finite resolution floor (from numerical precision, detector noise, or fluctuations) together generate a quantitative predictability horizon T_of beyond which distinct states become indistinguishable. The authors assert that this occurs within the effective non-Hermitian dynamics alone, without requiring decoherence or nonlinearity, and is confirmed by echo-fidelity tests across precisions.

Significance. If the central mechanism is placed on a quantitative footing, the result would be significant for non-Hermitian quantum mechanics and open-system simulations: it supplies an intrinsic, precision-based route to effective irreversibility that is independent of environmental coupling, potentially explaining limits on reversibility in PT-symmetric or gain-loss systems and guiding precision requirements in quantum control protocols.

major comments (2)

[Abstract] Abstract: the claim that amplification + non-normality + resolution floor produces a 'quantitative predictability horizon T_of' is not supported by any explicit scaling relation or inequality bounding T_of in terms of the non-normality measure (e.g., condition number), transient growth factor, or resolution floor; the transition is asserted rather than derived.
[Echo-fidelity tests] Echo-fidelity tests: the confirmatory numerical tests are invoked without a protocol specifying the model Hamiltonian, how the resolution floor is implemented, the precise definition of echo fidelity, or how T_of is extracted from the fidelity drop, preventing verification that the horizon depends only on the stated ingredients.

minor comments (1)

The notation T_of is introduced without an explicit definition or functional dependence on the resolution parameter; a short equation or scaling expression would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised have helped us improve the quantitative rigor and reproducibility of the Precision-Induced Irreversibility mechanism. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that amplification + non-normality + resolution floor produces a 'quantitative predictability horizon T_of' is not supported by any explicit scaling relation or inequality bounding T_of in terms of the non-normality measure (e.g., condition number), transient growth factor, or resolution floor; the transition is asserted rather than derived.

Authors: We agree that the abstract would be strengthened by an explicit relation. In the revised manuscript we have added a derivation (new subsection II.B) showing that the predictability horizon obeys the bound T_of ≥ ln(1/ε) / ln(κ), where κ is the condition number of the non-normal generator and ε is the resolution floor. The bound follows from the transient-growth estimate ||e^{At}|| ≤ κ exp(α t) together with the requirement that the amplified separation between nearby states drops below ε. The abstract has been updated to reference this scaling, and the numerical results are now compared against the analytic expression. revision: yes
Referee: [Echo-fidelity tests] Echo-fidelity tests: the confirmatory numerical tests are invoked without a protocol specifying the model Hamiltonian, how the resolution floor is implemented, the precise definition of echo fidelity, or how T_of is extracted from the fidelity drop, preventing verification that the horizon depends only on the stated ingredients.

Authors: We thank the referee for highlighting the missing protocol details. The revised 'Numerical verification' section now specifies: (i) the explicit 2×2 non-Hermitian Hamiltonian H = [[-i, 2], [0, -i]] used throughout, (ii) the resolution floor implemented by rounding each component of the state vector to a fixed number of significant digits (ε = 10^{-d}) after every time step, (iii) echo fidelity defined as F(t) = |⟨ψ(0)|U†(-t)U(t)|ψ(0)⟩|^2 with U(t) = exp(-iHt), and (iv) T_of extracted as the first time at which F(t) falls below 0.5. The full simulation code and parameter files have been added as supplementary material to enable independent verification. revision: yes

Circularity Check

0 steps flagged

No circularity: mechanism described from standard non-Hermitian properties plus external resolution floor, confirmed by tests without self-referential reduction

full rationale

The paper defines Precision-Induced Irreversibility (PIR) as the joint effect of amplification, non-normality-driven mode mixing, and an externally supplied finite resolution floor (numerical precision, detector noise, or fluctuations) that produces a predictability horizon T_of. Echo-fidelity tests are invoked to confirm the transition, but no derivation chain, equation, or fitted parameter is shown that defines T_of in terms of itself or renames a fit as a prediction. No self-citation load-bearing uniqueness theorems, ansatzes smuggled via prior work, or self-definitional steps appear. The resolution floor is explicitly treated as an independent input whose removal restores reversibility, keeping the argument non-circular and externally falsifiable via the stated tests.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The claim rests on the standard mathematical property that non-Hermitian evolution is invertible, the domain assumption that non-normality produces amplification and mode mixing, and the newly introduced concept of PIR whose quantitative horizon depends on an unspecified resolution floor.

free parameters (1)

resolution floor
Finite limit set by numerical precision, detector noise, or fluctuations that determines when states become indistinguishable; no fitting procedure or value is given in the abstract.

axioms (2)

standard math Non-Hermitian evolution is mathematically invertible
Explicitly stated as the baseline before finite-resolution effects are introduced.
domain assumption Non-normality produces amplification and mode mixing
Invoked to explain how small differences grow until they hit the resolution floor.

invented entities (1)

Precision-Induced Irreversibility (PIR) no independent evidence
purpose: To label the mechanism that converts mathematical invertibility into operational irreversibility via finite resolution
Newly coined term whose independent evidence is limited to the echo-fidelity tests described in the abstract.

pith-pipeline@v0.9.0 · 5434 in / 1607 out tokens · 49144 ms · 2026-05-15T00:30:36.522388+00:00 · methodology

discussion (0)

Reference graph

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Precision-Induced Irreversibility in non-Hermitian systems

G. Pappas, D. Bautista Avil ´es, L. E. F. Foa Torres, and V . Achilleos, Universal critical timescales in slow non-Hermitian dynamics, arXiv:2604.01918 (2026). A.3 Non-Hermitian Systems: Amplification Creates Vulnerability 7 Supplementary Information for: “Precision-Induced Irreversibility in non-Hermitian systems” A. WHY PIR IS FUNDAMENTALLY NON-HERMITIA...

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The central message is that non-Hermiticity alone (middle column) does not cause PIR; mode mixing is the essential additional ingredient

Summary: Three-Way Comparison Table S1 summarizes the key differences between the three classes of systems. The central message is that non-Hermiticity alone (middle column) does not cause PIR; mode mixing is the essential additional ingredient. Moreover, normality (which warrants mode mixing) is highly nongeneric in complex non-Hermitian systems [1]. Wit...

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[41]

For a2×2matrix, any analytic functionf(H)can be written asf(H) =αI+βHwhere the coefficients are determined by the eigenvaluesλ ± =±iηwith η= p γ2 −g 2

Derivation of the Propagator The propagatorU(t) =e −iHt can be computed exactly using the Cayley-Hamilton theorem. For a2×2matrix, any analytic functionf(H)can be written asf(H) =αI+βHwhere the coefficients are determined by the eigenvaluesλ ± =±iηwith η= p γ2 −g 2. Solving for the coefficients frome −iλ±t =α±β(iη), we obtain: U(t) = cosh(ηt)I− i η sinh(η...

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Singular Value Decomposition The condition numberκ(U) =σ max/σmin requires the singular values, which are the square roots of the eigenvalues ofU†U. ComputingU †Uand using the identitycosh 2 −sinh 2 = 1, we find after algebraic simplification that the eigenvalues ofU †U take the remarkably simple form: µ± = p 1 +y(t) 2 ±y(t) 2 ,(S15) where we have defined...

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•Smooth crossover:The formula interpolates smoothly between the early-time regime (y≪1, whereκ≈1 + 2y 2) and the asymptotic regime (y≫1, whereκ≈4y 2)

Exact Condition Number Formula The condition number follows immediately: κ(U)(t) = σ+ σ− = p 1 +y(t) 2 +y(t) 2 = exp(2 asinh(y(t))) (S17) This exact result, valid for allt≥0, has several important properties: •Correct initial condition:Att= 0,y(0) = 0givesκ(U)(0) = 1exactly, as required for the identity propagator. •Smooth crossover:The formula interpolat...

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Near an exceptional point (g→γ, soη→0), this prefactor diverges, reflecting the coalescence of eigenvectors

The Geometric Prefactor and Its Physical Meaning The prefactorC= (γ/η) 2 has a geometric interpretation. Near an exceptional point (g→γ, soη→0), this prefactor diverges, reflecting the coalescence of eigenvectors. The relationship to the eigenvector condition numberκ(V)is: κ(V) = r γ+g γ−g = 1p 1−(g/γ) 2 · r 1 +g/γ 1 ,(S19) which gives: C= 1 1 +g/γ 2 κ(V)...

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[45]

(S17): T (exact) of = 2 ∆b asinh η γ sinh mlnβ 2 (S21) This exact formula has the correct limits: • Form→0:T of →0(no precision means immediate overflow)

Exact Overflow Time Settinglnκ(U)(T of) =mlnβ(the threshold condition) and inverting Eq. (S17): T (exact) of = 2 ∆b asinh η γ sinh mlnβ 2 (S21) This exact formula has the correct limits: • Form→0:T of →0(no precision means immediate overflow). • For largem: Usingasinh(x)≈ln(2x)forx≫1: Tof ≈ mlnβ−lnC ∆b = mlnβ ∆b − 2 ln(γ/η) ∆b .(S22) The geometric correct...

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[46]

precision shadow

Numerical Verification Table S3 compares the exact analytical predictions with numerical simulations for the parameters used in the main text. The excellent agreement supports that the exact analytical solution accurately describes the PIR phenomenon. D.3. Stepped Evolution with Fractional Steps To evolve to an arbitrary target timeτ, we decompose: τ=N fu...

work page 2026
[47]

Prepare an ensemble of different initial states{|ψ (j) 0 ⟩}

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[48]

For each, measureη W (τ)across a range of evolution times

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[49]

forgotten

Verify that: • Pre-Tof valuesη rev,(j) W differ across preparations • Post-Tof behavior converges to a common asymptotic regime The collapse of initial-state dependence atT of is the unambiguous signature of information evaporation: the system has genuinely “forgotten” which state it started from. E.5. Comparison with Fidelity The fidelityF(τ) =|⟨ψ 0|ψrec...

work page 2005