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arxiv: 2606.02743 · v1 · pith:QELBXVVUnew · submitted 2026-06-01 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el

Geometric Decoherence Time in Lindbladian Dynamics

Rishabh Jha , Stephan Haas , Abhinav Prem This is my paper

Pith reviewed 2026-06-28 13:58 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-el
keywords geometric decoherence timeLindbladian dynamicslogarithmic negativityRényi entropyopen quantum systemsdecoherenceKitaev chainXXZ chain
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The pith

The geometric decoherence time is the first moment the exact relation between logarithmic negativity and Rényi-1/2 entropy breaks under Lindbladian evolution.

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

This paper defines the geometric decoherence time as the earliest instant when the monotone relation between logarithmic negativity and Rényi-1/2 entropy, which holds exactly for any bipartition of a pure state, ceases to hold during open-system evolution. The breakdown marks entropy growth that is no longer accompanied by entanglement growth, thereby identifying the onset of decoherence on a dynamical scale grounded in bipartite entanglement. The authors establish the definition for single-particle Gaussian dynamics and many-body Lindbladian systems, then apply it to the Kitaev chain with gain and loss, where the topological phase yields longer times than the trivial phase at fixed dissipation strength. They further show that quantum mutual information serves as a long-time diagnostic whose vanishing signals factorization of the steady state, a condition stricter than separability, and that this tracking fails when strong symmetries allow residual classical correlations to persist.

Core claim

The geometric decoherence time is introduced as the earliest moment the monotone relation between logarithmic negativity and Rényi-1/2 entropy breaks down under open-system evolution, signaling entropy growth without accompanying entanglement growth. This criterion is established in both single-particle Gaussian dynamics and many-body Lindbladian evolution. Quantum mutual information provides a complementary long-time diagnostic whose asymptotic vanishing is equivalent to factorization of the steady state across the bipartition. In the Kitaev chain with balanced gain and loss a closed-form solution shows that the topological phase sustains longer coherence times than the trivial phase at ide

What carries the argument

The geometric decoherence time, defined as the earliest time at which the monotone relation between logarithmic negativity and Rényi-1/2 entropy breaks under Lindbladian evolution.

If this is right

  • Quantum mutual information vanishes asymptotically if and only if the steady state factorizes across the bipartition.
  • When a product steady state is approached exponentially in trace norm, negativity and mutual information share the same decay rate.
  • Strong symmetries allow residual classical correlations to survive after entanglement has vanished, breaking the mutual-information tracking of negativity.
  • In the Kitaev chain with balanced gain and loss the topological phase exhibits longer geometric decoherence times than the trivial phase at identical dissipation.
  • Local Z-dephasing in the XXZ chain preserves residual classical correlations while gain and loss restore mutual-information tracking of negativity.

Where Pith is reading between the lines

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

  • The geometric decoherence time could serve as a practical figure of merit for tuning dissipation strengths in engineered open quantum systems to prolong coherence.
  • The concept may generalize to other entanglement monotones beyond logarithmic negativity, yielding alternative decoherence timescales.
  • Experimental protocols that measure both negativity and Rényi entropy in real time could directly extract the geometric decoherence time in platforms with controllable Lindblad operators.
  • The distinction between topological and trivial phases in coherence time suggests that topological protection may extend to open-system timescales in driven-dissipative settings.

Load-bearing premise

The assumption that the monotone relation between logarithmic negativity and Rényi-1/2 entropy holds exactly for all pure states and that its first breakdown under Lindbladian evolution is both well-defined and physically meaningful as the onset of decoherence.

What would settle it

Numerical integration of the Lindbladian for the Kitaev chain showing whether the computed geometric decoherence time is longer in the topological phase than in the trivial phase at the same dissipation strength, as predicted by the closed-form solution.

Figures

Figures reproduced from arXiv: 2606.02743 by Abhinav Prem, Rishabh Jha, Stephan Haas.

Figure 1
Figure 1. Figure 1: Decoherence-time landscape for the balanced gain/loss Kitaev chain in the ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometric decoherence time in the topological phase of the open Kitaev chain ( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mutual information I(A:B) and fermionic Gaussian negativity EF as functions of time for the same parameters as [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Fixed-µ comparison within the topological phase. Left: topological reference point, µ = 0.50 and γ = 0.15. Right: weaker dissipation, µ = 0.50 and γ = 0.02. The weak-dissipation panel shows the clear overshoot of τ peak d relative to τ g d where the peak is at￾tained at much later time and is comparatively flat. In both panels τ g d is extracted from Definition 1 via the entropy–negativity trajectory, whic… view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of exact logarithmic negativity [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Entropy–negativity trajectories Γ(t) = (S1/2(t), EN (t)) for the two protocols of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Finite-size and finite-cut dependence of the [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: Fixed-γ comparison at γ = 0.15. Left: topo￾logical reference point, µ = 0.50. Right: chiral point, µ = 0.00. Both panels show only a slight undershoot of τ peak d relative to τ g d . Although the entropy–negativity trajectory is not shown, τ g d is extracted from Definition 1 using the full trajectory Γ(t) = S1/2(t), N (t)  . Appendix H: Finite-Size and Subsystem Scaling of Gaussian Diagnostics To assess … view at source ↗
Figure 9
Figure 9. Figure 9: Small-system benchmark of the balanced Gaussian correlation-matrix solution against exact many-body [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Relative Frobenius error ∥CCMDE(t) − CCMS(t)∥F /∥CCMDE(t)∥F between the numerical CMDE solution and the closed-form CMS expression Eq. (21). For γ+ = γ− the error stays at numerical precision; for γ+ ̸= γ− it grows, confirming that the closed-form solu￾tion is specific to the balanced case and that the full differential equation must be integrated in the imbal￾anced regime. Parameters: open Kitaev chain, … view at source ↗
Figure 11
Figure 11. Figure 11: Same diagnostic as Fig [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Same diagnostic as Fig [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Entropy–negativity trajectories for the open XXZ chain with [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Entropy–negativity trajectories for the open XXZ chain with [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
read the original abstract

The onset of decoherence in open many-body systems lacks a dynamical timescale grounded in the loss of bipartite entanglement. Here, we introduce the $geometric$ $decoherence$ $time$, defined as the earliest moment the monotone relation between logarithmic negativity and R\'{e}nyi-$\tfrac{1}{2}$ entropy -- exactly equal across any bipartition for pure states -- breaks down under open-system evolution, signaling entropy growth without accompanying entanglement growth. We establish this criterion in both single-particle Gaussian dynamics and many-body Lindbladian evolution. We show that quantum mutual information provides a complementary long-time diagnostic: its asymptotic vanishing is equivalent to factorization of the steady state across the bipartition, a condition strictly stronger than separability, and whenever a product steady state is approached exponentially in trace norm, negativity and mutual information share the same decay rate. In the presence of a strong symmetry, this tracking can fail -- residual classical correlations can survive after entanglement has vanished. In the Kitaev chain with balanced gain and loss, we derive a closed-form solution and show that the topological phase sustains longer coherence times than the trivial phase at identical dissipation, with a local minimum at the chiral-symmetric point. In the interacting XXZ chain, exact many-body evolution shows that local $Z$-dephasing preserves residual classical correlations, whereas gain and loss restore the mutual-information tracking of negativity. Our results establish the geometric decoherence time as a dynamical scale tracking the onset of decoherence.

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 / 0 minor

Summary. The manuscript introduces the geometric decoherence time as the earliest moment at which the equality between logarithmic negativity E_N and Rényi-1/2 entropy S_{1/2} (which holds exactly for pure states across any bipartition) breaks down under Lindbladian evolution, interpreted as the onset of decoherence via entropy growth without entanglement growth. It derives closed-form expressions for single-particle Gaussian and many-body Kitaev-chain dynamics with balanced gain/loss, reports that the topological phase exhibits longer coherence times than the trivial phase at fixed dissipation (with a local minimum at the chiral-symmetric point), presents exact many-body numerics for the interacting XXZ chain under Z-dephasing versus gain/loss, and shows that quantum mutual information provides a complementary long-time diagnostic whose exponential decay tracks negativity when the steady state factorizes.

Significance. If the central definition is made rigorous, the geometric decoherence time supplies a new, entanglement-grounded dynamical scale for decoherence onset that can distinguish topological from trivial phases under identical dissipation. The closed-form Kitaev solution and the exact XXZ evolution constitute concrete, falsifiable results that strengthen the contribution.

major comments (1)
  1. [Abstract and Kitaev section] Abstract and § on Kitaev chain: the definition of geometric decoherence time as the 'earliest moment' the monotone relation E_N = S_{1/2} breaks down is not equipped with an explicit, threshold-independent detection rule. Both quantities are continuous in t, coincide at t=0, and generically separate for any t>0 under nonzero Lindblad operators; without a stated criterion (e.g., first nonzero derivative of |E_N(t)−S_{1/2}(t)| or |E_N−S_{1/2}| exceeding a fixed ε independent of numerical cutoff), the reported phase-dependent times rest on an incompletely specified procedure and are therefore not yet load-bearing for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for a more precise operational definition of the geometric decoherence time. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and Kitaev section] Abstract and § on Kitaev chain: the definition of geometric decoherence time as the 'earliest moment' the monotone relation E_N = S_{1/2} breaks down is not equipped with an explicit, threshold-independent detection rule. Both quantities are continuous in t, coincide at t=0, and generically separate for any t>0 under nonzero Lindblad operators; without a stated criterion (e.g., first nonzero derivative of |E_N(t)−S_{1/2}(t)| or |E_N−S_{1/2}| exceeding a fixed ε independent of numerical cutoff), the reported phase-dependent times rest on an incompletely specified procedure and are therefore not yet load-bearing for the central claim.

    Authors: We agree that the original definition requires an explicit, reproducible detection rule to be fully rigorous. In the revised manuscript we will define the geometric decoherence time τ_g as the infimum of times t>0 at which |E_N(t)−S_{1/2}(t)| exceeds a fixed numerical threshold ε=10^{-10}, chosen once and for all to lie well above machine precision and independent of model parameters or numerical cutoff. This criterion will be stated in the abstract, the introductory section, and the Kitaev-chain section. For the closed-form Gaussian solutions of the Kitaev chain the analytic expressions for E_N(t) and S_{1/2}(t) permit direct evaluation of τ_g without ambiguity; the same threshold will be applied uniformly to the numerical XXZ data. We will also add a short robustness check confirming that the reported ordering of coherence times between topological and trivial phases is insensitive to modest variations of ε around the chosen value. revision: yes

Circularity Check

0 steps flagged

No circularity: definition relies on independently established pure-state equality without reduction to fits or self-citations

full rationale

The paper defines the geometric decoherence time directly from the breakdown of the known equality between logarithmic negativity and Rényi-1/2 entropy that holds exactly for pure states (due to the relation ||ρ^{T_A}||_1 = [Tr √ρ_A]^2). This equality is a standard fact independent of the present work. The subsequent analysis applies this criterion to explicit Lindbladian models (Gaussian dynamics, Kitaev chain with closed-form solution, XXZ chain) and derives results such as phase-dependent timescales and mutual-information tracking without any fitted parameters renamed as predictions, self-citation chains, or ansatzes smuggled from prior author work. No load-bearing step reduces by construction to the inputs; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the mathematical properties of logarithmic negativity and Rényi-1/2 entropy for pure states (standard) and on the interpretation of their first breakdown as the onset of decoherence (ad hoc to the paper). No free parameters or new physical entities with independent evidence are introduced.

axioms (2)
  • standard math Logarithmic negativity and Rényi-1/2 entropy are exactly equal (monotone) across any bipartition for pure states
    Invoked in the definition of the geometric decoherence time (abstract, first sentence).
  • ad hoc to paper The first breakdown of this relation under Lindbladian evolution signals entropy growth without entanglement growth and therefore the onset of decoherence
    This interpretive step is required to promote the mathematical breakdown into a physically meaningful timescale.
invented entities (1)
  • geometric decoherence time no independent evidence
    purpose: Dynamical scale marking the onset of decoherence via breakdown of negativity-Rényi relation
    Newly defined quantity whose only evidence is the paper's own calculations; no external falsifiable prediction supplied.

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    Collecting all 2Loperators into the Nambu spinor Ψ = (c 1,

    Equation of motion forC(t)and the balanced solution We consider a chain ofLsites with fermionic cre- ation and annihilation operators satisfying{c i, c† j}=δ ij and{c i, cj}= 0. Collecting all 2Loperators into the Nambu spinor Ψ = (c 1, . . . , cL, c† 1, . . . , c† L)T , the Hamil- tonian takes the formH= 1 2Ψ†HBdGΨ up to a constant, whereH BdG is the 2L×...

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    Its spectrum is read- ily obtained

    Covariance-sector Liouvillian gap The homogeneous equation (F4) is governed by the linear super-operator Lcov(X) :=−i[H BdG, X]−2γX,(F7) whereX=C(t)− 1 2 12L is the traceless deviation from the infinite-temperature fixed point. Its spectrum is read- ily obtained. SinceH BdG is Hermitian, it admits an or- thonormal eigenbasis{ϕ n}with real eigenvaluesE n. ...

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    Perturbative decoherence time at smallγ We derive the leading-order shift ofτ g d away from the unitary maximumt ∗ whenγis small. Throughout,t ∗ is taken to be the first local maximum ofs(t) =E F (t;γ= 0) on (0,∞); if the unitary trajectory has multiple local maxima, the formula applies at each in turn, and the geometric decoherence time is the shift of t...

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    Balanced Gaussian dynamics versus exact many-body evolution For sufficiently small chains, the full Lindblad equation can be solved in the many-body Hilbert space, providing a direct comparison against the Gaussian correlation- matrix treatment. Figure 9 shows this comparison at the level of subsystem entropies: the time dependence ofS 1/2,S 2, andS vN ob...

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    (19) admits the closed-form solution (CMS) of Eq

    Balanced versus imbalanced gain/loss Whenγ + =γ −, the correlation-matrix equation of mo- tion Eq. (19) admits the closed-form solution (CMS) of Eq. (21), and the time evolution is obtained spectrally after a single diagonalization of the BdG matrix. When γ+ ̸=γ −, the dynamics remains Gaussian but no analytic simplification of this form is available; one...

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    3 against both time-step resolution and bipartition choice

    Convergence of the mutual-information tracking We verify the robustness of the tracking results shown in Fig. 3 against both time-step resolution and bipartition choice. For the baseline parametersL= 128,L A = 11, µ= 0.5,J= ∆ = 1, andγ + =γ − = 0.15, the geo- metric decoherence time isτ g d = 0.73 and the peak-based estimate isτ peak d = 0.66. The peak fe...