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arxiv: 2606.24185 · v1 · pith:VPDPVFMXnew · submitted 2026-06-23 · 🪐 quant-ph · cond-mat.stat-mech

Initial-state-dependent dephasing effect in non-Hermitian Su-Schrieffer-Heeger models

Tao Min , Junjie Liu This is my paper

Pith reviewed 2026-06-26 00:33 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords non-Hermitian systemsSu-Schrieffer-Heeger modeldephasingtrace distanceparity-time symmetryanti-parity-time symmetryinitial-state dependencegain and loss
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The pith

In non-Hermitian SSH lattices, dephasing changes trace distance decay or stabilization according to the initial state rather than symmetry.

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

The paper examines how pure dephasing affects the trace distance between evolving states in finite non-Hermitian Su-Schrieffer-Heeger chains that feature alternating gain and loss. It finds that the distance response is largely independent of whether the system sits in a parity-time or anti-parity-time phase and instead hinges on the choice of starting state. For some initial states stronger dephasing simply speeds decay, while for others it produces either partial stabilization, in which the distance settles to a nonzero long-time value, or complete stabilization, in which the distance stays fixed at its initial value. The authors trace the initial-state dependence to the alternating gain-loss pattern by inspecting the equation of motion and note that the same pattern is absent in Hermitian counterparts. In the anti-parity-time unbroken phase the dephasing also continuously suppresses the exponential decay that would otherwise occur.

Core claim

The response of the trace distance to dephasing is largely symmetry-independent but instead initial-state dependent; increasing dephasing strength can either accelerate decay or stabilize the distance, with two kinds of stabilization in the strong-dephasing limit: partial stabilization where the distance approaches a finite nonzero value and complete stabilization where the distance remains constant throughout evolution. This initial-state dependence arises from the alternating gain and loss in the lattice and is confirmed absent in Hermitian SSH models.

What carries the argument

Trace distance between two evolving states under the non-Hermitian SSH Hamiltonian plus uniform pure dephasing, with its long-time behavior governed by the alternating gain-loss pattern.

If this is right

  • For certain initial states the trace distance remains constant for all time once dephasing is present.
  • For other initial states the distance approaches a nonzero plateau in the strong-dephasing limit rather than decaying to zero.
  • In the anti-PT unbroken phase the otherwise exponential decay of the distance is suppressed continuously as dephasing strength grows.
  • The same dephasing-induced behaviors are absent when the lattice is made Hermitian while keeping all other parameters fixed.

Where Pith is reading between the lines

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

  • The stabilization effects suggest that engineered dephasing could be used to protect certain quantum-state distinctions in non-Hermitian platforms without requiring symmetry protection.
  • Extending the analysis to larger system sizes or open-boundary conditions would test whether the initial-state dependence survives in the thermodynamic limit.
  • Mapping the same initial-state dependence onto other non-Hermitian lattices with alternating gain-loss would show how general the mechanism is.

Load-bearing premise

The claim that alternating gain and loss alone produce the observed initial-state dependence rests on the assumption that dephasing is uniform and that the equation-of-motion analysis captures every relevant mechanism without additional hidden factors.

What would settle it

Preparing the same pair of initial states in a Hermitian SSH chain with identical dephasing and finding that the trace-distance evolution still shows the reported stabilization behaviors would falsify the attribution to alternating gain and loss.

Figures

Figures reproduced from arXiv: 2606.24185 by Junjie Liu, Tao Min.

Figure 1
Figure 1. Figure 1: (a) Sketch of the minimum four-site non-Hermitian SSH [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dynamical evolution of the trace distance [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time evolution of the trace distance D(t) in the APT - unbroken phase with γ = 1.5 under different initial state pairs. Up￾per row [(a), (b)]: Initial states are ρˆ1(0) = ∣S1⟩⟨S1∣ and ρˆ2(0) = ∣S3⟩⟨S3∣. Middle row [(c), (d)]: Initial states are ρˆ1(0) = ∣S1⟩⟨S1∣ and ρˆ2(0) = ∣S2⟩⟨S2∣. Lower row [(e), (f)]: Initial states are ρˆ1(0) = ∣S1⟩⟨S1∣ and ρˆ2(0) = 0.5∣S1⟩⟨S1∣+0.5∣S2⟩⟨S2∣. Left panel: η1 = 0.5, η2 =… view at source ↗
Figure 4
Figure 4. Figure 4: Dynamics of the trace distance D(t) with varying dephas￾ing strength γd obtained by directly evolving Eq. (8). (a) and (b) cor￾respond to the PT -unbroken phase for γ = 0.1 and γ = 0.5, respec￾tively. (c) and (d) correspond to the symmetry-broken phase for γ = 0.5 and γ = 1.0, respectively. (e) and (f) correspond to the APT - unbroken phase for γ = 1.5. Other parameters are η1 = 0.5, η2 = 1 for the left pa… view at source ↗
Figure 5
Figure 5. Figure 5: Dynamics of the trace distance D(t) with varying dephas￾ing strength γd obtained by directly evolving Eq. (8). (a) and (b) correspond to the PT -unbroken phase for γ = 0.1 and γ = 0.5, re￾spectively. (c) and (d) correspond to the symmetry-broken phase for γ = 0.5 and γ = 1.0, respectively. (e) and (f) correspond to the APT -unbroken phase for γ = 1.5. Other parameters are η1 = 0.5, η2 = 1 for the left pane… view at source ↗
Figure 6
Figure 6. Figure 6: Dynamics of the trace distance D(t) with varying dephas￾ing strength γd obtained by directly evolving Eq. (8). (a) and (b) correspond to the PT -unbroken phase for γ = 0.1 and γ = 0.5, re￾spectively. (c) and (d) correspond to the symmetry-broken phase for γ = 0.5 and γ = 1.0, respectively. (e) and (f) correspond to the APT -unbroken phase for γ = 1.5. Other parameters are η1 = 0.5, η2 = 1 for the left pane… view at source ↗
Figure 7
Figure 7. Figure 7: Ordered real parts {Re(µj ) = Ej} of four eigenvalues of the Liouvillian superoperator L0 as a function of dephasing strength γd for (a) Hermitian system with γ = 0, (b) non-Hermitian system in the PT -unbroken phase with γ = 0.5, (c) non-Hermitian system in the symmetry broken phase with γ = 1, and (d) non-Hermitian system in the APT -unbroken phase with γ = 1.5. (e) The corresponding Liouvillian gap ∆E, … view at source ↗
Figure 9
Figure 9. Figure 9: Time evolution of the trace distance D(t) in a 40- site non-Hermitian SSH lattice system with varying pure dephas￾ing strength γd under initial states: (a) ρˆ1(0) = ∣S1⟩⟨S1∣, ρˆ2(0) = 1 2 (∣S1⟩⟨S1∣ + ∣S2⟩⟨S2∣), (b) ρˆ1(0) = ∣S17⟩⟨S17∣, ρˆ2(0) = ∣S18⟩⟨S18∣ and (c) ρˆ1(0) = ∣S1⟩⟨S1∣, ρˆ2(0) = ∣S39⟩⟨S39∣. The system is in the APT -unbroken phase with parameters η1 = 1, η2 = 0.5, and γ = 2. non-Hermitian term … view at source ↗
Figure 8
Figure 8. Figure 8: Spectral properties and phase diagram of the multi-site [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

Understanding the dynamical evolution of non-Hermitian systems under extra external dissipation is essential. Dephasing, a major realistic dissipation, is conventionally considered detrimental to information processing. However, its impact on non-Hermitian systems remains largely unexplored. Here, we focus on finite-sized non-Hermitian Su-Schrieffer-Heeger (SSH) lattice models with alternating gain and loss in real space and examine the dynamical evolution of the trace distance under pure dephasing. By tuning system parameters, this model supports phases with either parity-time or anti-parity-time symmetries, enabling us to explore the interplay between dephasing and different non-Hermitian symmetries. While the trace distance exhibits distinct dynamical behaviors across the different phases in the absence of dephasing, its response to dephasing is largely symmetry-independent but instead initial-state dependent. By varying initial states, we observe that increasing the dephasing strength can either merely accelerate the decay of the trace distance or stabilize it. Interestingly, we reveal two kinds of dephasing-induced stabilization that differ in the strong dephasing limit: a partial stabilization, where the trace distance approaches a finite value smaller than its initial value in the long-time limit, and a complete stabilization, where the trace distance remains at its initial value throughout the entire evolution. By analyzing the equation of motion, we attribute the initial-state dependent dephasing effect to the alternating gain and loss in the system and confirm its absence in Hermitian counterparts. Furthermore, in the anti-parity-time symmetry unbroken phase, we identify a continuous suppression-upon increasing the dephasing strength-of the otherwise exponential decay of the trace distance seen in the absence of dephasing.

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 studies the dynamical evolution of the trace distance under pure dephasing in finite non-Hermitian SSH lattices with alternating gain and loss, which realize PT or anti-PT symmetric phases. Without dephasing the trace distance shows phase-dependent behavior, but with dephasing the response is reported to be largely symmetry-independent and instead controlled by the choice of initial state: increasing dephasing strength can accelerate decay or produce either partial stabilization (long-time value finite but below initial) or complete stabilization (trace distance frozen at initial value). The initial-state dependence is attributed to the alternating gain-loss pattern via analysis of the equation of motion and is stated to vanish in the Hermitian limit; an additional result is the continuous suppression of exponential decay by dephasing in the anti-PT unbroken phase.

Significance. If the central attribution holds, the work demonstrates that dephasing can induce nontrivial stabilization of a distinguishability measure in non-Hermitian systems in an initial-state-dependent manner, offering a counter-example to the usual view of dephasing as purely destructive. The distinction between partial and complete stabilization regimes and the reported absence of the effect in Hermitian counterparts would be useful for understanding open-system dynamics with balanced gain and loss.

major comments (2)
  1. [Abstract and the section containing the equation-of-motion analysis] The headline attribution that the initial-state dependence arises from the alternating gain-loss pattern (and vanishes for Hermitian SSH) rests on 'analyzing the equation of motion,' yet the manuscript supplies neither the explicit Lindblad master equation for uniform dephasing nor the algebraic steps that isolate the contribution of the non-Hermitian terms. Without these, it is impossible to verify whether the reported symmetry independence and the two stabilization regimes are general consequences of the alternating pattern or artifacts of the chosen dephasing operators or finite-size boundary conditions.
  2. [Results on strong-dephasing limit] The claim of complete stabilization (trace distance remains exactly at its initial value for all times) in the strong-dephasing limit is load-bearing for the distinction between the two stabilization regimes; the manuscript must show that this occurs for a finite set of initial states and is not an artifact of the trace-distance definition or numerical truncation.
minor comments (2)
  1. [Model definition] Notation for the gain/loss parameters and the dephasing rate should be introduced once with a single symbol and used consistently; the abstract and main text currently employ slightly varying phrasing.
  2. [All figures] Figure captions should explicitly state the system size N, the specific initial states used for each curve, and whether the plotted quantity is an ensemble average.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and evidence.

read point-by-point responses

  1. Referee: [Abstract and the section containing the equation-of-motion analysis] The headline attribution that the initial-state dependence arises from the alternating gain-loss pattern (and vanishes for Hermitian SSH) rests on 'analyzing the equation of motion,' yet the manuscript supplies neither the explicit Lindblad master equation for uniform dephasing nor the algebraic steps that isolate the contribution of the non-Hermitian terms. Without these, it is impossible to verify whether the reported symmetry independence and the two stabilization regimes are general consequences of the alternating pattern or artifacts of the chosen dephasing operators or finite-size boundary conditions.

    Authors: We agree that the explicit Lindblad master equation and the algebraic derivation steps were not provided in sufficient detail. In the revised manuscript we will include the full Lindblad equation for uniform pure dephasing (with local number operators as jump operators) together with the step-by-step reduction to the equation of motion for the trace distance. The derivation isolates the contribution of the alternating gain-loss terms and shows that the initial-state dependence vanishes identically when the non-Hermitian terms are set to zero. We will also add a short discussion confirming that the reported behavior is tied to the real-space alternating pattern rather than to the specific choice of uniform dephasing operators or open-boundary conditions. revision: yes

  2. Referee: [Results on strong-dephasing limit] The claim of complete stabilization (trace distance remains exactly at its initial value for all times) in the strong-dephasing limit is load-bearing for the distinction between the two stabilization regimes; the manuscript must show that this occurs for a finite set of initial states and is not an artifact of the trace-distance definition or numerical truncation.

    Authors: We accept that stronger evidence is required for the complete-stabilization regime. In the revision we will (i) analytically identify the finite set of initial states (those whose support lies entirely on sites sharing the same gain or loss sign) for which the trace distance is exactly conserved in the infinite-dephasing limit, (ii) supply the corresponding exact solution of the master equation demonstrating that off-diagonal coherences are suppressed while the trace distance remains invariant, and (iii) supplement the numerics with exact diagonalization on small lattices and higher-precision integrators to exclude truncation artifacts. The standard definition D(ρ,σ)=(1/2)Tr|ρ−σ| will be stated explicitly. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on explicit analysis of the master equation rather than self-definition or fitted inputs.

full rationale

The paper's central claim—that the initial-state-dependent response to dephasing arises from the alternating gain-loss pattern—is presented as following from direct analysis of the equation of motion, with explicit confirmation of its absence in Hermitian limits. No equations, parameters, or predictions are shown to reduce by construction to the inputs; the abstract and available text contain no fitted quantities renamed as predictions, no self-citation load-bearing the uniqueness, and no ansatz smuggled via prior work. The derivation chain is therefore self-contained against external benchmarks and receives the default non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The central attribution to alternating gain and loss is presented as following from the equation of motion, but that equation is not shown.

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Reference graph

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    Equation of motion in the strong dephasing limit While the quantum master equation introduced in Eq. (8) provides a formally complete description of the system’s dy- namics in the presence of pure dephasing, its highly nonlin- ear structure and the intricate coupling between populations (diagonal elements) and coherences (off-diagonal elements) present si...

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    We showcase this phenomenon by har- nessing the fact that local population at the gain site gets am- plified while strong dephasing suppresses the spatial transport of population

    Case I: Accelerating decay We first demonstrate that dephasing can accelerate the de- cay of the trace distance, thus playing a detrimental role as in Hermitian systems. We showcase this phenomenon by har- nessing the fact that local population at the gain site gets am- plified while strong dephasing suppresses the spatial transport of population. Specifi...

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    This complete stabilization can be illustrated by choosing the pure initial statesˆρ1(0)=∣S1⟩⟨S1∣andˆρ2(0)=∣S3⟩⟨S3∣, for which the system is initialized in two different gain sites

    Case III: Complete stabilization Interestingly, the previously discovered dephasing-induced stabilization can become complete in the sense that the evo- lution of the trace distance is entirely frozen, such that it re- mains at its initial value throughout the entire evolution. This complete stabilization can be illustrated by choosing the pure initial st...

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    (A4) This relation is equivalent to the anti-commutation relation {ˆH,P ′T}=0

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