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Dynamics of monitored SSH Model in Krylov Space: From Complexity to Quantum Fisher Information

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that time-averaged quantum Fisher information evaluated in Krylov space can detect both the PT transition at $\gamma=1$ and the measurement-induced entanglement phase near $\gamma=2$ in the non-Hermitian SSH model…

desk verdict Useful Krylov-space study of the non-Hermitian SSH model; the kCoP order parameter for the PT transition is solid, but the QFI probe of the measurement-induced phase is under-supported until checked against entanglement entropy. read the letter →

arxiv 2502.03434 v3 pith:TUYB6PRC submitted 2025-02-05 quant-ph cond-mat.str-elhep-th

Nilachal Chakrabarti , Neha Nirbhan , Arpan Bhattacharyya This is my paper · ORCID
classification quant-phcond-mat.str-elhep-th
keywords Krylovcomplexityspreadnon-HermitianSSHmodelPTsymmetrytransitionmeasurement-inducedphasequantumFisherinformationbi-Lanczosalgorithmmonitoreddynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the no-click limit of a monitored Su-Schrieffer–Heeger (SSH) chain, a non-Hermitian fermionic model with two known transitions: a PT transition at measurement rate $\gamma=1$ and a measurement-induced entanglement transition near $\gamma=2$. It asks whether complexity measures built in Krylov space can act as probes of these transitions. The authors find that spread complexity, spread entropy, and related quantities distinguish the PT transition but remain insensitive to the entanglement transition, while the time-averaged quantum Fisher information (QFI) in Krylov space changes slope at $\gamma=1$ and saturates near $\gamma=2$. The conclusion is that Krylov-space QFI, which structurally resembles the complexity functional but carries correlation information, is a promising probe of both phases.

What carries the argument

The tool that carries the argument is the bi-Lanczos algorithm, which generates two bi-orthonormal Krylov bases $\{|r_j\rangle\}$ and $\{|l_j\rangle\}$ for the non-Hermitian Hamiltonian, making it tridiagonal in this basis. Spread complexity is the average Krylov index weighted by the normalized probability amplitudes $|\tilde{\Psi}^r_j(t)^*\tilde{\Psi}^l_j(t)|$. Quantum Fisher information, when expanded in this basis, can be written as a double sum over Krylov indices whose structure mimics the complexity functional; keeping only the diagonal part gives $F^{\rm diag}_Q = \sum_n \phi^{R*}_n \phi^L_n f_n$, with $f_n$ encoding operator correlations. The paper compares the time-averaged versions of these quantities as functions of $\gamma$ against the known transition points $\gamma=1$ and $\gamma=2$.

What would settle it

Compute the bipartite entanglement entropy for the single-particle density matrix in this non-Hermitian SSH model as a function of $\gamma$ and overlay it with the time-averaged Krylov-space QFI. If the plateau of $\bar{F}_Q$ occurs at a value of $\gamma$ clearly different from the entropy's volume-to-area transition point, or if the same saturation appears in a model with no entanglement transition, the central probe claim is falsified.

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Extended reading notes

Core claim

The central claim is that the time-averaged quantum Fisher information, evaluated by expanding the state or the measurement operator in the bi-orthonormal Krylov basis, can serve as a probe of the measurement-induced phase in the non-Hermitian SSH model and not just of its PT transition. As the measurement rate $\gamma$ increases, $\bar{F}_Q$ decreases, its slope changes sharply at the PT transition $\gamma=1$, and it saturates to a small value around $\gamma=2$, the point where the model's entanglement entropy crosses from volume law to area law. The same behaviour is shown by the diagonal part of the QFI, which the authors argue is structurally similar to the Krylov complexity functional but contains correlation information through the factors $f_n$. Because the QFI in Krylov space carries this correlation landscape, it succeeds where spread complexity and Krylov complexity of purification fail.

Load-bearing premise

The paper takes the location of the measurement-induced transition at $\gamma=2$ from the earlier entanglement-entropy study and interprets the saturation of the time-averaged QFI around that value as a probe of this transition, even though the QFI slope begins to flatten just before $\gamma=2$ and no independent entanglement-entropy calculation is performed.

Editorial extensions

If this is right

  • In the PT-symmetric phase ($\gamma<1$) the spread complexity and related measures oscillate; in the PT-broken phase they saturate, and the saturation timescale grows with the measurement rate.
  • The scaling exponent of the late-time Krylov complexity of purification (kCoP) with subsystem size shows a sharp dip at $\gamma=1$, making kCoP a sensitive indicator of the PT transition but not of the measurement-induced transition.
  • The time-averaged QFI in Krylov space, as well as its diagonal part, is largely independent of system size and shows a slope change at $\gamma=1$ followed by saturation near $\gamma=2$, so it can probe both transitions.
  • These features persist under both open and periodic boundary conditions, and for different choices of the initially localized state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The saturation of $\bar{F}_Q$ near $\gamma=2$ could in principle be a generic signature of strong non-Hermitian localization in Krylov space rather than a specific witness of the entanglement transition; a direct computation of bipartite entanglement entropy in the same Krylov basis would separate the two.
  • If the QFI probe is genuine, it suggests a subsystem-free diagnostic for measurement-induced transitions in other monitored free-fermion chains, where subsystem partitioning is expensive.
  • The authors show that $f_n$ is positive but not monotonic, so QFI is not a strict complexity measure; understanding how $f_n$ tracks the correlation landscape could reveal why the diagonal part alone detects the transitions.
  • Going beyond the no-click limit to full stochastic Schrödinger evolution would test whether the QFI probe survives the noise that real measurements introduce.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper studies the non-Hermitian SSH model that arises in the no-click limit of a monitored SSH chain, using Krylov-space techniques. The authors compute spread complexity, spread entropy, entropic complexity, Krylov inverse participation ratio, and a purification-based subsystem complexity (kCoP) under the bi-Lanczos algorithm, and then compute time-averaged quantum Fisher information in Krylov space. They find that spread complexity oscillates below the PT transition at gamma=1 and saturates above it; that the kCoP scaling exponent shows a dip at gamma=1; and that time-averaged QFI decreases with measurement rate, changes slope at gamma=1, and saturates around gamma=2. They interpret the latter as a promising probe of the measurement-induced phase, while acknowledging they cannot pinpoint the transition.

Significance. The paper extends Krylov-complexity diagnostics to a non-Hermitian model that hosts both a PT transition and a measurement-induced entanglement transition. The qualitative dynamics (oscillation below gamma=1, saturation above) agree with earlier tight-binding results and are likely robust. The numerical workflow is described in sufficient detail that the main computations could be reproduced in principle. However, the central claim that time-averaged QFI probes the measurement-induced phase is not yet established: the identification of gamma=2 is imported from Ref. [167], no entanglement entropy is computed for the states studied here, and the authors explicitly state they cannot pinpoint the transition. If the missing comparison with entanglement entropy is supplied, the QFI result would be a useful addition; at present it rests on an untested identification and should be treated as a conjecture.

major comments (4)
  1. [Section 4, Figs. 15-16 and footnote 8] The paper's central claim—that time-averaged QFI probes the measurement-induced phase near gamma=2—is not supported by the evidence presented. The identification of gamma=2 as the entanglement transition is imported entirely from Ref. [167], and no entanglement entropy is computed for the states evolved here. Moreover, footnote 8 concedes that the slope of the QFI begins to saturate before gamma=2 and that the authors cannot pinpoint the exact transition. As a result, the observed saturation could reflect single-particle non-Hermitian localization onto one sublattice rather than the volume-to-area entanglement transition. A direct comparison of the time-averaged QFI with the bipartite entanglement entropy computed for the same initial states and parameters, or a collapse of the QFI data across system sizes at a critical gamma, is needed to make the probe claim.
  2. [Section 4, Eq. (4.1)] For the measurement operator n_A (or I−n_B) acting on a single-particle state, F_Q = 4 Var(n_A) is bounded by 1 and is a single-body sublattice-occupancy variance, not a multipartite entanglement witness. Calling the time-averaged QFI a witness of multipartite entanglement in the discussion around Fig. 16 is therefore unjustified as stated. The structural similarity with the complexity functional does not make it a many-body entanglement measure. Please either qualify the interpretation or provide a many-body (multi-particle) computation.
  3. [Section 3, 'Spread Complexity for a subsystem'] The definition of the subsystem in Krylov space is not specified. The text says the reduced density matrix is obtained by 'taking the reduced trace over a subspace in Krylov space,' but it does not state which Krylov indices form the subsystem A and which form the complement B. Since the kCoP scaling results in Figs. 10–14 depend on this partition, the procedure must be stated explicitly (e.g., first l Krylov basis vectors versus the remaining K−l) and, ideally, justified independently of the final-state localization properties.
  4. [Section 4, Eq. (4.9) and footnote 7] The symmetrization of the QFI for non-Hermitian evolution—replacing ⟨O^2⟩ by (1/2)(O†O+OO†) and ⟨O⟩^2 by ⟨O†⟩⟨O⟩—is introduced without derivation or justification. All subsequent QFI results depend on this choice. Please provide a derivation from a measurement-theoretic definition of QFI for non-unitary dynamics, or at least benchmark the symmetrized quantity against the standard QFI in a Hermitian limit and against the exact single-particle variance.
minor comments (6)
  1. [Footnote 8 and Section 5] The word 'slop' appears where 'slope' is intended; please correct the typo and rephrase the awkward clause 'the slope of it starts to saturate.'
  2. [Figs. 10 and 12] Please provide the fitting ranges, the number of subsystem sizes used, and the goodness-of-fit (e.g., R²) for the power-law fits that produce the scaling exponents alpha(γ); the current figures show only the extracted exponent.
  3. [Section 3, Eq. (3.8)] The definition of the time average with parameters t_ref and T should specify how t_ref and T are chosen; the results in Fig. 11 may depend on this averaging window.
  4. [Section 4, Eq. (4.12)] The statement that off-diagonal contributions are negligible under time averaging is not demonstrated; please include a representative comparison of the full and diagonal QFI as a function of time.
  5. [Fig. 17] The caption says 'we have done the numerical calculation with a time step of 0.1 and shown the results for each time-step,' but the plotted quantity f_n is a function of n; please clarify what is actually plotted, for example f_n at a fixed late time or averaged over time.
  6. [Abstract and Section 4] The abstract says QFI 'can be a promising probe of the measurement-induced phase,' while footnote 8 says the authors cannot pinpoint the transition; please adjust the wording to match the actual strength of the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the Krylov-space numerics are independent computations and the QFI-probe claim is an interpretation, not a fitted or definitionally forced result.

full rationale

The paper's new quantities — spread complexity, spread entropy, KIPR, kCoP, and time-averaged QFI — are computed directly from the non-Hermitian SSH Hamiltonian via the bi-Lanczos algorithm and the normalized no-click evolution (Eqs. 2.5, 3.3, 4.1-4.2), without fitting any of these outputs to the transition locations. The PT transition at gamma=1 follows from the exact single-particle spectrum (Eq. 2.8), and the measurement-induced transition at gamma=2 is imported from Ref. [167] (Gal-Turkeshi-Schiro), an external source, not from the authors' own prior work; the paper explicitly does not compute entanglement entropy for its evolved states, so the QFI-saturation claim is an interpretive probe claim rather than a derived equivalent of its input. The authors also explicitly verify (Fig. 17 and surrounding text) that f_n is not monotonic, and therefore decline to call F_diag a generalized complexity functional, avoiding the self-definitional 'complexity by construction' pattern. Footnote 8 concedes that the QFI slope saturates before gamma=2, so the paper does not overstate a sharp transition prediction. The numerous self-citations of A. Bhattacharyya are background/review citations for circuit complexity and are not load-bearing for the new Krylov-space numerics. The skeptic's worry that the QFI saturation may reflect single-particle non-Hermitian localization rather than multipartite entanglement is a physical-interpretation/correctness risk, not a circularity detectable from the paper's own equations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the no-click equivalence from [167], the locations of the PT and entanglement transitions, the bi-Lanczos construction, and two ad hoc steps: the kCoP purification scheme and the symmetrized QFI for non-Hermitian evolution. The hopping parameters set the transition positions in gamma.

free parameters (3)
  • hoppings w and v = w=1.5, v=0.5
    Chosen to set the PT transition at |w-v|=1 and the presumed entanglement transition at w+v=2. The quantitative locations of the claimed features depend on these choices.
  • time-averaging window (t_ref, T) = not specified precisely
    The time-averaged kCoP and QFI depend on the integration window; the paper does not quantify the sensitivity to these choices.
  • subsystem sizes for scaling fits = l = 4, 8, 10, 20, 40, 50, 100
    The exponents alpha(gamma) are fitted over this limited set without error bars.
assumptions (5)
  • domain assumption The no-click limit of the monitored SSH model is described by the non-Hermitian Hamiltonian in Eq. (2.1) with gamma as the measurement rate.
    Invoked in Section 2 and relies on the mapping established in [167]; not re-derived here.
  • domain assumption The PT transition occurs at gamma=|w-v| and the entanglement transition at gamma=w+v (here gamma=1 and gamma=2).
    Taken from [167]; the paper does not independently compute the entanglement entropy to confirm the gamma=2 transition.
  • standard math The bi-Lanczos algorithm defines a valid Krylov basis and spread complexity for non-Hermitian evolution.
    Adopted from [107, 110, 111]; this is the standard construction in the field.
  • ad hoc to paper The reduced density matrix in Krylov space is obtained by tracing over a subspace of Krylov indices, and the purified state's kCoP can be computed with the earlier bi-Lanczos coefficients.
    This step in Section 3 is not fully defined: the purified state lives in a doubled Hilbert space and no generator for its Krylov space is specified.
  • ad hoc to paper For the non-Hermitian case, the QFI is symmetrized by replacing <O^2> with 1/2(O^dagger O + O O^dagger) and <O>^2 with <O^dagger><O>.
    Stated in footnote 7 without derivation; affects the numerical values of F_Q.

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Pith. "Pith review of Dynamics of monitored SSH Model in Krylov Space: From Complexity to Quantum Fisher Information." pith.science (2026). https://pith.science/paper/TUYB6PRC

@misc{pith2026250203434,
  author       = {Pith},
  title        = {Pith review of: Dynamics of monitored SSH Model in Krylov Space: From Complexity to Quantum Fisher Information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUYB6PRC}},
  note         = {Machine review of arXiv:2502.03434}
}
abstract

In this paper, we investigate the dynamics of a non-Hermitian SSH model that arises out of the no-click limit of a monitored SSH model in the Krylov space. We find that the saturation timescale of the complexity associated with the spread of the state in the Krylov subspace increases with the measurement rate, and late time behaviour differs across the $\mathrm{PT}$ symmetry transition point. Furthermore, extending the notion of this complexity for subsystems in Krylov space, we find that the scaling of its late time value with subsystem size shows a discontinuous jump across the $\mathrm{PT}$ transition point, indicating that it can be used as a suitable order parameter for such transition but not for the measurement-induced transition. Finally, we show that a generalized measure in the Krylov subspace, which contains information about the correlation landscape, such as Quantum Fisher information, which also possesses some structural similarity with the complexity functional, can be a promising probe of the measurement-induced phase.

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Forward citations

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