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REVIEW 3 major objections 6 minor 45 references

Indirect Communication Between Non-Markovian Baths

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Two baths exchange information without touching

desk verdict A useful HEOM extension for undamped mode plus overdamped bath, but the information-transfer claim rests on non-additivity that doesn't establish it. read the letter →

arxiv 2412.14727 v2 pith:VFII3FFN submitted 2024-12-19 quant-ph

Ben S. Humphries , Dale Green , Garth A. Jones This is my paper · ORCID
classification quant-ph
keywords non-MarkovianHEOMundampedoscillatorbathLorentz-Drudebath-system-bathcoupling2Delectronicspectroscopyinformationtransferopenquantumsystems
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 develops a new hierarchical equations of motion model, LDUO-HEOM, in which a quantum system couples to two baths: an overdamped Lorentz-Drude bath and an undamped oscillator bath, combined by a direct sum rather than a tensor product. The authors aim to establish that information is transferred between these two baths even though they do not interact directly, the quantum system serving as the mediator through a bath-system-bath coupling. They also argue that this model removes the spurious damping of their earlier two-bath bath vibration model while producing qualitatively similar 2D electronic spectra at about 0.56% of the computational cost. If correct, the result implies that multi-bath open quantum systems can exhibit hidden channels of information exchange that are invisible in single-bath descriptions.

What carries the argument

The central object is the LDUO-HEOM hierarchy, a set of equations of motion for auxiliary density operators obtained from a path-integral influence functional whose spectral density is split into an overdamped Lorentz-Drude component and an undamped oscillator component $J(\omega)=J_{\mathrm{LD}}(\omega)+J_{\mathrm{UO}}(\omega)$, with $J_{\mathrm{UO}}$ proportional to $\omega(\delta(\omega-\omega_{\mathrm{UO}})+\delta(\omega+\omega_{\mathrm{UO}}))$. Each component is Matsubara-decomposed separately, giving one hierarchy axis for the undamped pair of imaginary frequencies $\pm i\omega_{\mathrm{UO}}$ and additional axes for the overdamped and Matsubara terms; the undamped axis is never Markovianized, which is what removes the spurious damping. Information flow between the baths is then analysed by projecting the collective bath coordinate expectation values $X^{(1)}$ and $X^{(2)}$ onto the separate hierarchy faces (UO and LD planes) and comparing the full model with isolated single-bath runs.

What would settle it

Run the same LDUO-HEOM simulation but compute the mutual information between the collective coordinates of the LD and UO baths directly from the full density matrix; if that mutual information stays at zero while the difference-plots in Figures 5 and 6 remain nonzero, the claim of bath-system-bath information transfer would be falsified. Alternatively, replace the quantum system with a classically driven linear oscillator coupled to the two baths; if the same residual pattern appears, the 'communication' is a classical driving artefact, not quantum information transfer.

Watch

Extended reading notes

Core claim

The paper claims that in the LDUO-HEOM model, the undamped oscillator bath and the overdamped Lorentz-Drude bath exchange information indirectly, despite having no direct coupling term, because each is coupled to the same quantum system. This is detected through first- and second-order bath coordinate expectation values: subtracting the isolated-bath responses from the full two-bath response leaves substantial nonzero residuals, showing that the full bath coordinate is not a sum of independent parts. The paper further claims that the LDUO-HEOM reproduces qualitatively the 2D electronic spectra of the earlier BVM while eliminating the extra damping that the finite linewidth of the underdamped mode had introduced, at a computational saving of at least 99.4%.

Load-bearing premise

The analysis assumes that a nonzero difference between the full two-bath response and the sum of the isolated-bath responses demonstrates information transfer between the baths; if the residuals only reflect that the system evolves differently when a second bath is present, the communication claim is not established.

Editorial extensions

If this is right

  • If the claim holds, two-bath open-system simulations must treat bath-system-bath coupling as a genuine physical channel, not as an artifact of the system-bath split.
  • The model gives a computationally cheap route (about 0.56% of BVM cost) for 2D electronic spectra of systems with an undamped intramolecular vibration in an overdamped environment, making realistic spectra calculations far more accessible.
  • The existence of indirect bath-bath information transfer means that non-Markovianity in multi-bath systems can be amplified or redirected through the system, with implications for quantum heat engines and refrigerators where multiple reservoirs are present.
  • The removal of superfluous damping sharpens the earlier conclusion that the Hamiltonian vibration model and the bath vibration model are not equivalent in practice, clarifying when a vibration can be safely moved from the Hamiltonian into the bath.

Reading between the lines

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

  • This result suggests that any open system coupled to two independent reservoirs may generically mediate information exchange between them, so the usual assumption of statistically independent baths should be checked whenever the system is non-Markovian.
  • A natural test would be to compute a direct information-theoretic measure, such as the mutual information between the two bath coordinates or a quantum channel capacity, and see whether it matches the residual signal; the paper uses expectation-value residuals as a proxy rather than a direct measure.
  • The difference plots in Figures 5 and 6 might be reconciled with a classical analogue: if a single driven oscillator coupled to two damped degrees of freedom reproduces the same residuals, the 'communication' could be a semiclassical interference effect rather than quantum information exchange.
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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

3 major / 6 minor

Summary. This paper develops a new hierarchical equations of motion (HEOM) variant, LDUO-HEOM, for a two-level system coupled to a bath whose spectral density is a sum of a Lorentz-Drude overdamped component and an undamped oscillator delta-function component. The derivation follows standard Matsubara/path-integral methods and is presented in detail. The model is used to compute 2D electronic spectra, which are compared qualitatively to the authors' earlier bath vibration model (BVM), and a large computational speedup is reported. The paper then uses first- and second-order bath-coordinate expectation values, following Zhu et al., to argue that the two baths exchange information indirectly through the system, a claim framed as bath-system-bath communication.

Significance. If the derivation and implementation are correct, LDUO-HEOM is a useful and computationally efficient alternative to underdamped-bath HEOM for systems with a well-defined vibrational mode in a dissipative environment; the reported speedup of roughly two orders of magnitude is striking, and the derivation is self-contained and reproducible in structure. The absence of parameter fitting is a strength. However, the central conceptual claim of information transfer between baths is not established by the presented evidence: the difference plots in Section V demonstrate non-additivity of bath-coordinate moments, not communication in an information-theoretic sense. The manuscript is publishable after the central claim is either supported by a direct information-flow analysis or substantially reframed.

major comments (3)
  1. [Section V, Figures 5 and 6] The residual plots in the bottom-right panels of Figs. 5 and 6 are presented as evidence that strong bath-system-bath coupling occurs and that information is transferred between the baths; this inference is not warranted. The quantities X^(1) and X^(2) are first and second moments of collective bath coordinates evaluated along three different trajectories: the full two-bath model and the two isolated single-bath models. Whenever the reduced system dynamics is not additive in the two couplings, the full-model expectation value will generically differ from the sum of the isolated values, because the system is a common cause that is itself affected by both baths. A nonzero residual is therefore necessary but not sufficient for inter-bath information transfer; a classical system driven by two independent stochastic forces would produce the same qualitative pattern. To support the headline claim, the authors should either perform an interventional test (e.g., decouple one bath at some time and show that the other bath's subsequent state changes) or compute a genuine measure of correlation and directionality between the two bath degrees of freedom, such as the mutual information of their reduced states or a bath-bath covariance. As written, the evidence does not distinguish 'communication through the system' from the trivial statement that two baths coupled to the same system both influence, and are influenced by, the system's state.
  2. [Section III, Eq. (42)] No convergence analysis is reported for the hierarchy truncation criterion Gamma_max = 10 max(I(gamma_k)). For the undamped component the Matsubara frequencies are purely imaginary, so the usual argument that high-tier ADOs decay and decouple is not available; the secular growth seen in the LDUO panels of Figs. 5 and 6 could in principle be a truncation artifact. Because these long-time bath-coordinate behaviours are the central evidence for the paper's main claim, the authors should show that X^(1), X^(2), and the residuals converge with respect to Gamma_max and hierarchy depth, and ideally compare with an independent numerically exact method or a substantially larger hierarchy for at least one test case.
  3. [Abstract and Section V] The term 'information transfer' is never operationally defined. The abstract and Section V state that information exchange occurs between the baths, but no information-theoretic quantity is introduced, and the bath-coordinate moments used are not information measures. Consequently, the central claim is not falsifiable as stated. The authors should either define the measure of information transfer and compute it, or rewrite the abstract and conclusions to claim non-additive bath dynamics or system-mediated bath-coordinate correlations, which is what the present data actually support.
minor comments (6)
  1. [Section IV, Figure 2] The claim of qualitative agreement between BVM and LDUO spectra is based on visual inspection; a quantitative measure, such as a normalized root-mean-square difference or spectral overlap, would make the comparison more convincing and reproducible.
  2. [Section IV, computational cost paragraph] The reported speedup of 0.56% would be more convincing if the authors specified the number of cores, software/library versions, and how the equilibration and evolution times were measured, since these details affect reproducibility.
  3. [Section V, Figure captions] The captions of Figures 5 and 6 should state explicitly which difference is plotted in the bottom-right panels: for example, X_LDUO - X_UO - X_LD for both X^(1) and X^(2); the current wording is ambiguous.
  4. [Section II] The statement that the baths are combined via a 'direct sum' is confusing because the total Hamiltonian is simply a sum of independent bath oscillators with additive system-bath coupling; clarifying what 'direct sum' adds beyond this standard construction would help readers.
  5. [Throughout] Please correct typographical errors, including 'incredibility powerful' (Introduction), 'superfluious' (Conclusion), 'bof' (Section I), and 'sate' (Eq. (28)).
  6. [Section V, Eqs. (51)-(56)] The recursion relations for L^(n+1)_i in Eq. (56) are stated without derivation; a brief explanation or a more explicit reference to the combinatorial factors would improve reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the LDUO-HEOM derivation is self-contained, and the central claims rest on independent computation rather than fitted or self-defined quantities.

full rationale

The LDUO-HEOM equations of motion (Eq. 39) follow from a standard path-integral/HEOM derivation (Eqs. 10-38) applied to the spectral density J = J_LD + J_UO (Eqs. 16-18). No simulation parameter is fitted to the target 2D spectra: eta, Lambda, and lambda_UO are stated inputs rather than free parameters tuned to reproduce the reported outputs. The 2DES comparison to the authors' earlier BVM is self-referential in the sense that the BVM is their own prior model, but the LDUO spectra are computed independently from the newly derived HEOM, so the qualitative agreement is an empirical finding rather than a construction. The computational cost comparison is a measured runtime, not a derived identity. For the bath-communication analysis, X^(1) and X^(2) are computed from the Zhu et al. formulas (Eqs. 51-56) on the full LDUO model and on the isolated LD and UO models; the difference plots are actual outputs, not quantities defined to equal the conclusion. The step from 'nonzero residual' to 'information transfer between baths' is an interpretive leap, and one could argue that non-additivity of bath-coordinate expectation values is necessary but not sufficient evidence for genuine information transfer. However, that is an evidentiary or correctness concern, not circularity: the residual is not defined as 'information transfer' by construction. The only notable self-citation is ref. 22 for the BVM's superfluous damping, which motivates the present model but is not the load-bearing step of the new derivation or of the bath-communication analysis. Overall, the paper's central derivation is self-contained against standard HEOM results, and its predictions do not reduce to fitted inputs or to self-citation chains.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The model depends on standard open-quantum-system approximations (Born factorization, Matsubara decomposition, Markovian cutoff) plus several hand-chosen parameters and one interpretive assumption about what the bath-coordinate difference plots mean. The model-specific free parameters are eta, Lambda, and lambda_UO; the hierarchy truncation factor is an ad hoc convergence choice. No independent evidence for an extra entity is required because none is posited.

free parameters (6)
  • eta_LD (Lorentz-Drude coupling strength) = 50 cm^-1
    Chosen for simulations; sets the system-bath coupling strength of the overdamped bath.
  • Lambda_LD (Lorentz-Drude cutoff) = 100 cm^-1
    Chosen to match the overdamped component of the BVM comparison; controls the bath memory time.
  • lambda_UO (undamped oscillator reorganisation energy) = 0.5 cm^-1
    Chosen small; sets the amplitude of the undamped delta-function spectral density via S_HR^UO = lambda_UO / omega_UO. No fitting procedure or independent physical justification is given.
  • omega_UO (undamped vibration frequency) = 500 cm^-1
    Physical model parameter for the vibrational mode; also used in the system Hamiltonian before canonical transformation.
  • omega_eg (electronic transition frequency) = 3000 cm^-1
    Two-level system parameter used in the 2DES simulations; standard for molecular models.
  • BVM comparison parameters = eta=(50,50) cm^-1, gamma=(100,2500) cm^-1
    Used only for benchmarking the LDUO result; not part of the new model but central to the qualitative agreement and speedup claims.
assumptions (6)
  • standard math Feynman-Vernon influence functional path integral (Eqs. 10-13).
    Background formalism for tracing out harmonic baths; assumed without proof.
  • domain assumption Born approximation and factorized initial state, rho0 = rhoS rhoB (Eq. 9).
    Assumes the bath starts thermally equilibrated and uncorrelated with the system; standard in HEOM but a restriction.
  • domain assumption Lorentz-Drude spectral density with Lambda = omega0^2 / gamma_LD requires gamma_LD >> omega0 (Eq. 17).
    The simplification from the more general spectral form to the Lorentz-Drude form relies on this separation of timescales.
  • domain assumption Markovian truncation condition nu_K = 2 pi K / (beta hbar) >> omega0 for the LD Matsubara tail (Eq. 40).
    High-frequency Matsubara terms are replaced by a delta function; standard but an approximation.
  • ad hoc to paper Hierarchy truncation at Gamma_max = 10 max(I(gamma_k)) (Eq. 42).
    No convergence study is shown; the factor of 10 is chosen by hand.
  • ad hoc to paper Non-additivity of bath coordinate expectation values is interpreted as evidence of bath-bath information transfer (Section V, Figs. 5-6).
    The comparison subtracts isolated single-bath trajectories rather than projections within the same two-bath model, so the residual may reflect common-cause coupling to the system rather than a direct information channel.

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Cite this review

Pith. "Pith review of Indirect Communication Between Non-Markovian Baths." pith.science (2026). https://pith.science/paper/VFII3FFN

@misc{pith2026241214727,
  author       = {Pith},
  title        = {Pith review of: Indirect Communication Between Non-Markovian Baths},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFII3FFN}},
  note         = {Machine review of arXiv:2412.14727}
}
read the original abstract

In this work we develop a model for an undamped vibration in the presence of an overdamped bath. This two-bath model involves a new derivation of the hierarchical equations of motion (HEOM) for an overdamped Lorentz-Drude (LD) environment that is summed together with an undamped oscillator (UO) bath, termed LDUO-HEOM. We show that information transfer occurs between the two baths, even in the absence of a direct coupling between the baths. This bath-system-bath, mediated information transfer leads to intricate non-Markovian dynamics. The model is analysed using expectation values of the bath coordinates and generates 2D electronic spectra that are in qualitative agreement with single-bath models. Furthermore, the model eliminates the additional superfluous damping introduced by the finite spectral width of the underdamped bath in our previous two-bath underdamped-overdamped, `bath vibration model' [J. Chem. Phys. 156, 084103 (2022)].

Figures

Figures reproduced from arXiv: 2412.14727 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic of one and two bath models. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: A generalised hierarchy diagram showing the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: A schematic of taking the projection onto the XY plane constituent of the collective bath mode resulting in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Tier 1 expectation values. Column 1, the bath coordinate of the full LDUO, projected onto the UO plane, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Tier 2 expectation values. Column 1, the bath coordinate of the full LDUO, projected onto the UO plane, [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.