Unveiling coherent dynamics in non-Markovian open quantum systems: exact expression and recursive perturbation expansion

Alessandra Colla; Giulio Gasbarri; Heinz-Peter Breuer

REVIEW 3 major objections 4 minor 2 cited by

Unveiling coherent dynamics in non-Markovian open quantum systems: exact expression and recursive perturbation expansion

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes a direct commutator expression and a recursive perturbation expansion for the effective Hamiltonian of a non-Markovian open quantum system, uniquely fixed by the minimal dissipation principle, and shows how the…

desk verdict Clean closed formula for the effective Hamiltonian; perturbative expansion needs a complete derivation and a normalization fix. read the letter →

arxiv 2506.04097 v2 pith:W4D5XOJP submitted 2025-06-04 quant-ph

Alessandra Colla , Heinz-Peter Breuer , Giulio Gasbarri This is my paper

classification quant-ph

keywords effectiveHamiltoniannon-Markovianopenquantumsystemsminimaldissipationprincipletime-convolutionlessmasterequationbathorderedcumulantsenergyrenormalizationspin-bosonmodelthermodynamics

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

The paper aims to make the coherent part of non-Markovian open-system dynamics computable in practice. It gives a closed expression that recovers the effective Hamiltonian $K(t)$ directly from any time-local generator of the reduced dynamics, and a recursive perturbation series that expresses $K(t)$ at every order in the system-bath coupling through system operators and bath correlation functions. Energy renormalization and eigenbasis rotations thus become accessible beyond the weak-coupling, Markovian regime, without first solving for the full generator. Applications to spin systems show that pure dephasing produces only a time-dependent frequency shift, while an orthogonal coupling can rotate the eigenbasis at odd orders and shift energies at even orders.

What carries the argument

The carrying objects are the minimal-dissipation projection onto commutator superoperators, the commutator formula $K = \frac{1}{2 i d} \sum_\alpha [F_\alpha^\dagger, \mathcal{L}[F_\alpha]]$, and the recursive family of bath ordered cumulants $D$ generated by the recursion formula. The projection fixes the otherwise ambiguous split of the time-local generator into a Hamiltonian part and a dissipator, and the commutator formula turns that projection into an explicit sum over any orthonormal operator basis. The recursion builds the ordered cumulants from products of $n$-point bath correlation functions, and the main series assembles them with interaction-picture system operators to give the $n$-th order effective Hamiltonian $K_n$. The time-ordering step functions inside each cumulant enforce the correct 'chunked' ordering of operators, in analogy with the standard ordered-cumulant formalism.

What would settle it

For a small exactly solvable model, such as a two-level system coupled to a two-level bath, reconstruct the exact time-local generator numerically, compute $K(t)$ both by direct numerical minimization of the dissipator and by the commutator formula, and compare the truncated series at order $n=3$ or higher; disagreement in either comparison would falsify the claimed expressions.

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Extended reading notes

Core claim

The central claim is that the effective Hamiltonian introduced by the minimal dissipation principle is not just a conceptual construction but a directly computable object. For any Hermiticity-preserving, trace-annihilating generator $\mathcal{L}$ of the reduced dynamics and any Hilbert-Schmidt orthonormal operator basis $\{F_\alpha\}$, the effective Hamiltonian is $K = \frac{1}{2 i d} \sum_\alpha [F_\alpha^\dagger, \mathcal{L}[F_\alpha]]$. When the generator is known only perturbatively, the paper provides $K(t)=\sum_n \lambda^n K_n$, with each $K_n$ written as integrals of bath ordered cumulants times interaction-picture system operators; the cumulants themselves obey a recursion. The even and odd terms have opposite time-reversal symmetry structure: even orders pair bath and system pieces of opposite symmetry, odd orders pair pieces of the same symmetry. This structure explains why a purely dephasing spin gets only a $\sigma_z$ energy shift, why a spin coupled through $\sigma_x$ can undergo an eigenbasis rotation, and why the finite-temperature spin-boson model shows only frequency renormalization.

Load-bearing premise

The expansion stands or falls on the recursive formula that builds higher-order bath correlation terms from lower-order ones, whose full derivation is left to a companion paper; if that recursion is wrong, every conclusion about energy shifts and eigenbasis rotations collapses.

Editorial extensions

If this is right

For a spin under pure dephasing ($A=\sigma_z$), the effective Hamiltonian is always proportional to $\sigma_z$ with a time-dependent frequency; all even-order terms vanish, so only odd bath cumulants can shift the energy.
For an unbiased spin with $A=\sigma_x$, odd orders of $K(t)$ rotate the eigenbasis while even orders only renormalize the energy; at finite temperature the spin-boson model exhibits only a frequency renormalization, confirming an earlier numerical conjecture.
Order by order, the expansion expresses $K(t)$ using only interaction-picture system operators and $n$-point bath correlation functions, so coherent dynamics can be obtained without an explicit closed-form generator.
Because the decomposition into symmetric and antisymmetric time-ordering classes fixes the even/odd structure, environments with symmetric two-point correlations (Gaussian, zero-mean baths) produce simpler renormalization effects than environments with nonvanishing odd-order cumulants.
The second-order term is written explicitly through the response function and the symmetrized correlation function, linking the coherent shift to the fluctuation-dissipation structure of the environment.

Reading between the lines

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

The commutator formula could be applied to generators obtained numerically or experimentally (for example, from process tomography or hierarchical equations of motion), giving a practical extraction of the time-dependent effective Hamiltonian in models where no analytic generator exists.
The recursion implies a combinatorial classification: any bath whose odd ordered cumulants vanish cannot produce eigenbasis rotations at leading order in couplings orthogonal to the free Hamiltonian, a testable prediction for non-Gaussian reservoirs.
In strong-coupling quantum thermodynamics, the even/odd structure suggests that work and heat contributions from the effective Hamiltonian are controlled by different time-symmetry classes of environmental correlations, which could be probed in a driven quantum thermal machine.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: the paper's first main result—the closed expression Eq. (6) for the effective Hamiltonian—is clean, general, and usable; the perturbative expansion advertised in the title is the part that worries me. And the end matter contains a normalization error that the authors should fix before this goes anywhere.

What's genuinely new: I don't know another place where the minimal-dissipation effective Hamiltonian is written directly in terms of the time-local generator as a sum over commutators with an arbitrary orthonormal operator basis. That's a genuinely operational formula. The even/odd symmetry decomposition of the expansion (23)-(24) is also a nice result, and the spin examples do real work: they explain why pure dephasing shows no Lamb shift when the bath is Fock-diagonal, and they reproduce the numerical conjecture from [43] about finite-temperature spin-boson models. The paper doesn't fit any parameters; the spin conclusions are checked against known exact results. That's good evidence.

The soft spots: First, Eq. (A.37). With F_j = σ_j/√d, plugging into (6) yields a prefactor 1/(2 i d²) on Σ[σ_j, L[σ_j]], not 1/(2 i d) as printed. For d=2 and L = -i[σ_z,·] the printed version gives 2σ_z instead of σ_z. That's a concrete inconsistency in the formal section, and it makes me want to see the full derivation, not just the claims.

Second, Eq. (17) is not a recursion as written. D appears on both sides, there's no base case, and Ddot is itself defined in terms of D times delta functions. The Letter calls it a recursion and says the construction follows van Kampen, but the reader can't reproduce any order beyond n=1 from the text. All the spin structure—the vanishing even orders, the rotation in odd orders—sits on top of Eq. (16), which is exactly where the missing derivation is. That's a load-bearing gap in the presentation, even if the companion paper [34] fills it.

None of this is fatal conceptually. The projection idea is solid and Eq. (6) is provable in the end matter. But the perturbative engine is not yet self-contained. A referee can't check the central tool without the companion. Citation practice looks fair; the self-citations are to genuinely related prior work.

My recommendation: send it to review, but explicitly ask for the derivation of (16)-(18) to be included (or the companion made available) and for A.37 to be corrected. The paper is worth a serious referee's time—the closed formula and the spin applications are valuable—but it shouldn't be accepted on the strength of the advertised perturbative expansion until that gap is closed.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a framework to define and compute the effective Hamiltonian K(t) of a non-Markovian open quantum system via the minimal dissipation principle. Its first result is an explicit formula, Eq. (6), for K(t) from the time-local generator, together with the useful trace formula Eq. (8). The second result is a recursive perturbative expansion, Eqs. (15)-(17), expressing K(t) to arbitrary order in the system-bath coupling in terms of bath ordered cumulants. The authors apply the expansion to spin-boson models, deriving structural statements about even and odd orders: pure dephasing has vanishing even orders and only sigma_z renormalization, while orthogonal coupling produces diagonal even-order and off-diagonal odd-order contributions.

Significance. If the perturbative expansion is valid, the paper provides a practical route to coherent energy renormalization beyond weak coupling and gives falsifiable structural predictions for spin systems, several of which are checked against known models. The derivation of Eq. (6) is largely self-contained and Eq. (8) is operational; the spin-system results, such as the absence of renormalization in the pure-dephasing spin-boson model of Eqs. (20)-(21), are consistent with existing exact solutions. The main bottleneck is verification of the central expansion (16)-(17), which is deferred to an inaccessible companion, together with a normalization error in the su(d) basis formula (A.37). These issues are fixable within the manuscript's scope, so I do not see them as undermining the overall approach.

major comments (3)

[Systematic perturbation expansion, Eq. (17)] Equation (17) is not a well-defined recursion as printed. The symbol D appears both as the object being defined (the bath ordered cumulant on the left-hand side) and in the double sum on the right-hand side, while the raw correlation function in Eq. (12) carries the same symbol; no base cases (e.g., D(empty,empty), D(tau,empty), D(empty,s)) or an explicit distinction between the two objects is supplied. Consequently Eq. (16) cannot be evaluated for any order n from the information in the Letter. Please provide the missing base and initial conditions, clarify which D appears on the right-hand side, and show at least the first two orders n=1 and n=2 explicitly, including the connection to Eq. (25).
[End matter, basis choice for su(d), Eq. (A.37)] With the stated normalization F_0=1/sqrt(d) and F_j=sigma_j/sqrt(d), substituting into Proposition (6) yields K = 1/(2 i d^2) sum_{j=1}^{d^2-1} [sigma_j, L[sigma_j]], not the printed factor 1/(2 i d). For d=2 and L=-i[sigma_z/2, .], Eq. (A.37) gives sigma_z instead of the correct sigma_z/2, whereas Eq. (8) and Eq. (A.38) give the correct value. Please correct the normalization and check all related formulas for possible residual factors of d.
[Systematic perturbation expansion, Eq. (16)] The central perturbative result, Eq. (16), is stated without derivation, and all technical details are referred to the companion paper [34], which is not provided and has no published identifier. Because the even/odd structural statements in Eqs. (23)-(24) and the spin-model conclusions all depend on Eq. (16), the main claim of the Letter is not independently verifiable as submitted. Please include a derivation or a sufficiently detailed proof sketch in the Letter or in supplementary material, and verify Eq. (16) against a known exact case, for instance the pure-dephasing spin-boson model of Eqs. (20)-(21).

minor comments (4)

[Effective Hamiltonian, Eq. (4)] The text says the basis {H_j} is orthonormal under the Hilbert-Schmidt product, while the end matter defines {H_j} with Tr{H_j^2}=d(d+1)/2 and then rescales to obtain a normalized basis; please align the terminology so that 'orthonormal' applies to the rescaled basis.
[After Eq. (3)] The word 'dimentions' should be 'dimensions', and 'apriori' should be written as 'a priori' in the two places where it appears.
[Definition of D, Eq. (12)] The sentence introducing Eq. (12) says the time-ordering functions theta inside D are responsible for the time ordering of all operators and functions, but the displayed definition of D(tau^k_1,s^{n-k}_1) only states ordering within each variable set; please specify the operator order in the string B_R(s^{n-k}_1)B_L(tau^k_1) and explain how the relative ordering of tau and s variables enters.
[Notation for products] The shorthand A(t^k_1) and D(tau^k_1,s^{n-k}_1) is used before an explicit product ordering is given; please define, for example, A(t^k_1)=A(t_1)...A(t_k) and state how empty variable sets are handled.

Circularity Check

2 steps flagged · score 4.0 of 10

The central perturbative expansion (Eq. 16) is delegated to an unpublished same-author companion [34], and the printed recursion (17) is self-referential without a base case; the Proposition (6) itself is derived in the text, so the circularity is partial rather than total.

self citation load bearing [Systematic perturbation expansion, Eqs. (14)-(17); companion paper [34]]
"Here, we provide a systematic perturbation expansion for K in powers of the coupling strength λ. This constitutes the second main result of this work. Below, we outline the strategy; technical details are provided in [34]. ... where the functions D are bath n-ordered cumulants and are linear combinations of products of the n-point correlation functions (see companion paper [34]) which can be recovered via the following recursion formula:"

The second main result, Eq. (16), is not derived in the Letter; its validity is delegated entirely to [34], an unpublished companion by the same three authors. The recursion (17) that defines the ordered cumulants D, and therefore every K_n, the even/odd structural statements, and the spin-system conclusions, is likewise sourced only to [34]. No independent derivation, base case, or external verification is supplied in this manuscript, so the load-bearing step of the claimed recursive expansion reduces to a self-citation that is not itself established in the text.
self definitional [Eq. (17) and surrounding text]
"D(τk1,sn−k1) = ˙D(τk1,sn−k1) − Σ_{l=0}^k Σ_{r=0}^{n-k} D(τl1,sr1)D(τkl+1,sn−kr+1) (17) where ˙D(τk1,sn−k1) = D(τk1,sn−k1)(δτ1,t + δs1,t), (18) and we have taken the convention ∫0^t dτ δτ,t f(τ) = f(t)."

As printed, Eq. (17) is not a constructive recursion: the unknown D(τk1,sn−k1) appears on both sides, and the diagonal term l=k, r=n−k contributes D(τk1,sn−k1)D(∅,∅). No value for D(∅,∅) or any other base/initial condition is stated. The formula is therefore an implicit, self-referential equation rather than a rule that 'recovers' the ordered cumulants. Since Eq. (16) is expressed entirely through these D functions, the claimed systematic computation of K(t) is not determined by the manuscript unless an unstated convention is imported.

full rationale

The independently checkable core is the closed-form expression for K in terms of a generator, Eq. (6), which is derived in the end matter from the Hayden-Sorce projection and Haar averages; no fitted parameter is hidden there, and the spin applications are checked against known solvable models such as the dephasing spin-boson model. I do not find fitted-input circularity or renaming of a known result. However, the recursive perturbation expansion, which is the advertised second main result, is not self-contained: Eq. (16) and the ordered-cumulant recursion (17) are deferred to a companion paper by the same authors [34], and Eq. (17) as printed is self-referential with no base case. The central structural claims about even and odd orders and the spin-model conclusions depend on this unverified cascade. That warrants a score of 4 rather than 0-2; it is not a 6-10 because the core Proposition (6) and the application checks have independent content, and the deficiency is best described as a load-bearing self-citation plus an ill-defined recursion, not a result forced by definition or by fitting.

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

No free parameters are fitted to data; the only inputs are the system-bath Hamiltonian and the bath state. The framework rests on five explicit assumptions listed above, the two most fragile being the Haar-averaged minimal-dissipation choice and the imported cumulant recursion from [34]. No new physical entities are postulated.

assumptions (5)

domain assumption The reduced dynamics possesses a time-local generator L_t (TCL master equation) for all times and coupling strengths.
Throughout the Letter, K(t) is defined from L_t; the existence of L_t is assumed, standard in open quantum systems, refs [1,13].
domain assumption The initial state factorizes as ρ_SE(0) = ρ_S ⊗ ρ_E.
Stated before Eq. (10); restricts to factorized initial conditions, excluding correlated initial states.
ad hoc to paper The minimal dissipation principle with the Haar-averaged scalar product (2) yields the physically correct effective Hamiltonian K(t).
This choice fixes the otherwise ambiguous decomposition in Eq. (1); the paper adopts it from [19] and cites experimental support [20], but it is not derived.
ad hoc to paper The bath ordered cumulants D satisfy the recursion (17) and yield the perturbation series (16).
This is imported from the companion paper [34] and is not proven in the Letter.
domain assumption The interaction Hamiltonian is factorized, H_I = A⊗B, with extensions to sums being 'straightforward'.
Assumed in Section 'Systematic perturbation expansion'; limits generality to factorized couplings.

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Pith. "Pith review of Unveiling coherent dynamics in non-Markovian open quantum systems: exact expression and recursive perturbation expansion." pith.science (2026). https://pith.science/paper/W4D5XOJP

@misc{pith2026250604097,
  author       = {Pith},
  title        = {Pith review of: Unveiling coherent dynamics in non-Markovian open quantum systems: exact expression and recursive perturbation expansion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4D5XOJP}},
  note         = {Machine review of arXiv:2506.04097}
}

read the original abstract

We introduce a systematic framework to derive the effective Hamiltonian governing the coherent dynamics of non-Markovian open quantum systems. By applying the minimal dissipation principle, we uniquely isolate the coherent contribution to the time-local generator of the reduced dynamics. We derive a general expression for the effective Hamiltonian and develop a recursive perturbative expansion that expresses it in terms of system-bath interaction terms and bath correlation functions. This expansion provides a systematic tool for analyzing energy renormalization effects across different coupling regimes. Applying our framework to paradigmatic spin systems, we reveal how environmental correlations influence energy shifts and eigenbasis rotations, offering new insights into strong-coupling effects and non-Markovian quantum thermodynamics.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Finite-Bath Open Quantum Systems: Exact Dynamics
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quant-ph 2025-06 accept novelty 5.0 of 10

In a solvable dephasing model, local and global quantum thermodynamics give very different heat, work, and entropy production, with the difference persisting at weak coupling.

Reference graph

Works this paper leans on

49 extracted references · 18 canonical work pages · cited by 2 Pith papers

[43]

Quantum Reservoir Computing (QuReCo)

due to numerical evidence. Commonly studied scenarios are those in which the dynamics is characterized by two-point correlation func- tions, as in the case of linearly coupled, zero-mean Gaus- sian baths. The present analysis clarifies why such mod- els exhibit simpler renormalization effects: in the case of pure dephasing, renormalization is entirely abs...

work page 2022
[34]

Mascherpa, A

F. Mascherpa, A. Smirne, A. D. Somoza, P. Fernández- Acebal, S.Donadi, D.Tamascelli, S.F.Huelga,andM.B. Plenio, Phys. Rev. A101, 052108 (2020)

work page 2020
[1]

Breuer and F

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2007)

2007
[2]

Vacchini, Open Quantum Systems (Springer Cham, 2024)

B. Vacchini, Open Quantum Systems (Springer Cham, 2024)

2024
[3]

H. A. Bethe, Phys. Rev.72, 339 (1947)

1947
[4]

W. E. Lamb and R. C. Retherford, Phys. Rev.72, 241 (1947)

1947
[5]

S. H. Autler and C. H. Townes, Phys. Rev.100, 703 (1955)

work page 1955
[6]

Hossein-Nejad, E

H. Hossein-Nejad, E. J. O’Reilly, and A. Olaya-Castro, New Journal of Physics17, 075014 (2015)

work page 2015

Show all 49 references

[7]

A. H. A. Malavazi and F. Brito, Entropy 24, 10.3390/e24111645 (2022)

2022 doi
[8]

Hernandes Alves Malavazi, Ph

A. Hernandes Alves Malavazi, Ph. D. Thesis (2022)

2022
[9]

Nicacio and R

F. Nicacio and R. N. P. Maia, Phys. Rev. A108, 022209 (2023)

2023
[10]

L. R. T. Neves and F. Brito, Phys. Rev. A108, 042209 (2023)

2023
[11]

L. R. Neves and F. Brito, arXiv preprint arXiv:2408.04111 (2024)

2024
[12]

Tiwari, B

D. Tiwari, B. Bose, and S. Banerjee, The Journal of Chemical Physics162, 114104 (2025)

2025
[13]

Breuer, B

H.-P. Breuer, B. Kappler, and F. Petruccione, Annals of Physics 291, 36 (2001)

2001
[14]

M. J. W. Hall, J. D. Cresser, L. Li, and E. Andersson, Phys. Rev. A89, 042120 (2014)

2014
[15]

Breuer, J

H.-P. Breuer, J. Phys. B45, 154001 (2012)

2012
[16]

Rivas, S

A. Rivas, S. F. Huelga, and M. B. Plenio, Reports on Progress in Physics77, 094001 (2014)

2014
[17]

Breuer, E.-M

H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys.88, 021002 (2016)

2016
[18]

Chruściński, Physics Reports992, 1 (2022), dynami- cal maps beyond Markovian regime

D. Chruściński, Physics Reports992, 1 (2022), dynami- cal maps beyond Markovian regime

2022
[19]

Sorce and P

J. Sorce and P. M. Hayden, J. Phys. A: Math. Theor. 10.1088/1751-8121/ac65c2 (2022)

2022 doi
[20]

Colla, F

A. Colla, F. Hasse, D. Palani, T. Schaetz, H.-P. Breuer, and U. Warring, Nature Communications 16, 2502 (2025)

2025
[21]

Colla and H.-P

A. Colla and H.-P. Breuer, Phys. Rev. A105, 052216 (2022)

2022
[22]

I. A. Picatoste, A. Colla, and H.-P. Breuer, Phys. Rev. Res. 6, 013258 (2024)

2024
[23]

Huang and W.-M

W.-M. Huang and W.-M. Zhang, Phys. Rev. Res. 4, 023141 (2022)

2022
[24]

Colla and H.-P

A. Colla and H.-P. Breuer, Quantum Science and Tech- nology 10, 015047 (2024)

2024
[25]

Smirne and B

A. Smirne and B. Vacchini, Phys. Rev. A82, 022110 (2010)

2010
[26]

B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D45, 2843 (1992)

1992
[27]

Ferialdi, Phys

L. Ferialdi, Phys. Rev. Lett.116, 120402 (2016)

2016
[28]

Łuczka, Physica A: Statistical Mechanics and its Ap- plications 167, 919 (1990)

J. Łuczka, Physica A: Statistical Mechanics and its Ap- plications 167, 919 (1990)

1990
[29]

R. Doll, D. Zueco, M. Wubs, S. Kohler, and P. Hänggi, Chemical Physics 347, 243 (2008), ultrafast Photoin- duced Processes in Polyatomic Molecules

2008
[30]

faithfully

or pseudomodes [31, 32], or experimentally through quantum process tomography. In well-behaved regimes, a time-convolutionless (TCL) perturbation expansion for weak-coupling parameters may instead be sufficient for an appropriate description of the system dynamics. In all thes...
[31]

Tamascelli, A

D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett.120, 030402 (2018)

2018
[32]

Tanimura and R

Y. Tanimura and R. Kubo, Journal of the Physical Soci- ety of Japan58, 101 (1989)

1989
[33]

Colla, N

A. Colla, N. Neubrand, and H.-P. Breuer, New Journal of Physics24, 123005 (2022)

2022
[35]

Diósi, Physica A: Statistical Mechanics and its Appli- cations 199, 517 (1993)

L. Diósi, Physica A: Statistical Mechanics and its Appli- cations 199, 517 (1993)

1993
[36]

Colla, H.-P

A. Colla, H.-P. Breuer, and G. Gasbarri, Recursive per- turbation approach to time-convolutionless master equa- tions: Explicit construction of generalized Lindblad gen- erators for arbitrary open systems
[37]

Chou, Z.-b

K.-c. Chou, Z.-b. Su, B.-l. Hao, and L. Yu, Phys. Rept. 118, 1 (1985)

1985
[38]

Diósi, Foundations of Physics20, 63 (1990)

L. Diósi, Foundations of Physics20, 63 (1990)

1990
[39]

Van Kampen, Physica74, 215 (1974)

N. Van Kampen, Physica74, 215 (1974)

1974
[40]

Gasbarri and L

G. Gasbarri and L. Ferialdi, Phys. Rev. A97, 022114 (2018)

2018
[41]

Seegebrecht and T

A. Seegebrecht and T. Schilling, Journal of Statistical Physics 191, 34 (2024)

2024
[42]

Van Kampen, Physica74, 239 (1974)

N. Van Kampen, Physica74, 239 (1974)

1974
[44]

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. Fisher, A.Garg,andW.Zwerger,ReviewsofModernPhysics 59, 1 (1987)

1987
[45]

Gatto, A

S. Gatto, A. Colla, H.-P. Breuer, and M. Thoss, Phys. Rev. A110, 032210 (2024)

2024
[46]

Kubo, Journal of the Physical Society of Japan12, 570 (1957)

R. Kubo, Journal of the Physical Society of Japan12, 570 (1957)

1957
[47]

M. A. Nielsen, Physics Letters A303, 249 (2002)

2002
[48]

Gessner and H.-P

M. Gessner and H.-P. Breuer, Phys. Rev. E87, 042128 (2013)

2013
[49]

Gorini, A

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys.17, 821 (1976). END MATTER K in terms of generator fidelity The scalar product defined in [19] for two generators reads ⟨L1,L2⟩ = ∫ dϕ ∫ dψ⟨ϕ|L 1 [|ψ⟩⟨ψ|]L2 [|ψ⟩⟨ψ|]|ϕ⟩ , (A.26) where ψ and ϕ are Haar-random gene...

1976

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [43]

Quantum Reservoir Computing (QuReCo)

due to numerical evidence. Commonly studied scenarios are those in which the dynamics is characterized by two-point correlation func- tions, as in the case of linearly coupled, zero-mean Gaus- sian baths. The present analysis clarifies why such mod- els exhibit simpler renormalization effects: in the case of pure dephasing, renormalization is entirely abs...

work page 2022

[2] [34]

Mascherpa, A

F. Mascherpa, A. Smirne, A. D. Somoza, P. Fernández- Acebal, S.Donadi, D.Tamascelli, S.F.Huelga,andM.B. Plenio, Phys. Rev. A101, 052108 (2020)

work page 2020

[3] [1]

Breuer and F

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2007)

2007

[4] [2]

Vacchini, Open Quantum Systems (Springer Cham, 2024)

B. Vacchini, Open Quantum Systems (Springer Cham, 2024)

2024

[5] [3]

H. A. Bethe, Phys. Rev.72, 339 (1947)

1947

[6] [4]

W. E. Lamb and R. C. Retherford, Phys. Rev.72, 241 (1947)

1947

[7] [5]

S. H. Autler and C. H. Townes, Phys. Rev.100, 703 (1955)

work page 1955

[8] [6]

Hossein-Nejad, E

H. Hossein-Nejad, E. J. O’Reilly, and A. Olaya-Castro, New Journal of Physics17, 075014 (2015)

work page 2015