Pith. sign in

REVIEW 1 major objections 5 minor 34 references

Quantum Markovian master equations: resonance theory shows validity for all time scales

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that a Markovian master equation approximates exact open-system dynamics for all times, with uniform error of order the squared coupling strength.

desk verdict Uniform-in-time Davies approximation (1.24) is a genuine advance and looks correct; Result 3 is not established because it reuses the very lemma the paper admits is broken. read the letter →

arxiv 1908.01984 v2 pith:BLV4IEOW submitted 2019-08-06 quant-ph math-phmath.MP

Marco Merkli This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81S2281Q12 PACS 03.65.Yz
keywords openquantumsystemsMarkovianmasterequationDaviesgeneratorweakcouplinglimitresonancesspectraldeformationcompletepositivityasymptoticallyexactsemigroup
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 removes the traditional time restriction from the Markovian master equation for a quantum system weakly coupled to a heat bath. For Hamiltonians of the form $H = H_S + H_R + \lambda G\otimes \varphi(g)$ with an analytic coupling form factor, it proves that the exact reduced dynamics $V_t$ is approximated by the Davies semigroup $e^{t(L_S+\lambda^2 K)}$ with error at most $C\lambda^2$ uniformly in $t\ge 0$, for fixed small $\lambda$. Previously only the regime $\lambda^2 t \le \text{constant}$ was covered. Under the Fermi Golden Rule condition, the paper also constructs a renormalized completely positive semigroup whose generator is analytic in $\lambda$ and whose approximation error decays to zero as $t\to\infty$, making it asymptotically exact; a fully proven version is given for the populations. A reader should care because master equations are the standard workhorse for open quantum systems, and the result says that their use can be trusted at arbitrarily long times, not just on the $\lambda^2 t$-bounded transient window.

What carries the argument

The engine is spectral deformation, or 'translation analyticity,' imposed on the reservoir form factor in equation (2.22): after a complex shift $\theta$ of the frequency variable, the function $g_\beta(u,\Sigma)$ must extend analytically to $0<\operatorname{Im}\theta<\theta_0$. This makes the reservoir correlation function decay exponentially and turns the Liouville operator $L_\lambda$ into a non-self-adjoint operator whose spectrum has isolated resonances. A resolvent-contour deformation then produces the resonance expansion (2.40), in which each resonance contributes $e^{it\epsilon_e^{(s)}(\lambda)}$ times a spectral projection, plus an exponentially decaying remainder. The quadratic corrections $\epsilon_e^{(s)}(\lambda)=e+\lambda^2 a_e^{(s)}+O(\lambda^4)$ are eigenvalues of the level-shift operators $\Lambda_e$ in (2.26); the Davies generator $K$ is assembled from these $a_e^{(s)}$ and the associated projections. Complete positivity of the approximate dynamics is shown by writing the approximating group as the composition of the free system evolution and a weak-coupling limit of completely positive maps.

What would settle it

Pick a two-level system, $\beta>0$, and an analytic form factor satisfying (2.22), e.g. $g(|\mathbf{k}|,\Sigma)=|\mathbf{k}|^{1/2}e^{-|\mathbf{k}|}g_1(\Sigma)$. Numerically simulate the exact reduced dynamics on a finely discretized reservoir to times far beyond $1/\lambda^2$ at fixed small $\lambda$, and check whether $\sup_{t\ge0}\|V_t-e^{t(L_S+\lambda^2 K)}\|/\lambda^2$ stays bounded; if it grows without bound as $\lambda^2 t\to\infty$, the central inequality (1.24) is false.

Watch

Extended reading notes

Core claim

The central claim is stated as inequality (1.24): for $|\lambda|\le \lambda_0$, $\sup_{t\ge0}\|V_t - e^{t(L_S+\lambda^2 K)}\| \le C\lambda^2$, where $V_t$ is the reduced dynamical map of a finite-level system coupled to a bosonic reservoir, $L_S=-i[H_S,\cdot]$, and $K$ is the Davies generator built from second-order level-shift data. This upgrades the van Hove weak-coupling limit, which only controls $\lambda^2 t \le a$, to a uniform-in-time bound. The proof runs through a resonance expansion of the coupled Liouvillian: the deformed spectrum consists of resonances $\epsilon_j(\lambda)=E_j+\lambda^2\epsilon_j^{(2)}+O(\lambda^4)$, whose imaginary parts, under the Fermi Golden Rule condition $\gamma_{\rm FGR}>0$, give a decay gap used to control remainders. A companion resonance expansion, (1.15), keeps the exact final state $\rho_{S,\beta,\lambda}$ and gives an error that also decays exponentially in time. As a second result, a renormalized generator $M(\lambda)$, analytic in $\lambda$ and chosen so that the coupled equilibrium state is invariant, yields $e^{tM(\lambda)}$ with error $O((|\lambda|+\lambda^2 t)e^{-\lambda^2\gamma_{\rm FGR}t})$, hence the approximation becomes exact in the long-time limit. The paper notes that a prior outline of this asymptotic-exactness result had a gap, and gives the fully proven statement for the populations of the system in (1.32).

Load-bearing premise

The load-bearing premise is the analytic deformation condition on the reservoir coupling: the form factor $g_\beta(u,\Sigma)$ must extend analytically under a complex frequency shift, which forces exponential decay of reservoir correlations; for merely smooth, polynomially decaying form factors the theorems are not proven, and the semigroup results additionally need a strictly positive second-order decay rate.

Editorial extensions

If this is right

  • If (1.24) is correct, applications of the Davies master equation, from decoherence calculations to quantum thermometry, remain quantitatively reliable at times much longer than $1/\lambda^2$, with error still $O(\lambda^2)$ rather than growing with time.
  • The old weak-coupling limit (1.13) becomes a special case: taking $\lambda\to0$ with $\lambda^2 t$ fixed recovers the previous statement, so the new bound is strictly stronger.
  • The asymptotically exact semigroup $e^{tM(\lambda)}$ predicts the correct stationary state $\rho_{S,\beta,\lambda}$ of the coupled system, correcting the $O(\lambda^2)$ final-state error inherent in the Davies approximation.
  • For populations, (1.32) provides a Pauli-type master equation valid for all times with error $O((|\lambda|+\lambda^4 t)e^{-\lambda^2 t(\gamma_{\rm FGR}+O(\lambda^2))})$, so rate-equation descriptions of level occupation are justifiable beyond the weak-coupling window.

Reading between the lines

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

  • A natural next step is to prove the same uniform $O(\lambda^2)$ bound for reservoirs whose correlations decay only polynomially; the paper points to a different commutator technique for that case but does not execute it. If it works, the all-time Markovian regime would cover a much larger class of physical environments.
  • Because the resonance expansion (2.40) is stated for entangled initial states in the equilibrium folium, one could extend Result 2 to those states, yielding all-time Markovian approximations for initially correlated system-bath preparations.
  • The renormalization producing $M(\lambda)$ suggests a hierarchy: truncating the generator at higher orders in $\lambda$ should give completely positive semigroups accurate on successively longer time scales, with the Davies generator as the lowest-order member.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This paper studies the reduced dynamics of an N-level system coupled to a bosonic thermal reservoir through a Hamiltonian of the form H = H_S + H_R + λ G ⊗ φ(g) in the continuous-mode limit. The author develops a resonance (spectral deformation) expansion of the coupled Liouvillian and derives three results. Result 1 (1.15) is a resonance expansion of the dynamical map V_t with remainder O(λ² e^{−γ(λ)t}) for disentangled initial states. Result 2 (1.24) removes the usual λ² t ≤ constant restriction: assuming the Fermi Golden Rule condition (1.22), there is λ₀ such that for |λ| ≤ λ₀ and all t ≥ 0, ‖V_t − e^{t(L_S+λ²K)}‖ ≤ Cλ², where e^{t(L_S+λ²K)} is the Davies CPT semigroup. Result 3 (1.29) claims an asymptotically exact CPT semigroup e^{tM(λ)} with generator analytic in λ and error (|λ|+λ²t)e^{−λ²γ_FGR t}, together with a separate population result (1.32). The proofs use translation analyticity of the form factor (condition (A), Eq. (2.22)) and the level shift operators of Eq. (2.26). The manuscript explicitly states in Section 2.1 that the original proof of the main result in [18] has a gap and that the erratum restores it only for populations; nevertheless, the proof of the coherence part of Result 3 in Section 3.2.2 is delegated to Lemma 3.4 of [18].

Significance. Result 2, if correct, is a significant improvement: it converts the weak-coupling Markov approximation into a uniform-in-time approximation with O(λ²) error for fixed small λ, while preserving complete positivity. The proof strategy is conceptually clean: the error estimate (3.6) is explicit, and the argument in Section 3.1.1 that limits of CP maps are CP is persuasive. The paper is also honest about the scope of its hypotheses, explicitly assuming translation analyticity and the Fermi Golden Rule condition, and it clearly flags the gap in [18]. However, the advertised 'asymptotically exact CPT semigroup' claim for coherences (Result 3) is not supported by the present manuscript, because its proof rests on an unproved lemma from a paper whose gap the author acknowledges. The population version (1.32) appears to be proved independently and is not affected by this gap. Thus the core Result 2 is strong and likely correct, but the full set of claims as stated requires additional work.

major comments (1)
  1. [Section 3.2.2, Eq. (3.27) and footnote 12] The full statement of Result 3, including coherences, is not established. The proof requires the equality ~Ω₀ = D′Ω_{SR,β,λ} with D′ in the commutant and D′ = 1 + O(λ), and then uses this to replace D′ by 1 in the passage from (3.30) to (3.32). This identity is delegated to Lemma 3.4 of [18], but Section 2.1 states that [18] has a gap in its main result and that the erratum restores the result only for populations. No proof of Lemma 3.4 appears in the present manuscript, and the footnote's assertion that the technicalities are not too severe is not a substitute. Consequently, the coherence part of (1.29) is unsupported, while the population bound (1.32) does not require the full D′ identity and is not affected. The authors should either supply a proof of Lemma 3.4 within this manuscript or restrict Result 3 to the population statement.
minor comments (5)
  1. [Title] The title on the arXiv metadata, 'Quantum Markovian master equations: resonance theory shows validity for all time scales', differs from the title on the first page of the manuscript, 'Quantum Markovian master equations: Resonance theory overcomes the weak coupling regime'. Please harmonize them.
  2. [Eq. (2.22)] In the definition of g_β(u,Σ), the factors √u and |u|^{1/2} are redundant and the expression as written is not real for u < 0. Please clarify whether the intended prefactor is √|u|/(1−e^{−βu}) and whether the extra |u|^{1/2} factor is a typographical artifact.
  3. [Eqs. (2.32) and (2.37)] The symbol Q^{(s)}_e is used both for spectral projections on the doubled system Hilbert space in (2.32) and for maps on system observables in (2.37). This dual use is confusing; please introduce separate notation or explicitly state the identification.
  4. [Section 2.1] The sentence describing [18] as a two-page outline of a proof of Results 1 and 2 is inconsistent with the following sentence, which says that [18] focuses on the asymptotically exact Markovian approximation that is part of Result 3. Please rephrase for clarity.
  5. [Eq. (1.29)] The statement of (1.29) should explicitly say that the constant C is independent of λ and t; this is clear from the proof but should be part of the theorem statement.

Circularity Check

1 steps flagged · score 6.0 of 10

Result 3's coherence bound (1.29) is delegated to Lemma 3.4 of the authors' [18], whose main proof the paper admits has a gap and whose erratum restores only populations; the asymptotically exact CPT semigroup for coherences is therefore carried by an unverified self-citation.

  1. self citation load bearing [Section 3.2.2, eq. (3.27) and footnote; Section 2.1]
    "Some care has to be taken here as D′ is not a bounded operator, but the technicalities of this difficulty are not too severe to overcome, see Lemma 3.4 of [18]. ... However, there is a gap in the proof of the main result in [18]. This is explained in an erratum to [18], where it is also announced that we can still show the result in its full strength for the dynamics of the populations of the system (but not the coherences)."

    The derivation of (1.29) for coherences needs the exact vector identity (3.27), ~Ω0 = D′Ω_SR,β,λ with D′ in the commutant; this identity is neither proved here nor supplied by an independent argument. It is assigned to Lemma 3.4 of the authors' own [18], while Section 2.1 concedes that [18]'s main proof has a gap and that the erratum restores the result only for populations, not coherences. The subsequent step before (3.32), 'we replace D′ by 1 (see (3.27))', is exactly where the unsupported identity is used, so the claimed asymptotically exact CPT semigroup for coherences is imported from a self-citation whose needed content is admittedly unverified. The population statement (1.32), proved in this paper, matches the erratum's restricted scope and does not repair the coherence gap.

full rationale

Results 1 and 2 are derived in this paper from the resonance expansion (2.40) and the explicit level-shift calculation; no fitted data or parameter renamed as prediction is involved, and the Davies generator K is computed standardly and yields genuinely time-uniform O(λ²) bounds. The construction of H̃_S(λ) via (1.28) is a deliberate choice ensuring the semigroup's invariant state is the exact coupled equilibrium; while that makes the asymptotic final-state equivalence true by design, the intermediate-time approximation and the CP property are still nontrivial, so this is not circular. The circularity concern is confined to the coherence part of Result 3: the proof of (1.29) leans on Lemma 3.4 of the authors' [18] via identity (3.27), and the present paper itself concedes that [18] has a gap and that the erratum recovers only populations. Since the needed lemma is neither proved here nor independently verified, that subclaim is carried by an unverified self-citation. This is a partial circularity: it affects the asymptotically exact CPT semigroup for coherences, while the population version (1.32) is proved in the paper.

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

No free parameters fitted to data: the paper is a theorem-proof contribution; λ, β and the form factor g are model inputs fixed before the theorems. The error constants C are existential bounds whose values are not claimed. The renormalized Hamiltonian H̃_S(λ), eq. (3.14), and the semigroup generators M(λ), M_d(λ) are mathematical constructions defined from the physical equilibrium state ρ_S,β,λ and the resonance data; they introduce no new physical entities. The load-bearing assumptions are the six axioms listed; of these, translation analyticity (A), the Fermi Golden Rule Condition, and level-shift diagonalizability are stated hypotheses restricting the model class, while the resonance-strip theorem is inherited from prior work [16, 4, 26].

assumptions (6)
  • domain assumption Translation analyticity of the form factor g_β, condition (A), eq. (2.22): the map θ → T_θ g_β extends analytically from R to L²(R × S²) in 0 < Im θ < θ0, continuous at Im θ → 0+.
    This hypothesis makes reservoir correlations decay exponentially and enables the spectral deformation U_θ behind the resonance expansion (2.27). All three results are proved only under this condition; weaker (real-differentiable) form factors are explicitly deferred to the Mourre-theory technique of [19].
  • domain assumption Fermi Golden Rule Condition, eq. (1.22): γ_FGR = min_j Im ε_j^(2) > 0, where ε_j^(2) is the second-order resonance correction.
    Results 2 and 3 are stated under this condition (Sections 1.2, 1.3). It supplies the positive λ² decay gap in the bound (3.6) and in the remainders (1.24), (1.29), (1.32); without it the paper proves no decay rate and no uniform estimate.
  • domain assumption Diagonalizability and simplicity of the level shift operators Λ_e, eq. (2.32): Λ_e = Σ_s a_e^(s) Q_e^(s) with rank-one spectral projections; the resonances ε_e^(s) are distinct for λ ≠ 0.
    Needed for the rank-one expansion (2.35) of resonance projections and for the spectral decompositions of W_t and M(λ). Stated as a generic condition in Section 2.4; symmetric or degenerate systems are excluded.
  • domain assumption Resonance-strip structure of the deformed Liouvillian L_{λ,θ}: in {0 ≤ Im z < θ0/2} its spectrum consists of θ-independent eigenvalues ε_e^(s)(λ) (analytic in λ, expanding as e + λ² a_e^(s) + O(λ⁴)); all other spectrum lies in {Im z > 3θ0/4}.
    Imported from prior resonance theory [16, 4, 26], Section 2.4; the present paper gives a sketch ('analytic perturbation theory and the fact that L_{0,θ} = L_0 + θN... show') but does not prove this theorem.
  • domain assumption Araki-Woods representation, purification of the reservoir equilibrium state, and cyclicity/separability of the coupled equilibrium vector Ω_SR,β,λ.
    Sets up the Hilbert-space reformulation (Sections 2.2, 2.3) and the representation of arbitrary initial vectors as B'Ω_SR,β,λ in (2.18); the infinite-dimensional cyclicity/separability is asserted by analogy with the finite-dimension argument (2.11).
  • standard math Standard spectral theory and analytic perturbation theory: Riesz projections, complex deformation, Kato perturbation expansions.
    Background tooling used throughout Sections 2.4 and 3; not specific to the model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Markovian master equations: resonance theory shows validity for all time scales." pith.science (2026). https://pith.science/paper/BLV4IEOW

@misc{pith2026190801984,
  author       = {Pith},
  title        = {Pith review of: Quantum Markovian master equations: resonance theory shows validity for all time scales},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLV4IEOW}},
  note         = {Machine review of arXiv:1908.01984}
}
abstract

Quantum systems coupled to environments exhibit intricate dynamics. The master equation gives a Markov approximation of the dynamics, allowing for analytic and numerical treatments. It is ubiquitous in theoretical and applied quantum sciences. The accuracy of the master equation approximation was so far proven for small values of the system-environment interaction coupling strength $\lambda$, under the additional constraint that time $t$ must not exceed an upper bound, $\lambda^2 t\le$ constant. Here, we show that the Markov approximation is valid for fixed small coupling strength and for all times. We also construct a new approximate Markovian dynamics -- a completely positive, trace preserving semigroup -- which is asymptotically in time exact, to all orders in the coupling.

Figures

Figures reproduced from arXiv: 1908.01984 by the authors.

Figure 1
Figure 1. Fig.1: The eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [18]

    K¨ onenberg, M

    M. K¨ onenberg, M. Merkli: Completely positive dynamical semigroups and quantum resonance theory, Lett. Math. Phys. 107, Issue 7, 1215-1233 (2017)

    work page 2017

  2. [1]

    Alicki, K

    R. Alicki, K. Lendi: Quantum Dynamical Semigroups and Applications, Lect. Notes Phys. 717, Springer Verlag 2007

    work page 2007

  3. [2]

    Alicki: The Markov Master Equations and the Fermi Golden Rule , International Journal of Theoretical Physics, 16, No.5, 351-355 (1977)

    R. Alicki: The Markov Master Equations and the Fermi Golden Rule , International Journal of Theoretical Physics, 16, No.5, 351-355 (1977)

    work page 1977

  4. [3]

    Araki, E.J

    H. Araki, E.J. Woods: Representation of the canonical commutation relations describ- ing a nonrelativistic infinite free bose gas , J. Math. Phys. 4, 637-662 (1963)

    work page 1963

  5. [4]

    V. Bach, J. Fr¨ ohlich, I.M. Sigal:Return to equilibrium, J. Math. Phys. 41, 3985-4060 (2000) 13Use that J(Z⊗ 1 lS)J commutes with X⊗ 1 lS and JΩS,β = ΩS,β and J(Z⊗ 1 lS)ΩS,β =e−βHS/2(Z⊗ 1 lS)ΩS,β. 28

    work page 2000

  6. [5]

    Benatti, R

    F. Benatti, R. Floreanini: Open Quantum Dynamics: Complete Positivity and En- tanglement, Int. J. Mod. Phys. B19 3063 (2005)

    work page 2005

  7. [6]

    Benatti, M

    F. Benatti, M. Fannes, R. Floreanini, D. Petritis (Eds): Quantum Information, Com- putation and Cryptography, Springer Lecture Notes in Physics 808, 2010

    work page 2010

  8. [7]

    Breuer, F

    H.-P. Breuer, F. Petruccione: The theory of open quantum systems, Oxford Univer- sity Press 2002

    work page 2002

Show all 34 references
  1. [8]

    Chru´ sci´ nski, S

    D. Chru´ sci´ nski, S. Pascazio:A Brief History of the GKLS Equation , Open Systems & Information Dynamics, Vol. 24, No. 03, 1740001 (2017)

    2017

  2. [9]

    Davies: Markovian Master Equations , Commun

    E.B. Davies: Markovian Master Equations , Commun. Math. Phys. 39, 9-110 (1974)

    1974

  3. [10]

    Davies: Markovian Master Equations, II , Math

    E.B. Davies: Markovian Master Equations, II , Math. Ann. 219, 147-158 (1976)

    1976

  4. [11]

    D¨ umke, H

    R. D¨ umke, H. Spohn: The Proper Form of the Generator in the Weak Coupling Limit, Z. Physik B, 34, 419-422 (1979)

    1979

  5. [12]

    Facchi, S

    P. Facchi, S. Pascazio: Deviations from exponential law and Van Hove’s λ2t limit, Physica A 271 133-146 (1999)

    1999

  6. [13]

    Facchi, M

    P. Facchi, M. Ligab` o, K. Yuasa:On the Derivation of the GKLS Equation for Weakly Coupled Systems, Open Systems & Information Dynamics 24, No.4

  7. [14]

    Gorini, A

    V. Gorini, A. Kossakowski, E.C.G. Sudarshan: Completely positive dynamical semi- groups of N-level systems , J. Math. Phys. 17, 821 (1976)

    1976

  8. [15]

    Haroche, J.-M

    S. Haroche, J.-M. Raimond: Exploring the Quantum. Atoms, Cavities and Photons, Oxford University Press 2006

    2006

  9. [16]

    Jak˘ si´ c, C.-A

    V. Jak˘ si´ c, C.-A. Pillet:On a model for quantum friction. II. Fermi’s golden rule and dynamics at positive temperature, Comm. Math. Phys. 176, no. 3, 619-644 (1996)

    1996

  10. [17]

    Jak˘ si´ c, C.-A

    V. Jak˘ si´ c, C.-A. Pillet:From resonances to master equations, Ann. Inst. H. Poincar´ e 67, no.4, 425-445 (1997)

    1997

  11. [19]

    K¨ onenberg, M

    M. K¨ onenberg, M. Merkli: On the irreversible dynamics emerging from quantum resonances, J. Math. Phys. 57, 033302 (2016)

    2016

  12. [20]

    Lindblad: On the generators of quantum dynamical semigroups , Comm

    G. Lindblad: On the generators of quantum dynamical semigroups , Comm. Math. Phys. 48, No.2, 119-130 (1976)

    1976

  13. [21]

    Merkli: The ideal quantum gas , Lecture Notes in Mathematics, 1880, 183-233 (2006) 29

    M. Merkli: The ideal quantum gas , Lecture Notes in Mathematics, 1880, 183-233 (2006) 29

    2006

  14. [22]

    Merkli, G.P

    M. Merkli, G.P. Berman, R.T. Sayre, S. Gnanakaran, M. K¨ onenberg, A.I. Nesterov, H. Song: Dynamics of a Chlorophyll Dimer in Collective and Local Thermal Envi- ronments, J. Math. Chem. 54(4), 866-917 (2016)

    2016

  15. [23]

    Mohseni, Y

    M. Mohseni, Y. Omar, G.S. Engel, and M.B. Plenio (Eds): Quantum Effects in Biology, Cambridge University Press, 2014

    2014

  16. [24]

    Merkli, G.P

    M. Merkli, G.P. Berman, F. Borgonovi, K. Gebresellasie: Evolution of Entanglement of Two Qubits Interacting through Local and Collective Environments , Quantum In- formation & Computation 11, no. 5 & 6 , 390-419 (2011)

    2011

  17. [25]

    Merkli, I.M

    M. Merkli, I.M. Sigal, G.P. Berman: Decoherence and Thermalization, Phys. Rev. Lett. 98, 130401 (2007)

    2007

  18. [26]

    Merkli, I.M

    M. Merkli, I.M. Sigal, G.P. Berman: Resonance Theory of Decoherence and Ther- malization, Annals of Physics 323, 373-412 (2008)

    2008

  19. [27]

    Rivas: Refined weak-coupling limit: Coherence, entanglement, and non- Markovianity, Phys

    `A. Rivas: Refined weak-coupling limit: Coherence, entanglement, and non- Markovianity, Phys. Rev. A 95, 042104, 10pp (2017)

    2017

  20. [28]

    Rivas, A.D.K

    `A. Rivas, A.D.K. Plato, S. F. Huelga, M. B. Plenio: Markovian master equations: a critical study, New J. Phys. 12 113032, 38pp, (2010)

    2010

  21. [29]

    Rivas, S

    `A. Rivas, S. F. Huelga, M.B. Plenio: Quantum non-Markovianity: characterization, quantification and detection, Reports on Progress in Physics 77, no.9, 094001, 26pp (2014)

    2014

  22. [30]

    Schaller, T

    G. Schaller, T. Brandes: Preservation of positivity by dynamical coarse graining , Phys. Rev. A 78, 022106, 17pp (2008)

    2008

  23. [31]

    Tasaki, K

    S. Tasaki, K. Yuasa, P. Facchi, G. Kimura, H. Nakazato, I. Ohba and S. Pascazio:On the Assumption of Initial Factorization in the Master Equation for Weakly Coupled Systems I: General Framework, Annals of Physics 322 631-656 (2007)

    2007

  24. [32]

    Van Hove: Quantum-mechanical perturbations giving rise to a statistical transport equation, Physica 21, Issue 1-5, 517-540 (1955)

    L. Van Hove: Quantum-mechanical perturbations giving rise to a statistical transport equation, Physica 21, Issue 1-5, 517-540 (1955)

    1955

  25. [33]

    Van Kampen: Stochastic behavior of quantum systems , Chapter XVII of Stochastic Processes in Physics and Chemistry, North-Holland Personal Library, 3rd edition, 422-456, 2007

    N.G. Van Kampen: Stochastic behavior of quantum systems , Chapter XVII of Stochastic Processes in Physics and Chemistry, North-Holland Personal Library, 3rd edition, 422-456, 2007

    2007

  26. [34]

    Yuasa, S

    K. Yuasa, S. Tasaki, P. Facchi, G. Kimura, H. Nakazato, I. Ohba and S. Pascazio:On the Assumption of Initial Factorization in the Master Equation for Weakly Coupled Systems II: Solvable Models , Annals of Physics 322 657-676 (2007) 30

    2007

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.