Pith. sign in

REVIEW 3 major objections 5 minor 48 references

An open harmonic chain: Exact vs global and local reduced dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Local master equations beat global ones for weak chain coupling, but global wins above a temperature-dependent critical strength.

desk verdict Solid analytic core, plausible crossover claim, but the numerical evidence is under-specified and the missing spectral density undermines reproducibility. read the letter →

arxiv 2505.16669 v1 pith:OGQGCD6O submitted 2025-05-22 quant-ph

Melika Babakan , Fabio Benatti , Laleh Memarzadeh This is my paper · ORCID
classification quant-ph PACS 05.30.-d03.65.Yz03.65.-w
keywords openquantumsystemsGKSLmasterequationlocalapproachglobalharmonicchainGaussianstatesfidelitycriticalcoupling
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 studies a three-oscillator harmonic chain whose end sites each couple to an independent thermal bath at different temperatures. Because the total Hamiltonian is quadratic, the exact system-plus-bath dynamics can be solved numerically as a Gaussian map, and the authors compare that exact evolution with two standard approximations: the 'global' GKSL master equation, built from the full chain eigenmodes, and the 'local' one, built from the uncoupled oscillators. The central claim is that there exists a temperature-dependent critical inter-oscillator coupling strength $g_c$ such that for $g < g_c$ the local steady state deviates less from the exact dynamics, while for $g > g_c$ the global steady state is more accurate. If correct, this gives a practical rule for choosing between the two approaches in open harmonic networks: use the global equation once internal coupling is strong, and the simpler local equation for weak coupling, with the crossover shifting upward as the bath temperatures become more unequal.

What carries the argument

The analysis is carried out in the Gaussian-state formalism: quadratic Hamiltonians generate symplectic evolution, so all three approaches (exact, local, global) are reduced to linear equations for the covariance matrix $C(t)$ and displacement vector $d(t)$ of the form $\dot{C} = M C + C M^\dagger + N$. The central object is the covariance-matrix steady-state equation $M C_\infty + C_\infty M^\dagger = -N$, which is solved exactly for the global approach in the normal-mode basis and perturbatively for the local approach by treating the inter-oscillator coupling $g$ as a small parameter. The comparison quantity is the Gaussian-state fidelity, evaluated between each approximate steady state and its own time-evolved state under the exact dynamics.

What would settle it

Run the same fidelity comparison with $M=100$ and $M=200$ modes per bath at the same parameters ($\lambda=0.1$, $\omega_0=1$, $\omega_c=3$, $T_\ell=10$, $T_r=8$). If the curves of $F(\rho_{\rm loc}^{\infty}, e^{t\mathcal{L}_{\rm exc}}[\rho_{\rm loc}^{\infty}])$ and the global analogue cross at a value of $g$ that shifts by more than the spacing between the $g$ values plotted, or if the apparent quasi-stationarity at $t=50$ breaks down at larger $M$, the reported $g_c$ values in Fig. 4 would be an artifact of the finite bath and finite-time cutoff rather than a physical crossover.

Watch

Extended reading notes

Core claim

The paper claims to reveal a temperature-dependent critical inter-oscillator coupling strength $g_c$ that sorts the regimes of validity of the local and global GKSL master equations for a dissipative harmonic chain. They compute the steady states of the local and global reduced dynamics (analytically for the global one, perturbatively to second order in $g$ for the local one), and compare how each steady state evolves under the exact unitary dynamics generated by the coupled chain plus two finite thermal baths. The finding is that when $g < g_c$, the fidelity $F(\rho_{\rm loc}^{\infty}, e^{t\mathcal{L}_{\rm exc}}[\rho_{\rm loc}^{\infty}])$ exceeds the analogous global fidelity, so the local steady state stays closer to the exact evolution; when $g > g_c$, the inequality reverses and the global approach is the better approximation. The critical value $g_c$ grows as the temperature difference between the two baths increases, and vanishes when the bath temperatures are equal, at which point the global approach is preferred for all $g$ considered.

Load-bearing premise

The finite-bath simulation with $M=50$ modes per bath and evaluation at time $t=50$ is treated as a faithful stand-in for the infinite-reservoir quasi-stationary dynamics, without checking convergence in $M$ or that $t=50$ lies inside the quasi-stationary window for every coupling $g$.

Editorial extensions

If this is right

  • If the claim is correct, practitioners can select between local and global GKSL master equations for an open harmonic chain based solely on whether the inter-oscillator coupling lies below or above a temperature-dependent threshold $g_c$.
  • The result implies that the computationally cheaper local approach is not merely an approximation to the global approach; for weak coupling it can be a genuinely more accurate description of the exact reduced dynamics.
  • The finding suggests that in transport calculations for bosonic chains, where bath temperatures differ, a crossover in accuracy will occur as internal coupling is increased, so conclusions drawn from a single master-equation choice may flip sign across that crossover.
  • The analytical forms of the steady-state covariance matrices (exact in the global case, second-order perturbative in the local case) can be reused as benchmarks for other approximations in open bosonic systems.

Reading between the lines

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

  • The paper's crossover criterion is stated for a single three-oscillator chain; a natural extension is to test whether an analogous temperature-dependent $g_c$ appears for longer chains or for chains with different inter-site couplings, where the local approach is often the only tractable option.
  • The criterion compares only the steady states under exact evolution; one could extend this to full-time fidelity curves, which may reveal that the preferred approximation depends on the time window of interest, not just on $g$ and temperatures.
  • The claim that $g_c$ vanishes for equal bath temperatures suggests a symmetry-breaking mechanism: the temperature imbalance is what makes the local approximation competitive, and a testable prediction is that increasing the imbalance should monotonically raise the crossover coupling.
  • Because the exact dynamics is run with finite $M=50$ bath modes, a stronger test of the claimed crossover would be to check whether the curves of Fig. 3 remain stable as $M$ increases, before applying the result to infinite-reservoir settings.
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

3 major / 5 minor

Summary. The paper studies a three-site harmonic oscillator chain whose end sites are weakly coupled to two independent thermal baths at different temperatures. Within a Gaussian-state formalism, the authors derive the exact unitary dynamics of system plus finite baths, the global and local GKSL master equations, and their steady states (analytic for the global, second-order perturbative in the inter-oscillator coupling g for the local). They compare the fidelity between each approximate steady state and its evolution under the exact dynamics and claim the existence of a temperature-dependent critical coupling gc such that the local approach is more accurate for g<gc and the global one for g>gc. The numerical evidence for gc comes from finite-bath simulations with M=50 modes per bath at time t=50.

Significance. The question of local versus global master equations is of current interest, and the solvable Gaussian model offers a clean testbed. The authors provide explicit analytical formulas for the global steady state, a perturbative local steady state, and a Gaussian framework that is a valuable contribution. The claimed crossover, if backed by solid numerics, would be a useful practical criterion. However, the numerical validation is presently incomplete, and the paper would be considerably strengthened by convergence checks and a fully specified bath model.

major comments (3)
  1. [Section VII, Fig. 4, Eqs. (109)-(110)] The central claim about gc rests solely on finite-bath simulations with M=M'=50 modes per bath evaluated at t=50 for each g. The authors assert that when the baths contain a sufficiently large number of modes the dynamics reaches an almost stationary behavior before periodicity, but they do not demonstrate convergence in M, do not show a quasi-stationary plateau for every g used in Fig. 4, and report no error bars. If the crossing of the two fidelity curves shifts with M or with the evaluation time t, the reported gc values and their temperature dependence may be numerical artifacts. The authors should provide convergence checks (e.g., M=50 vs 100 vs 200) and verify that the ordering in Eqs. (109)-(110) is stable over a time window, not just at t=50.
  2. [Sections III, IV, VII] The bath spectral density J_alpha(omega) and the coupling functions h_{alpha,k} (or mode frequencies omega_{alpha,k}) used in the exact simulations are never specified. Equations (20) and (24) define the interaction via h_{ell,k}, h_{r,k}, but no explicit choice is given. The local and global master equations depend on J_alpha(omega_0) and J_alpha(epsilon_i), so without specifying the spectral density and its discretization, the exact simulation is not reproducibly tied to the GKSL rates. Moreover, with a hard cutoff omega_c=3 and M=50 modes per bath, the mode spacing is about 0.06, which is larger than the global frequency splitting sqrt(2) g near g=0.01; the finite bath may fail to resolve transitions on which the global master equation relies. Please specify the spectral density, the discretization prescription, and test sensitivity to the number of modes.
  3. [Section VII, Fig. 3] The quasi-stationary behavior is shown for only three values of g (0.01, 0.1, 0.6), and no quantitative criterion is given for identifying the plateau. For g=0.01 the two fidelity curves are very close, as the authors note, so the sign of the inequality in Eq. (109) may be sensitive to the chosen evaluation time. The authors should establish the time window where the ordering of the two fidelities is stable and show that the same ordering persists over that window, not merely at t=50.
minor comments (5)
  1. [Section VII, Fig. 4 caption and text] The text after Fig. 4 says the green curve is for T_l=15, while the Fig. 4 caption says the green line is for T_l=20. This inconsistency should be corrected.
  2. [Eq. (107)] The notation F^4(rho_loc^infty, rho_glb^infty) is confusing: it appears to denote the fourth power of the fidelity, but the exponent placement could be misinterpreted. Please clarify the notation throughout.
  3. [Title and Abstract] The arXiv title contains a typo: "dyn amics" should be "dynamics". There are also minor grammatical issues in the abstract, e.g., "the behavior of fidelity between them versus inter-oscillator coupling depends on the two bath temperatures" could be rephrased for clarity.
  4. [Section IV, Eq. (33)] It would be helpful to state explicitly how the reduced covariance matrix of the three-oscillator chain is extracted from the total covariance matrix C_tot(t), since this is the quantity used in subsequent fidelity calculations.
  5. [Introduction, references] The paper would benefit from a more explicit discussion of how its results extend earlier exactly solvable comparisons of local and global master equations, e.g., Ref. [30] by Gonz\'alez et al. and related works on bosonic chains.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the local and global steady states are benchmarked against an independently computed exact unitary dynamics, and no parameter is fitted to the fidelity comparison that defines gc.

full rationale

The derivation chain is self-contained and non-circular. The local and global GKSL generators are constructed from the same system-bath Hamiltonian by choosing different eigenprojections (Remark 1), and the steady states are solved from the Lyapunov equation (86), perturbatively for the local approach and analytically for the global approach. The exact dynamics is an independent finite-mode unitary simulation obtained from the symplectic evolution in Eqs. (27)-(28). The critical coupling gc in Eqs. (109)-(110) is read off from the crossing of F(rho_inf_loc, e^{tL_exc}[rho_inf_loc]) and F(rho_inf_glb, e^{tL_exc}[rho_inf_glb]); neither quantity is fitted to the other, and no parameter is adjusted to produce the crossing. The self-fidelity criterion (Section VII: 'the more accurate description is provided by the steady state which less deviate from itself when evolved in time by the exact evolution') is an assumption about how to assess accuracy, but it is not circular because the exact evolution is computed independently. References [35,36] are background citations to the authors' earlier local-versus-global studies and are not load-bearing for the derivation. The main weaknesses are numerical robustness issues rather than circularity: the paper asserts without convergence data that M=50 and t=50 effectively mimic infinite reservoirs and reach a quasi-stationary window (Section VII, Fig. 3), and it does not specify the bath coupling functions h_{alpha,k} or the spectral densities J_alpha(omega) used in the exact simulation, so consistency with the GKSL rates is not demonstrated. These concerns affect the reliability of gc but do not make the argument circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central result is a numerical crossover observation; it depends on model choices such as spectral density, bath size, and evaluation time rather than on newly postulated entities. No particles, forces, dimensions, or conserved quantities are invented.

free parameters (3)
  • Bath spectral density J_alpha(omega)
    Figures 2-4 are computed with cutoff omega_c=3, but the functional form of J_alpha(omega) is never stated; the fidelity curves and gc values depend on this unspecified choice.
  • Bath mode number M = 50
    The exact unitary simulation uses M=50 modes per bath; no convergence scan or recurrence-time analysis is shown.
  • Evaluation time t = 50
    The comparison between local and global steady states under exact evolution is evaluated at t=50, with no check that every g value is inside the quasi-stationary window.
assumptions (4)
  • standard math Gaussian states remain Gaussian and are determined by first and second moments under quadratic Hamiltonians and linear Lindblad operators.
    Used throughout Sections II-IV to reduce operator dynamics to covariance-matrix equations.
  • domain assumption The weak-coupling limit and secular approximation produce valid GKSL generators at lambda=0.1.
    Section V relies on the WCL to derive both local and global master equations from the same quadratic system-bath interaction.
  • ad hoc to paper A finite bath with M=50 modes per side approximates infinite reservoirs over the simulated time window.
    Section VII asserts quasi-stationary behavior before recurrence without convergence evidence.
  • ad hoc to paper The fidelity between a steady state and its exact-evolved version measures the accuracy of the corresponding approximation.
    The crossover criterion is built on this proxy; no direct comparison to an exact steady state is possible because the finite-M unitary dynamics has no true steady state.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An open harmonic chain: Exact vs global and local reduced dynamics." pith.science (2026). https://pith.science/paper/OGQGCD6O

@misc{pith2026250516669,
  author       = {Pith},
  title        = {Pith review of: An open harmonic chain: Exact vs global and local reduced dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGQGCD6O}},
  note         = {Machine review of arXiv:2505.16669}
}
read the original abstract

In the following, we study the dissipative time-evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the unitary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) approach to open quantum systems, one can perform the rotating wave approximation with respect to the interacting, or non-interacting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different time-scales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behaviour of fidelity between them versus inter-oscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical inter-oscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger inter-oscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of inter-oscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.

Figures

Figures reproduced from arXiv: 2505.16669 by the authors.

Figure 1
Figure 1. The schematic representation of a three in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. reveals that, for small values of g, the two bath temperatures affect the fidelity between the two steady states. On the other hand, for large values of g, the fidelities for different values of Tr coincide and decrease so that the two steady states deviate from each other no matter what the baths temperatures are. Up to here, our analysis has established a clear com￾parison between the steady states obtained from t… view at source ↗
Figure 4
Figure 4. Here, we plot gc versus Tr. The blue, orange and green lines are for Tℓ = 5, 10, 20, respectively. we set λ = 0.1, ω0 = 1, ωc = 3, M = M′ = 50 and t = 50. a critical gc, the global approach is to be preferred to the local one (see sub-figures in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: Fidelity between ρ∞ loc and the evolved ρ∞ loc under the evolution of the exact approach i.e F(ρ∞ loc, etLexc [ρ∞ loc]) (orange curve) and fidelity between ρ∞ glb and the evolved ρ∞ glb under the evolution of the ex￾act approach i.e F(ρ∞ glb, etLexc [ρ∞ glb]) (blue cur…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 38 canonical work pages

  1. [1]

    (56), (57), (76), and (77), M and N have the following form M = M ⊕ M, N = N ⊕ N

    (87) As it is seen in Eqs. (56), (57), (76), and (77), M and N have the following form M = M ⊕ M, N = N ⊕ N. (88) Therefore, to find C∞ we must solve Eq. (86) which yields the following equations for C1 and C2: MC1 + C1M† = −N, (89) MC2 + C2M = 0. (90) Because of the uniqueness of the steady state C2 = 0 is the only solution of Eq. Eq. (90). To solve Eq. (...

    work page

  2. [2]

    In such a case, due to the symmetry, we expect that the covariance matrix remains invariant under the change of labels 1 and 3

    A special case Here we derive the covariance matrix of the steady state for a special case: Jℓ(ω0) = Jr(ω0) =: J(ω0) and the two bath temperatures are equal (¯nℓ(ω0) = ¯nr(ω0) =: ¯n(ω0)). In such a case, due to the symmetry, we expect that the covariance matrix remains invariant under the change of labels 1 and 3. That gives the following form for the cov...

    work page

  3. [3]

    By expanding |C1⟩, which is the vectorized form of the first block of the local steady state, in powers of g |C1⟩ = ∞∑ n=0 gn ⏐ ⏐ ⏐C(n) ⟩ , (C3) and using Eq

    Perturbative steady state Here we drive the local steady state with the per- turbative approach up to the second order of the inter- oscillator coupling strength g. By expanding |C1⟩, which is the vectorized form of the first block of the local steady state, in powers of g |C1⟩ = ∞∑ n=0 gn ⏐ ⏐ ⏐C(n) ⟩ , (C3) and using Eq. (92) when F and |N⟩ are replaced b...

    work page

  4. [4]

    Brantut, C

    J. Brantut, C. Grenier, J. Meineke, D. Stadler, S. Krin- ner, C. Kollath, T. Esslinger, and A. Georges, A thermo- electric heat engine with ultracold atoms, Science 342, 713 (2013)

    work page 2013

  5. [5]

    J. T. Barreiro, M. M¨ uller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, An open-system quantum simulator with trapped ions, Nature 470, 486 (2011)

    2011

  6. [6]

    Caruso, A

    F. Caruso, A. W. Chin, A. Datta, S. F. Huelga, and M. B. Plenio, Highly efficient energy excitation trans- fer in light-harvesting complexes: the fundamental role of noise-assisted transport, J. Chem. Phys. 131, 105106 (2009)

    work page 2009

  7. [7]

    Killoran, S

    N. Killoran, S. F. Huelga, and M. B. Plenio, Enhancing light-harvesting power with coherent vibrational interac - tions: a quantum heat engine picture, J. Chem. Phys. 143, 155102 (2015)

    work page 2015

  8. [8]

    R. J. Glauber, Coherent and incoherent states of the ra- diation field, Phys. Rev. 131, 2766 (1963)

    work page 1963

Show all 48 references
  1. [9]

    E. C. G. Sudarshan, Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams, Phys. Rev. Lett. 10, 277 (1963)

    1963

  2. [10]

    Baumgratz, M

    T. Baumgratz, M. Cramer, and M. B. Plenio, Quantify- ing coherence, Phys. Rev. Lett. 113, 140401 (2014)

    2014

  3. [11]

    R. F. Werner, Quantum states with einstein-podolsky- rosen correlations admitting a hidden-variable model, Phys. Rev. A 40, 4277 (1989)

    1989

  4. [12]

    Peres, Separability criterion for density matrices, Phy s

    A. Peres, Separability criterion for density matrices, Phy s. Rev. Lett. 77, 1413 (1996)

    1996

  5. [13]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Separa- bility of mixed states: necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996)

    1996

  6. [14]

    Gemmer, Quantum thermodynamics: emergence of thermodynamic behavior within composite quantum sys- tems, 2nd ed., Lecture notes in physics (Springer, Berlin, 2009)

    J. Gemmer, Quantum thermodynamics: emergence of thermodynamic behavior within composite quantum sys- tems, 2nd ed., Lecture notes in physics (Springer, Berlin, 2009)

    2009

  7. [15]

    Binder, L

    F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, eds., Thermodynamics in the Quantum regime: Fundamental aspects and new directions , Vol. 195 (Springer International Publishing, Cham, 2018)

    2018

  8. [16]

    Deffner and S

    S. Deffner and S. Campbell, Quantum thermodynamics: An introduction to the thermodynamics of quantum infor- mation (Morgan & Claypool Publishind, 2019)

    2019

  9. [17]

    Spohn and J

    H. Spohn and J. L. Lebowitz, Irreversible thermodynam- ics for Quantum systems weakly coupled to thermal reser- voirs, in Advances in Chemical Physics , Vol. 38, edited by S. A. Rice (Wiley, 1978) 1st ed., pp. 109–142

    1978

  10. [18]

    E. B. Davies, A model of heat conduction, J. Stat. Phys. 18, 10.1007/BF01014307 (1978)

    1978 doi

  11. [19]

    Alicki, The quantum open system as a model of the heat engine, J

    R. Alicki, The quantum open system as a model of the heat engine, J. Phys. A: Math. Gen. 12, L103 (1979)

    1979

  12. [20]

    Zoubi, M

    H. Zoubi, M. Orenstien, and A. Ron, Dissipations in cou- pled quantum systems, Phys. Rev. A 67, 063813 (2003)

    2003

  13. [21]

    Rivas, A

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

    2010

  14. [22]

    Migliore, M

    R. Migliore, M. Scala, A. Napoli, K. Yuasa, H. Nakazato, and A. Messina, Dissipative effects on a generation 15 scheme of a w state in an array of coupled josephson junctions, J. Phys. B 44, 075503 (2011)

    2011

  15. [23]

    Werlang and D

    T. Werlang and D. Valente, Heat transport between two pure-dephasing reservoirs, Phys. Rev. E 91, 012143 (2015)

    2015

  16. [24]

    P. H. Guimar˜ aes, G. T. Landi, and M. J. de Oliveira, Nonequilibrium quantum chains under multisite lindblad baths, Phys. Rev. E 94, 032139 (2016)

    2016

  17. [25]

    Wichterich, M

    H. Wichterich, M. J. Henrich, H.-P. Breuer, J. Gemmer, and M. Michel, Modeling heat transport through com- pletely positive maps, Phys. Rev. E 76, 031115 (2007)

    2007

  18. [26]

    Rivas and M

    A. Rivas and M. Martin-Delgado, Topological heat trans- port and symmetry-protected Boson currents, Sci. Rep. 7, 6350 (2017)

    2017

  19. [27]

    S. F. Huelga and M. B. Plenio, Vibrations, quanta and biology, Contemp. Phys. 54, 181 (2013)

    2013

  20. [28]

    Gorini and A

    V. Gorini and A. Kossakowski, Completely positive dy- namical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976)

    1976

  21. [29]

    Lindblad, On the generators of quantum dynamical semigroups, Commun

    G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976)

    1976

  22. [30]

    Levy and R

    A. Levy and R. Kosloff, The local approach to quantum transport may violate the second law of thermodynamics, Europh. Lett. 107, 20004 (2014)

    2014

  23. [31]

    J. P. Santos and G. T. Landi, Microscopic theory of a nonequilibrium open bosonic chain, Phys. Rev. E 94, 062143 (2016)

    2016

  24. [32]

    A. S. Trushechkin and I. V. Volovich, Perturbative treat- ment of inter-site couplings in the local description of open quantum networks, Europ. Lett. 113, 30005 (2016)

    2016

  25. [33]

    J. O. Gonz´ alez, L. A. Correa, G. Nocerino, J. P. Palao, D. Alonso, and G. Adesso, Testing the va- lidity of the ‘local’ and ‘global’ gkls master equa- tions on an exactly solvable model, Open Sys- tems & Information Dynamics 24, 1740010 (2017), https://doi.org/10.1142/S12301...

    2017 doi

  26. [34]

    P. P. Hofer, M. Perarnau-Llobet, L. D. M. Miranda, G. Haack, R. Silva, J. B. Brask, and N. Brunner, Marko- vian master equations for quantum thermal machines: local versus global approach, New J. Phys. 19, 123037 (2017)

    2017

  27. [35]

    De Chiara, G

    G. De Chiara, G. Landi, A. Hewgill, B. Reid, A. Ferraro, A. J. Roncaglia, and M. Antezza, Reconciliation of quan- tum local master equations with thermodynamics, New J. Phys. 20, 113024 (2018)

    2018

  28. [36]

    Cattaneo, G

    M. Cattaneo, G. L. Giorgi, S. Maniscalco, and R. Zam- brini, Local versus global master equation with common and separate baths: superiority of the global approach in partial secular approximation, New J. Phys. 21, 113045 (2019)

    2019

  29. [37]

    Farina, G

    D. Farina, G. De Filippis, V. Cataudella, M. Polini, and V. Giovannetti, Going beyond local and global ap- proaches for localized thermal dissipation, Phys. Rev. A 102, 052208 (2020)

    2020

  30. [38]

    Benatti, R

    F. Benatti, R. Floreanini, and L. Memarzadeh, Bath- assisted transport in a three-site spin chain: global versu s local approach, Phys. Rev. A 102, 042219 (2020)

    2020

  31. [39]

    Benatti, R

    F. Benatti, R. Floreanini, and L. Memarzadeh, Exact steady state of the open X X -spin chain: Entangle- ment and transport properties, PRX Quantum 2, 030344 (2021)

    2021

  32. [40]

    Scali, J

    S. Scali, J. Anders, and L. A. Correa, Local master equa- tions bypass the secular approximation, Quantum 5, 451 (2021)

    2021

  33. [41]

    Konopik and E

    M. Konopik and E. Lutz, Local master equations may fail to describe dissipative critical behavior, Phys. Rev. Res. 4, 013171 (2022)

    2022

  34. [42]

    M. M. Wolf, G. Giedke, and J. I. Cirac, Extremality of gaussian quantum states, Phys. Rev. Lett. 96, 080502 (2006)

    2006

  35. [43]

    Ferraro, S

    A. Ferraro, S. Olivares, and M. G. A. Paris, Gaus- sian states in continuous variable quantum information (2005), arXiv:quant-ph/0503237

    2005 arXiv

  36. [44]

    Eisert and M

    J. Eisert and M. M. Wolf, Gaussian quantum channels (2005), arXiv:quant-ph/0505151

    2005 arXiv

  37. [45]

    Serafini, Quantum continuous variables: a primer of theoretical methods, first issued in paperback ed

    A. Serafini, Quantum continuous variables: a primer of theoretical methods, first issued in paperback ed. (CRC Press, Taylor & Francis Group, Boca Raton London New York, 2021)

    2021

  38. [46]

    Toscano and F

    F. Toscano and F. Nicacio, Thermal equilibrium in gaus- sian dynamical semigroups, Phys. Rev. A 106, 062207 (2022)

    2022

  39. [47]

    Spohn, An algebraic condition for the approach to equilibrium of an open N-level system, Lett

    H. Spohn, An algebraic condition for the approach to equilibrium of an open N-level system, Lett. Math. Phys. 2, 33 (1977)

    1977

  40. [48]

    Banchi, S

    L. Banchi, S. L. Braunstein, and S. Pirandola, Quantum Fidelity for Arbitrary Gaussian States, Phys. Rev. Lett. 115, 260501 (2015)

    2015

Pith tools

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