pith. sign in

arxiv: 2509.05206 · v2 · submitted 2025-09-05 · 🪐 quant-ph

Adiabatic preparation of thermal states and entropy-noise relation on noisy quantum computers

Etienne Granet , Henrik Dreyer This is my paper

Pith reviewed 2026-05-18 18:56 UTC · model grok-4.3

classification 🪐 quant-ph
keywords adiabatic preparationthermal statesquantum computersentropy conservationnoise resilienceIsing modelmirror circuits
0
0 comments X

The pith

Thermal states can be prepared on quantum computers by adiabatically evolving an initial Gibbs state of a simple Hamiltonian, with local entropy density conserved to determine the final temperature.

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

The paper shows how to prepare states that are locally in thermal equilibrium at finite temperature by starting from the thermal Gibbs state of a simple Hamiltonian and adiabatically evolving it under a time-dependent interpolating Hamiltonian. This works identically to the usual adiabatic method for ground states but targets excited thermal states instead. In the thermodynamic limit the entropy density of local density matrices stays constant, so the final state's energy and entropy can both be obtained and the temperature read off directly. The protocol is shown to be resilient to depolarizing noise because the measured energy-temperature relation stays largely unchanged, and mirror circuits provide a way to quantify the extra entropy introduced by hardware imperfections.

Core claim

States that are locally at thermal equilibrium can be prepared by evolving adiabatically an initial thermal Gibbs state of a simple Hamiltonian with an interpolating time-dependent Hamiltonian, identically to adiabatic ground state preparation. The entropy density of local density matrices is conserved during the adiabatic evolution in the thermodynamic limit, so that both the entropy and energy of the final state can be computed, and thus the final temperature too. In the presence of hardware noise the entropy created by the noisy evolution can be precisely benchmarked with mirror circuits, and numerical evidence indicates the energy-temperature curve is insensitive to depolarizing noise.

What carries the argument

Adiabatic evolution of an initial thermal Gibbs state under a time-dependent interpolating Hamiltonian, together with conservation of local entropy density in the thermodynamic limit.

If this is right

  • The energy-temperature curve remains insensitive to the strength of depolarizing noise during state preparation.
  • Entropy generated by hardware imperfections can be quantified exactly using mirror circuits.
  • Lack of adiabaticity in a Trotterized implementation can be estimated from the observed entropy per site.
  • A thermal state of the Ising model at temperature 2.56 ± 0.26 was prepared on a 5 × 4 lattice using 640 two-qubit gates on ion-trap hardware.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may let quantum simulators extract thermodynamic quantities without reconstructing the full density matrix.
  • Similar adiabatic ramps could be used for other lattice models that thermalize, provided the initial simple Hamiltonian is chosen appropriately.
  • Because the protocol tolerates depolarizing noise, it could be combined with error-mitigation techniques to reach larger system sizes on near-term devices.

Load-bearing premise

The entropy density of local density matrices is conserved during the adiabatic evolution in the thermodynamic limit.

What would settle it

A measurement on a large enough system showing that local entropy density changes measurably during the adiabatic process would mean the final temperature cannot be reliably inferred from the measured energy alone.

Figures

Figures reproduced from arXiv: 2509.05206 by Etienne Granet, Henrik Dreyer.

Figure 1
Figure 1. Figure 1: FIG. 1. Entropy per site as a function of time, for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Left: Energy density [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Left panel: entropy density measured through the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We consider the problem of preparing thermal equilibrium states at finite temperature on quantum computers. Assuming thermalization, we show that states that are locally at thermal equilibrium can be prepared by evolving adiabatically an initial thermal Gibbs state of a simple Hamiltonian with an interpolating time-dependent Hamiltonian, identically to adiabatic ground state preparation. We argue that the entropy density of local density matrices is conserved during the adiabatic evolution in the thermodynamic limit, so that both the entropy and energy of the final state can be computed, and thus the final temperature too. We show that in the presence of hardware noise, the entropy created by the noisy evolution can be precisely benchmarked with mirror circuits. We give numerical evidence that the resulting thermal state preparation protocol is noise-resilient for depolarizing noise, in the sense that the energy-temperature curve measured on a noisy quantum computer is remarkably insensitive to the amplitude of depolarizing noise in the state preparation. We finally propose a protocol to estimate the lack of adiabaticity in a given actual Trotter implementation of the dynamics. We test our protocol on Quantinuum's H1-1 ion-trap device. We measure that a circuit with $640$ two-qubit gates implemented on hardware generates an entropy per site of $0.166 \pm 0.0045$, giving a benchmark metric for this state preparation. We report the preparation of a thermal state with temperature $2.56 \pm 0.26$ of the Ising model in size $5\times 4$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes preparing finite-temperature thermal states on quantum computers via adiabatic evolution of an initial Gibbs state of a simple Hamiltonian under a time-dependent interpolating Hamiltonian. It argues that in the thermodynamic limit, the entropy density of local reduced density matrices is conserved, enabling computation of the final state's energy and entropy to determine its temperature. The work provides numerical evidence for resilience to depolarizing noise, uses mirror circuits to benchmark entropy production due to noise, and reports an experimental implementation on Quantinuum's H1-1 device for the Ising model on a 5×4 lattice, achieving a temperature of 2.56 ± 0.26 with a circuit of 640 two-qubit gates.

Significance. If the entropy density conservation holds as assumed, this protocol offers a promising route for preparing and verifying thermal states without prior knowledge of the target temperature, directly addressing challenges in quantum simulation of finite-temperature physics. The numerical demonstration of noise resilience and the concrete experimental benchmark with error bars on entropy per site (0.166 ± 0.0045) provide practical value for noisy intermediate-scale quantum devices.

major comments (2)
  1. Abstract and main text discussion of entropy conservation: the claim that the entropy density of local density matrices is conserved during the adiabatic evolution in the thermodynamic limit is load-bearing for deriving the final temperature from measured energy and entropy, yet the manuscript provides no rigorous finite-size proof, scaling analysis, or counter-example study for interpolation paths that may cross gapless points or phase transitions.
  2. Experimental results on the 5×4 Ising model: the reported benchmark uses a system size too small to test the thermodynamic-limit assumption required for the local entropy-temperature relation; the 640-gate circuit yields entropy per site 0.166 ± 0.0045 but does not address finite-size deviations that could decouple local entropy density from the global energy.
minor comments (1)
  1. Main text: the distinction between global von Neumann entropy (conserved by unitarity) and local entropy density s could be stated more explicitly when introducing the thermodynamic relation used to extract temperature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying key points that require clarification. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: Abstract and main text discussion of entropy conservation: the claim that the entropy density of local density matrices is conserved during the adiabatic evolution in the thermodynamic limit is load-bearing for deriving the final temperature from measured energy and entropy, yet the manuscript provides no rigorous finite-size proof, scaling analysis, or counter-example study for interpolation paths that may cross gapless points or phase transitions.

    Authors: We agree that the conservation of local entropy density is a central assumption and that the manuscript presents a physical argument rather than a fully rigorous proof. The argument rests on the locality of reduced density matrices and the expectation that, in the thermodynamic limit for gapped systems, long-range correlations generated during adiabatic evolution do not affect local entropy. We have included numerical checks on finite lattices in the supplementary material, but we acknowledge the absence of a systematic scaling analysis or explicit counter-example study for paths that cross gapless regions. In the revised manuscript we will add a dedicated subsection that (i) states the conditions under which the argument is expected to hold, (ii) presents finite-size scaling data for the Ising model showing convergence of local entropy density, and (iii) discusses the potential breakdown for interpolation paths that traverse critical points, including a brief numerical example. revision: yes

  2. Referee: Experimental results on the 5×4 Ising model: the reported benchmark uses a system size too small to test the thermodynamic-limit assumption required for the local entropy-temperature relation; the 640-gate circuit yields entropy per site 0.166 ± 0.0045 but does not address finite-size deviations that could decouple local entropy density from the global energy.

    Authors: We concur that a 5×4 lattice is modest and cannot by itself validate the thermodynamic-limit assumption. The experiment is intended as a hardware demonstration of the adiabatic preparation protocol together with mirror-circuit entropy benchmarking, rather than a direct test of the limit. Numerical simulations on larger lattices (presented in the supplementary material) indicate that local entropy density remains close to its thermodynamic value for this model even at moderate sizes. In the revision we will (i) explicitly state the limited scope of the experimental result, (ii) add classical tensor-network data illustrating finite-size corrections to the local entropy-energy relation, and (iii) clarify that the reported temperature 2.56 ± 0.26 is obtained under the working assumption that finite-size deviations are small for the chosen parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent assumptions and measurements

full rationale

The paper claims that local thermal states can be prepared adiabatically from an initial Gibbs state of a simple Hamiltonian, with final temperature obtained from conserved local entropy density and measured energy in the thermodynamic limit. This conservation is presented as an argued property of unitary evolution rather than a self-referential definition or fitted input. Entropy is benchmarked independently via mirror circuits on hardware, and temperature is computed post hoc without defining the target T by the result itself. No load-bearing self-citations, uniqueness theorems from the authors, or ansatzes smuggled via prior work reduce the central protocol to a tautology. The numerical noise-resilience evidence and 5x4 experimental test provide external validation paths. The derivation chain therefore remains independent of its outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption of local thermalization during adiabatic evolution and entropy-density conservation in the thermodynamic limit; no new particles or forces are introduced.

axioms (2)
  • domain assumption Local thermal equilibrium is reached and maintained during the adiabatic evolution
    Invoked in the abstract and introduction to justify that the final state remains locally thermal.
  • domain assumption Entropy density of local density matrices is conserved in the thermodynamic limit
    Stated explicitly as the basis for computing final temperature from energy and entropy.

pith-pipeline@v0.9.0 · 5797 in / 1401 out tokens · 39901 ms · 2026-05-18T18:56:11.455577+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 3 Pith papers

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

  1. Spectral functions on a quantum computer through system-environment interaction

    quant-ph 2026-05 unverdicted novelty 7.0

    A new quantum circuit method computes spectral functions A(k,ω) by simulating ARPES-like system-environment coupling, cutting sampling overhead by O(N) and demonstrated on a 54-qubit ion-trap processor for a 27-site chain.

  2. Universal scaling of finite-temperature quantum adiabaticity in driven many-body systems

    quant-ph 2026-02 unverdicted novelty 6.0

    Finite-temperature adiabatic threshold driving rate in gapped local Hamiltonians factorizes into zero-T system-size scaling plus a universal T-factor exponentially close to 1 at low T and linear at high T.

  3. Variational Thermal State Preparation on Digital Quantum Processors Assisted by Matrix Product States

    quant-ph 2025-10 unverdicted novelty 6.0

    A variational framework assisted by matrix product states prepares approximate thermal Gibbs states for 1D lattices up to 30 sites and 2D lattices up to 6x6 using up to 44 qubits, with a demonstration on IBM Heron hardware.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · cited by 3 Pith papers · 5 internal anchors

  1. [1]

    This logarithmic dependence of the best possible temper- ature with the error rate of the quantum computer, no- toriously difficult to lower down, might look pessimistic

    The dotted lines indicate the entropy density computed with (30). This logarithmic dependence of the best possible temper- ature with the error rate of the quantum computer, no- toriously difficult to lower down, might look pessimistic. However, at temperature lower than the gap of the sys- tem, all physical quantities will have exponentially small deviat...

    work page

  2. [2]

    For a givenβ0, we prepare the quantum computer in a Gibbs state ofH0 in (10). We evolve the state of the quantum computer with the time-dependent Hamiltonian(1− t T )H0+ t T Hf, fora given adiabatic timeTassumed to be large enough so that the evo- lution is adiabatic. We then measure the energy of the final HamiltonianHf obtained, denotedE(β 0)

    work page

  3. [3]

    For the sameβ 0, we prepare again the quantum computer in the Gibbs state ofH0. We implement the same adiabatic evolution, but only up to half of the timet=T /2, and then apply the exact inverse circuit, so as to come back exactly to the initial state for an ideal noiseless circuit. We measurem the average expectation value ofXon the qubits, and deduce th...

    work page

  4. [4]

    We deduce the curveE(β)

    Collecting the values ofE(β0)andS(β 0)for sev- eral values ofβ 0, we compute an estimated final temperatureβ(β 0) = dS(β0) dE(β0) when varyingβ 0. We deduce the curveE(β). If we assume that Eq (31) holds true, namely that the de- pendence of the measuredmdepends onβ0 only through (31), then it may suffice to perform step2of this protocol only for one valu...

    work page

  5. [5]

    hardware

    The adiabatic path is taken to be (9) with the initial HamiltonianH 0 therein being−H0 in (10). We trotterize 9 this adiabatic evolution into a unitary operatorUthat containsMTrotter steps U=V(J M−1 , fM−1 )V(J M−2 , fM−2 )...V(J 0, f0),(39) with the Trotter step operator V(J, f) =e if P j Xj eiJ P ⟨j,k⟩ Zj Zk .(40) We set the sequence fn = 1 (a+bn) c , J...

    work page 2025

  6. [6]

    A. M. Dalzell, S. McArdle, M. Berta, P. Bienias, C.- F. Chen, A. Gilyén, C. T. Hann, M. J. Kastoryano, E. T. Khabiboulline, A. Kubica,et al., arXiv preprint arXiv:2310.03011 (2023), 10.48550/arXiv.2310.03011

    work page doi:10.48550/arxiv.2310.03011 2023

  7. [7]

    R. P. Feynman, inFeynman and computation(cRc Press,

    work page

  8. [8]

    Ponsioen, S

    B. Ponsioen, S. S. Chung, and P. Corboz, Physical Re- view B100, 195141 (2019)

    work page 2019

  9. [9]

    Qin, C.-M

    M. Qin, C.-M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollwöck, S. R. White, S. Zhang, and S. C. on the Many-Electron Problem), Physical Review X10, 031016 (2020)

    work page 2020

  10. [10]

    Zhang, Science384, eadh7691 (2024)

    H.Xu, C.-M.Chung, M.Qin, U.Schollwöck, S.R.White, and S. Zhang, Science384, eadh7691 (2024)

    work page 2024

  11. [11]

    Digital quantum magnetism on a trapped-ion quantum computer

    R. Haghshenas, E. Chertkov, M. Mills, W. Kadow, S.-H. Lin, Y.-H. Chen, C. Cade, I. Niesen, T. Begušić, M. S. Rudolph,et al., arXiv preprint arXiv:2503.20870 (2025), 10.48550/arXiv.2503.20870

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2503.20870 2025

  12. [12]

    D. A. Abanin, R. Acharya, L. Aghababaie-Beni, G. Aigeldinger, A. Ajoy, R. Alcaraz, I. Aleiner, T. I. An- dersen, M. Ansmann, F. Arute,et al., arXiv preprint arXiv:2506.10191 (2025), 10.48550/arXiv.2506.10191

    work page doi:10.48550/arxiv.2506.10191 2025

  13. [13]

    T. I. Andersen, N. Astrakhantsev, A. H. Karamlou, J. Berndtsson, J. Motruk, A. Szasz, J. A. Gross, A. Schuckert, T. Westerhout, Y. Zhang,et al., Nature 638, 79 (2025)

    work page 2025

  14. [14]

    A. D. King, A. Nocera, M. M. Rams, J. Dziarmaga, R. Wiersema, W. Bernoudy, J. Raymond, N. Kaushal, N. Heinsdorf, R. Harris,et al., Science388, 199 (2025)

    work page 2025

  15. [15]

    Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zale- tel, K. Temme,et al., Nature618, 500 (2023)

    work page 2023

  16. [16]

    W. M. Foulkes, L. Mitas, R. Needs, and G. Rajagopal, Reviews of Modern Physics73, 33 (2001)

    work page 2001

  17. [17]

    Albash and D

    T. Albash and D. A. Lidar, Reviews of Modern Physics 90, 015002 (2018)

    work page 2018

  18. [18]

    A. Y. Kitaev, arXiv preprint quant-ph/9511026 (1995), 10.48550/arXiv.quant-ph/9511026

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.quant-ph/9511026 1995

  19. [19]

    S. Lu, M. C. Bañuls, and J. I. Cirac, PRX quantum2, 020321 (2021)

    work page 2021

  20. [20]

    A. N. Chowdhury and R. D. Somma, arXiv preprint arXiv:1603.02940 (2016), 10.48550/arXiv.1603.02940

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1603.02940 2016

  21. [21]

    Motta, C

    M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. Brandao, and G. K.-L. Chan, Na- ture Physics16, 205 (2020)

    work page 2020

  22. [22]

    Coopmans, Y

    L. Coopmans, Y. Kikuchi, and M. Benedetti, PRX Quantum4, 010305 (2023)

    work page 2023

  23. [23]

    Temme, T

    K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin, and F. Verstraete, Nature471, 87 (2011)

    work page 2011

  24. [24]

    J. Cohn, F. Yang, K. Najafi, B. Jones, and J. K. Freer- icks, Physical Review A102, 022622 (2020)

    work page 2020

  25. [25]

    C.-F. Chen, M. J. Kastoryano, and A. Gi- lyén, arXiv preprint arXiv:2311.09207 (2023), 10.48550/arXiv.2311.09207

    work page doi:10.48550/arxiv.2311.09207 2023

  26. [26]

    Shtanko and R

    O. Shtanko and R. Movassagh, arXiv preprint arXiv:2112.14688 (2021), 10.48550/arXiv.2112.14688

    work page doi:10.48550/arxiv.2112.14688 2021

  27. [27]

    Granet and H

    E. Granet and H. Dreyer, arXiv preprint arXiv:2401.02207 (2024), 10.48550/arXiv.2401.02207

    work page doi:10.48550/arxiv.2401.02207 2024

  28. [28]

    Bergamaschi, C.-F

    T. Bergamaschi, C.-F. Chen, and Y. Liu, in2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)(IEEE, 2024) pp. 1063–1085

    work page 2024

  29. [29]

    C.-F. Chen, M. J. Kastoryano, F. G. Brandão, and A. Gilyén, arXiv preprint arXiv:2303.18224 (2023), 10.48550/arXiv.2303.18224

    work page internal anchor Pith review doi:10.48550/arxiv.2303.18224 2023

  30. [30]

    Brunner, L

    E. Brunner, L. Coopmans, G. Matos, M. Rosenkranz, F. Sauvage, and Y. Kikuchi, arXiv preprint arXiv:2412.17706 (2024), 10.48550/arXiv.2412.17706

    work page doi:10.48550/arxiv.2412.17706 2024

  31. [31]

    P. Rall, C. Wang, and P. Wocjan, Quantum7, 1132 (2023)

    work page 2023

  32. [32]

    Hémery, K

    K. Hémery, K. Ghanem, E. Crane, S. L. Campbell, J. M. Dreiling, C. Figgatt, C. Foltz, J. P. Gaebler, J. Johansen, M. Mills,et al., PRX Quantum5, 030323 (2024)

    work page 2024

  33. [33]

    T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Jour- nal of Physics B: Atomic, Molecular and Optical Physics 51, 112001 (2018)

    work page 2018

  34. [34]

    Yarloo, H.-C

    H. Yarloo, H.-C. Zhang, and A. E. Nielsen, PRX Quan- tum5, 020365 (2024)

    work page 2024

  35. [35]

    R. L. Greenblatt, M. Lange, G. Marcelli, and M. Porta, Communications in Mathematical Physics405, 75 (2024)

    work page 2024

  36. [36]

    Il ‘in, A

    N. Il ‘in, A. Aristova, and O. Lychkovskiy, Physical Re- view A104, L030202 (2021)

    work page 2021

  37. [37]

    Y. Zuo, Q. Yang, B.-G. Liu, and D. E. Liu, Physical Review E110, L022105 (2024)

    work page 2024

  38. [38]

    Irmejs, M

    R. Irmejs, M. C. Bañuls, and J. I. Cirac, arXiv preprint arXiv:2505.20042 (2025), 10.48550/arXiv.2505.20042

    work page doi:10.48550/arxiv.2505.20042 2025

  39. [39]

    Plastina, A

    F. Plastina, A. Alecce, T. J. Apollaro, G. Falcone, G. Francica, F. Galve, N. Lo Gullo, and R. Zambrini, Physical review letters113, 260601 (2014)

    work page 2014

  40. [40]

    Francica, J

    G. Francica, J. Goold, and F. Plastina, Physical Review E99, 042105 (2019)

    work page 2019

  41. [41]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of Statistical Mechan- ics: Theory and Experiment2005, P04010 (2005)

    work page 2005

  42. [42]

    B. M. Spar, E. Guardado-Sanchez, S. Chi, Z. Z. Yan, and W. S. Bakr, Physical review letters128, 223202 (2022)

    work page 2022

  43. [43]

    Carcy, G

    C. Carcy, G. Hercé, A. Tenart, T. Roscilde, and D. Clé- ment, Physical Review Letters126, 045301 (2021)

    work page 2021

  44. [44]

    Bloch, J

    I. Bloch, J. Dalibard, and W. Zwerger, Reviews of mod- ern physics80, 885 (2008)

    work page 2008

  45. [45]

    Area laws for the entanglement entropy - a review

    J. Eisert, M. Cramer, and M. B. Plenio, arXiv preprint arXiv:0808.3773 (2008). 12

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  46. [46]

    Proctor, K

    T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, Nature Physics18, 75 (2022). Appendix A: Second order of entropy density evolution We have the expansion at second order inδH eit(H+δH) =eitH +i Z t 0 dseisH δHe i(t−s)H − Z t 0 ds Z s 0 dueiuH δHe i(s−u)H δHe i(t−s)H +O(δH 3). (1) So we get eit(H+δH) ρe−it(H+δH) =eitH ρe−itH +i Z t 0 dse...

    work page 2022