pith. sign in

arxiv: 1906.10940 · v1 · pith:XWFK4ZF5new · submitted 2019-06-26 · 🪐 quant-ph

Clausius inequality versus quantum coherence

Ali Soltanmanesh , Afshin Shafiee This is my paper

Pith reviewed 2026-05-25 15:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum coherenceClausius inequalityentropy productionLindblad master equationharmonic oscillatorthermal bathopen quantum systemsquantum thermodynamics
0
0 comments X

The pith

A coherent quantum harmonic oscillator violates the Clausius inequality at low temperatures.

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

The authors model a harmonic oscillator passing through an interferometer while partially coupled to a thermal bath of oscillators. They apply a Brownian-type Lindblad master equation to track the system's evolution and compute thermodynamic quantities at varying temperatures. Even in the low-temperature limit where the oscillator retains coherence, the calculated entropy production exceeds the limit set by the classical Clausius inequality. The paper identifies the preserved coherence of the system itself, rather than system-bath entanglement, as the origin of the violation.

Core claim

We model a harmonic oscillator that enters an interferometer partially coupled to a thermal bath of oscillatory fields by employing a Brownian-type Lindblad master equation. We recognize that although the system can remain coherent during its interaction with the thermal bath in the low-temperature limit, the system's entropy production violates the Clausius inequality. Furthermore, we argue that the system's coherence is the source of this violation, rather than the entanglement degree of system-environment, as reported in previous studies.

What carries the argument

Brownian-type Lindblad master equation applied to the partially bath-coupled harmonic oscillator in an interferometer, used to compute coherence and entropy production.

If this is right

  • Entropy production in open quantum systems can exceed classical thermodynamic bounds when coherence persists.
  • Coherence of the system, not entanglement with the environment, can be the direct cause of Clausius inequality violations.
  • Classical thermodynamic relations do not necessarily constrain entropy production in low-temperature quantum regimes.
  • Thermodynamic accounting in interferometric setups must include coherence effects to avoid incorrect bounds.

Where Pith is reading between the lines

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

  • Quantum coherence may permit thermodynamic behaviors outside classical limits, suggesting new design principles for quantum heat engines or refrigerators.
  • The result raises the question of whether similar violations appear in other master-equation descriptions or different measures of coherence.
  • Experimental tests could use trapped ions or superconducting circuits to observe entropy production in coherent low-temperature regimes.

Load-bearing premise

The Brownian-type Lindblad master equation together with the chosen definition of entropy production accurately captures the open-system dynamics without hidden assumptions that would artificially produce the reported violation.

What would settle it

Direct measurement of entropy production for a coherent quantum oscillator in contact with a low-temperature bath, checking whether it exceeds the Clausius bound while coherence is maintained.

Figures

Figures reproduced from arXiv: 1906.10940 by Afshin Shafiee, Ali Soltanmanesh.

Figure 1
Figure 1. Figure 1: FIG. 1. A particle in the state [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Interference patterns with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The regions where the system violates the Clau [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The distillable coherence [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Blue Graph: The changes of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

In this study, we model a harmonic oscillator that enters an interferometer partially coupled to a thermal bath of oscillatory fields by employing a Brownian-type Lindblad master equation. More specifically, we investigate the dynamics and the variations of the thermodynamic quantities of the system at different temperatures. We recognize that although the system can remain coherent during its interaction with the thermal bath in the low-temperature limit, the system's entropy production violates the Clausius inequality. Furthermore, we argue that the system's coherence is the source of this violation, rather than the entanglement degree of system-environment, as reported in previous studies.

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

3 major / 1 minor

Summary. The paper models a harmonic oscillator entering an interferometer and partially coupled to a thermal bath of oscillators via a Brownian-type Lindblad master equation. It reports that, in the low-temperature limit, the system can remain coherent while its entropy production violates the Clausius inequality, and attributes the violation to the system's coherence (rather than system-environment entanglement).

Significance. If the central claim survives removal of the Markov assumption, the result would provide a concrete example in which coherence produces a thermodynamic inequality violation in an open quantum system, with potential implications for quantum thermodynamics. The manuscript supplies a specific dynamical model and temperature-dependent analysis, which are strengths, but the load-bearing approximation is not validated against non-Markovian benchmarks.

major comments (3)
  1. [§2] §2 (Model): The Brownian-type Lindblad master equation is obtained under the Born-Markov and weak-coupling approximations. No quantitative check is given that the bath correlation time remains ≪ system evolution time at the low temperatures where the Clausius violation is reported; the correlation time scales as ħ/k_B T and the Markov condition therefore fails precisely in the regime of interest.
  2. [§4] §4 (Results and discussion): The entropy-production violation is computed directly from the Lindblad dynamics; because the master equation itself is uncontrolled at low T, any apparent violation could be an artifact of the approximation rather than a genuine effect of coherence. An independent non-Markovian calculation (HEOM, exact path integral, or non-Markovian master equation) is not reported.
  3. [§3] §3 (Thermodynamic quantities): The definition of entropy production used to test the Clausius inequality is tied to the same Lindblad generator; the paper does not demonstrate that the reported violation persists when the generator is replaced by a non-Markovian equivalent while keeping the coherence measure fixed.
minor comments (1)
  1. [§2] Notation for the partial coupling strength and the precise form of the system-bath interaction Hamiltonian should be stated explicitly in §2 rather than left implicit.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and the constructive comments, which correctly identify the central role of the Markov approximation in our analysis. We address each major comment below. We have revised the manuscript to add explicit discussion of the approximation's validity range and to qualify the scope of our claims, but we cannot supply a non-Markovian benchmark calculation.

read point-by-point responses

  1. Referee: §2 (Model): The Brownian-type Lindblad master equation is obtained under the Born-Markov and weak-coupling approximations. No quantitative check is given that the bath correlation time remains ≪ system evolution time at the low temperatures where the Clausius violation is reported; the correlation time scales as ħ/k_B T and the Markov condition therefore fails precisely in the regime of interest.

    Authors: We agree that the bath correlation time lengthens at low T. In the revised manuscript we have added a new paragraph in §2 that supplies the requested quantitative estimate: with the weak-coupling strength kept fixed (λ ≪ ω), the system relaxation timescale remains longer than the bath correlation time down to the temperatures at which the violation appears. This estimate is obtained directly from the parameters already used in the numerical simulations. revision: yes

  2. Referee: §4 (Results and discussion): The entropy-production violation is computed directly from the Lindblad dynamics; because the master equation itself is uncontrolled at low T, any apparent violation could be an artifact of the approximation rather than a genuine effect of coherence. An independent non-Markovian calculation (HEOM, exact path integral, or non-Markovian master equation) is not reported.

    Authors: We acknowledge that the reported violation is obtained inside the Lindblad framework and that non-Markovian corrections could modify the entropy-production balance. Our contribution is to exhibit the effect within this standard, widely employed model and to link it to the coherence term. We have added an explicit caveat in §4 and the conclusions stating that the result should be understood as a property of the Markovian dynamics and that non-Markovian extensions constitute important future work. revision: partial

  3. Referee: §3 (Thermodynamic quantities): The definition of entropy production used to test the Clausius inequality is tied to the same Lindblad generator; the paper does not demonstrate that the reported violation persists when the generator is replaced by a non-Markovian equivalent while keeping the coherence measure fixed.

    Authors: The entropy-production functional is defined consistently with the generator of the dynamics, following the standard construction for Markovian open systems. Replacing the generator with a non-Markovian one would also require a correspondingly generalized entropy-production expression, an issue that remains under active investigation. We have clarified in the revised text that our claim is restricted to the Lindblad case and that persistence under non-Markovian dynamics is an open question. revision: partial

standing simulated objections not resolved
  • Performing an independent non-Markovian calculation (HEOM, exact path integral, or non-Markovian master equation) to benchmark the low-temperature regime.

Circularity Check

0 steps flagged

No circularity: derivation uses external master equation without self-referential reduction

full rationale

The paper applies the Brownian-type Lindblad master equation to compute entropy production and coherence for a harmonic oscillator coupled to a thermal bath. Thermodynamic quantities are obtained by direct integration of the resulting dynamical equations at varying temperatures; no parameter is fitted to a data subset and then relabeled as a prediction, no quantity is defined in terms of itself, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The reported Clausius violation therefore emerges from the explicit time evolution under the chosen equation rather than by algebraic rearrangement of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the applicability of the Lindblad master equation to the interferometer setup and on the standard definition of entropy production in open quantum systems; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Brownian-type Lindblad master equation governs the partially coupled harmonic oscillator dynamics
    Invoked to model interaction with the thermal bath of oscillatory fields.

pith-pipeline@v0.9.0 · 5611 in / 1105 out tokens · 42022 ms · 2026-05-25T15:55:46.902365+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages · 2 internal anchors

  1. [1]

    Higher amounts of Ω /T cause more violation from the Clausius inequality

    Positive F ensures us that the system satisfies the inequality. Higher amounts of Ω /T cause more violation from the Clausius inequality. (b) The violation of Clausius inequality at different temperatures and |C2|2 with Ω = 1× 1012s−1, while the decoherence process is complete. Low temperatures and more transparency of the beamsplitter BS1 cause stronger vi...

    work page

  2. [2]

    Goldstein and F

    M. Goldstein and F. I. Goldstein, The Refrigerator and the Universe , Harvard University Press, Cambridge (1993)

    work page 1993

  3. [3]

    N. Liu, J. Goold, I. Fuentes, V. Vedral, K. Modi, D.E. Bruschi, Class. Quantum Grav. 33, 035003 (2016)

    work page 2016

  4. [4]

    Gemmer, M

    J. Gemmer, M. Michel and G. Mahler, Quantum Ther- modynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems , Lect. Notes Phys. 784, Springer, Berlin Heidelberg (2009)

    work page 2009

  5. [5]

    Goold, M

    J. Goold, M. Huber, A. Riera, L. del Rio and P. Skrzypczyk, J. Phys. A: Math. and Theo. 49, 143001 (2016)

    work page 2016

  6. [6]

    Vinjanampathy and J

    S. Vinjanampathy and J. Anders, Cont. Phys. 57, 545- 579 (2016)

    work page 2016

  7. [7]

    F. G. Brandao, M. Horodecki, J. Oppenheim, J. M. Renes and R. W. Spekkens, Phys. Rev. Lett.111, 250404 (2013)

    work page 2013

  8. [8]

    Binder, S

    F. Binder, S. Vinjanampathy, K. Modi and J. Goold, Phys. Rev. E 91, 032119 (2015)

    work page 2015

  9. [9]

    Deutsch, New J

    J. Deutsch, New J. Phys. 12, 075021 (2010)

    work page 2010

  10. [10]

    ÄĘwikliÅĎski, M

    P. ÄĘwikliÅĎski, M. StudziÅĎski, M. Horodecki, J. Op- penheim, Phys. rev. lett. 115, 210403 (2015)

    work page 2015

  11. [11]

    Popescu, A

    S. Popescu, A. J. Short and A. Winter, Nat. Phys. 2, 754-758 (2006)

    work page 2006

  12. [12]

    del Rio, J

    L. del Rio, J. Aberg, R. Renner, O. Dahlsten and V. Vedral, Nature 474, 61-63 (2011)

    work page 2011

  13. [13]

    Anders and V

    J. Anders and V. Giovannetti, New J. of Phys.15, 033022 (2013)

    work page 2013

  14. [14]

    T. D. Kieu, Phys. rev. lett. 93, 140403 (2004)

    work page 2004

  15. [15]

    Allahverdyan and T

    A. Allahverdyan and T. M. Nieuwenhuizen, Phys. Rev. E 71, 066102 (2005)

    work page 2005

  16. [16]

    Esposito and S

    M. Esposito and S. Mukamel, Phys. Rev. E 73, 046129 (2006)

    work page 2006

  17. [17]

    L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, Pergamon, London (1980)

    work page 1980

  18. [18]

    A. E. Allahverdyan and T. M. Nieuwenhuizen, Phys. Rev. Lett. 85, 1799 (2000)

    work page 2000

  19. [19]

    Yu. L. Klimontovich, Statistical Theory of Open Systems , Kluwer, Amsterdam (1997)

    work page 1997

  20. [20]

    Evans, E

    D .J. Evans, E. Cohen and G. Morriss, Phys. Rev. Lett. 71, 2401 (1993)

    work page 1993

  21. [21]

    Sagawa and M

    T. Sagawa and M. Ueda, Phys. rev. lett. 100, 080403 (2008)

    work page 2008

  22. [22]

    Jacobs, Phys

    K. Jacobs, Phys. Rev. E 86, 040106 (2012)

    work page 2012

  23. [23]

    S. R. De Groot and P. Mazur, Non-equilibrium thermo- dynamics, Courier Corporation (2013)

    work page 2013

  24. [24]

    Gogolin and J

    C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quan- tum systems , Rep. Prog. Phys. 79, 056001 (2016)

    work page 2016

  25. [25]

    T. M. Nieuwenhuizen and A. E. Allahverdyan, Phys. Rev. E 66, 036102 (2002)

    work page 2002

  26. [26]

    M. J. Henrich, G. Mahler and M. Michel, Phys. Rev. E 75, 051118 (2007)

    work page 2007

  27. [27]

    Weimer, M

    H. Weimer, M. J. Henrich, F. Rempp, H. SchrÃűder and G. Mahler, Europhys. Lett. 83, 30008 (2008)

    work page 2008

  28. [28]

    Esposito, K

    M. Esposito, K. Lindenberg and C. Van den Broeck, New J. Phys. 12, 013013 (2010)

    work page 2010

  29. [29]

    Esposito and C

    M. Esposito and C. Van den Broeck, Europhys. Lett. 95, 40004 (2011)

    work page 2011

  30. [30]

    Gemmer and R

    J. Gemmer and R. Steinigeweg, Phys. Rev. E 89, 042113 (2014)

    work page 2014

  31. [31]

    Wilming, R

    H. Wilming, R. Gallego and J. Eisert, Phys. Rev. E 93, 042126 (2016)

    work page 2016

  32. [32]

    J. P. Santos, G. T. Landi and M. Paternostro, Phys. Rev. Lett. 118, 220601 (2017)

    work page 2017

  33. [33]

    J. M. Deutsch, Phys. Rev. A 43, 2046 (1991)

    work page 2046

  34. [34]

    Rigol, V

    M. Rigol, V. Dunjko and M. Olshanii, Nature 452, 854 (2008)

    work page 2008

  35. [35]

    T. M. Cover and J. A. Thomas, Elements of information theory, John Wiley and sons (2006)

    work page 2006

  36. [36]

    Linden, S

    N. Linden, S. Popescu, A. J. Short and A. Winter, Phys. Rev. E 79, 061103 (2009)

    work page 2009

  37. [37]

    A. J. Short and T. C. Farrelly, New J. Phys. 14, 013063 (2012)

    work page 2012

  38. [38]

    Riera, C

    A. Riera, C. Gogolin and J. Eisert, Phys. Rev. Lett. 108, 08040 (2012)

    work page 2012

  39. [39]

    Horodecki and J

    M. Horodecki and J. Oppenheim, Int. J.Mod. Phys. 27, 1345019 (2012)

    work page 2012

  40. [40]

    Plenio and V

    M.B. Plenio and V. Vitelli, Contemp. Phys. 42, 25-60 (2001)

    work page 2001

  41. [41]

    Parrondo, J.M

    J.M. Parrondo, J.M. Horowitz and T. Sagawa, Nat. Phys. 11, 131-139 (2015)

    work page 2015

  42. [42]

    Hilt and E

    S. Hilt and E. Lutz, Phys. Rev. A 79, 010101 (2009)

    work page 2009

  43. [43]

    H¨ orhammer and H

    C. H¨ orhammer and H. B¨ uttner, J. Stat. Phys.133, 1161- 1174 (2008)

    work page 2008

  44. [44]

    Lostaglio, D

    M. Lostaglio, D. Jennings and T. Rudolph, Nat. commun. 6, 6383 (2015)

    work page 2015

  45. [45]

    Lostaglio, K

    M. Lostaglio, K. Korzekwa, D. Jennings and T. Rudolph, Phys. Rev. X 5, 021001 (2015)

    work page 2015

  46. [46]

    Kosloff and A

    R. Kosloff and A. Levy, Annu. Rev. Phys. Chem. 65, 365 (2014)

    work page 2014

  47. [47]

    Gelbwaser-Klimovsky, W

    D. Gelbwaser-Klimovsky, W. Niedenzu and G. Kurizki, Adv. At., Mol., Opt. Phys. 64, 329 (2015)

    work page 2015

  48. [48]

    H. T. Quan, Y. Liu, C. P. Sun and F. Nori, Phys. Rev. E 76, 031105 (2007)

    work page 2007

  49. [49]

    Roulet, S

    A. Roulet, S. Nimmrichter, J. M. Arrazola, S. Seah and V. Scarani, Phys. Rev. E 95 062131 (2017)

    work page 2017

  50. [50]

    Levy and R

    A. Levy and R. Kosloff, Phys. Rev. Lett. 108, 070604 (2012)

    work page 2012

  51. [51]

    M. T. Mitchison, M. Huber, J. Prior, M. P. Woods and M. B. Plenio, Quantum Sci. Technol. 1, 015001 (2016)

    work page 2016

  52. [52]

    Boyanovsky and D

    D. Boyanovsky and D. Jasnow, Phys. Rev. A 96, 012103 (2017)

    work page 2017

  53. [53]

    Erker P, M

    P. Erker P, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner and M. Huber, Phys. Rev. X 7, 031022 (2017)

    work page 2017

  54. [54]

    P. P. Hofer, M. Perarnau-Llobet, L. D. M. Miranda, G. Haack, R. Silva, J. B. Brask and N. Brunner, New J. Phys. 19, 123037 (2017)

    work page 2017

  55. [55]

    Maniscalco, J

    S. Maniscalco, J. Piilo, F. Intravaia, F. Petruccione and A. Messina, Phys. Rev. A 70, 032113 (2004)

    work page 2004

  56. [56]

    Gorini, A

    V. Gorini, A. Kossakowski, and E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976)

    work page 1976

  57. [57]

    H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems Oxford University Press (2002)

    work page 2002

  58. [58]

    Maniscalco, F

    S. Maniscalco, F. Intravia, J. Piilo, and A. Messina, J. Opt. B: Quantum Semiclassical Opt. 6, S98 (2004)

    work page 2004

  59. [59]

    M. A. Schlosshauer, Decoherence: and the Quantum-to- Classical Transition, Springer Science & Business Media (2007)

    work page 2007

  60. [60]

    Quantum Decoherence in System-Bath Interferometry

    A. Soltanmanesh and A. Shafiee, arXiv preprint arXiv:1802.07468 (2018). 10

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  61. [61]

    E. H. Lieb and M. B. Ruskai, Phys. Rev. Lett. 30, 434 (1973)

    work page 1973

  62. [62]

    Wehrl, Rev

    A. Wehrl, Rev. Mod. Phys. 50, 221 (1978)

    work page 1978

  63. [63]

    Quantifying Superposition

    J. ÃĚberg, arXiv:quant-ph/0612146 (2006)

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  64. [64]

    Baumgratz, M

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

    work page 2014

  65. [65]

    Streltsov, G

    A. Streltsov, G. Adesso and M.B. Plenio, Rev. Mod. Phys. 89, 041003 (2017)

    work page 2017

  66. [66]

    Winter and D

    A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)

    work page 2016

  67. [67]

    X. Yuan, H. Zhou, Z. Cao, and X. Ma, Phys. Rev. A 92, 022124 (2015)

    work page 2015

  68. [68]

    Streltsov, Quantum Correlations Beyond Entangle- ment and their Role in Quantum Information Theory , Springer Science & Business Media (2015)

    A. Streltsov, Quantum Correlations Beyond Entangle- ment and their Role in Quantum Information Theory , Springer Science & Business Media (2015)

    work page 2015

  69. [69]

    Allahverdyan, R

    A. Allahverdyan, R. Balian, and T. Nieuwenhuizen, Eu- rophys. Lett. 67, 565 (2004)

    work page 2004

  70. [70]

    Pusz and S

    W. Pusz and S. L. Woronowicz, Commun. Math. Phys. 58, 273 (1978)

    work page 1978

  71. [71]

    Perarnau-Llobet, K

    M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, J. Tura and A. Acin, Phys. Rev. E 92, 042147 (2015)

    work page 2015