Experimental investigation of coherence contributions to a nonequilibrium thermodynamic process in a driven quantum system

Arvind; Kavita Dorai; Krishna Shende

REVIEW 4 major objections 4 minor 40 references

Experimental investigation of coherence contributions to a nonequilibrium thermodynamic process in a driven quantum system

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read In a two-qubit NMR processor, irreversible entropy from unitary driving is shown to split into coherence and population-mismatch parts, with the total always exceeding a Bures-length lower bound.

desk verdict Real NMR data on the coherence/population split of irreversible entropy, but the two-qubit drive behind the theoretical curves is under-specified and the 'verification' claim is stronger than the evidence. read the letter →

arxiv 2411.17952 v1 pith:OGP7J43O submitted 2024-11-27 quant-ph cond-mat.stat-mech

Krishna Shende , Kavita Dorai , Arvind This is my paper · ORCID

classification quant-phcond-mat.stat-mech PACS 03.65.-w03.67.-a05.70.Ln

keywords irreversibleentropyproductionquantumcoherencegeneralizedClausiusinequalityBureslengthNMRprocessornonequilibriumthermodynamicsunitarydrivingstatetomography

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 reports an experiment on a two-qubit NMR processor in which a thermal equilibrium state is driven out of equilibrium by a time-dependent unitary Hamiltonian, with driving times from 100 to 800 microseconds. The authors measure the irreversible entropy produced in the process and show that it splits into two positive parts: one from quantum coherence created during the drive and one from a population mismatch between the actual final state and the target equilibrium state. They also verify a generalized Clausius inequality: in every run, the irreversible entropy remains above a nonzero lower bound set by the Bures length between the actual and equilibrium states. The result matters because it tests a thermodynamic bound that is much tighter than the classical statement that entropy production is merely nonnegative, on actual quantum hardware rather than in theory alone.

What carries the argument

The argument rests on two identities. First, the irreversible entropy $\Delta S_{\mathrm{irr}} = D(\rho_\tau \| \rho_f)$ is decomposed as $C(\rho_\tau) + D(\Delta_\tau[\rho_\tau] \| \rho_B)$, where $C(\rho_\tau) = S(\Delta_\tau[\rho_\tau]) - S(\rho_\tau)$ is the relative entropy of coherence of the actual state and the second term is the population mismatch with the target equilibrium state. Second, the generalized Clausius inequality bounds this entropy from below by $(8/\pi^2) L^2(\rho_\tau, \rho_f)$, where $L(\rho_\tau, \rho_f) = \arccos \sqrt{F(\rho_\tau, \rho_f)}$ is the Bures length built from the fidelity $F$. Experimentally, the machinery is an NMR processor that prepares a Gibbs state at fixed pseudospin temperature, applies a GRAPE-optimized unitary drive with time-dependent energy gap, and reconstructs the final density matrix by quantum state tomography so that all three quantities can be computed and compared.

What would settle it

Repeat the protocol with drive durations comparable to the qubits' decoherence time, or add a variable delay after the drive before tomography: the unitary prediction of monotonically decreasing entropy with $\tau$ will break down, and the bound would be falsified if the measured $\Delta S_{\mathrm{irr}}$ falls below $(8/\pi^2)L^2(\rho_\tau, \rho_f)$ within experimental uncertainty.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the unitary-driving protocol on 13C-labeled glycine qubits produces entropy production $\Delta S_{\mathrm{irr}} = D(\rho_\tau \| \rho_f)$ whose two contributions, coherence generation $C(\rho_\tau)$ and population mismatch $D(\Delta_\tau[\rho_\tau] \| \rho_B)$, follow the predicted decomposition (Eq. 3), and that the measured values satisfy the generalized Clausius inequality $\Delta S_{\mathrm{irr}} \geq (8/\pi^2) L^2(\rho_\tau, \rho_f)$ (Eq. 7) for both final energy gaps studied. The data show that entropy production decreases as driving time increases toward the quasi-static limit, that the coherence contribution tracks the total and becomes almost negligible at long times, and that driving farther from equilibrium produces more total entropy but a smaller share from coherence. Experimental points lie slightly above the theoretical curves, which the authors attribute to pulse calibration, state preparation, field inhomogeneities, and GRAPE optimization errors, all of which add extra entropy.

Load-bearing premise

The comparison to the unitary model assumes the drive is effectively unitary because 100 to 800 microseconds is far shorter than the qubits' decoherence times, so any dissipative error is neglected.

Editorial extensions

If this is right

Longer driving times monotonically reduce irreversible entropy production, so the quasi-static limit is approached from above as the control parameter is changed more slowly.
The coherence contribution to entropy production follows the same trend as the total, so reducing coherence generation is a practical route to lowering irreversibility in driven quantum processes.
Driving the system farther from equilibrium raises total entropy production but lowers the fraction of it that comes from coherence; population mismatch then dominates.
The Bures-length lower bound holds for both final gaps examined and for every driving time, making it a valid tighter alternative to the plain Clausius inequality in these unitary-driven processes.

Reading between the lines

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

A testable extension the authors leave implicit: deliberately adding pulse noise or using non-optimal drives should increase the gap between $\Delta S_{\mathrm{irr}}$ and its Bures-length bound while leaving the inequality intact, which would probe how tight the bound is.
If the decomposition of Eq. (3) is generic, control strategies that target coherence suppression separately from population targeting could be used to design low-dissipation unitary strokes for quantum engines.
Extending the same measurement to open-system settings, where a bath is present during the drive, would require modifying the second term to include bath-induced transitions; whether the Bures bound survives that generalization is not addressed by this experiment.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read this one if you want a concrete NMR check of the Francica/Santos decomposition. The experiment is straightforward: pseudopure state on 13C-glycine, thermal initialization, GRAPE pulses that drive the energy gap from 2000 Hz to 3600/5000 Hz, tomography, and plots of ΔSirr, C(ρτ), and 8L²/π² versus driving time. The data track the theoretical curves, the bound is satisfied at every point, and the data points sitting slightly above the curves are consistent with the stated near-unitary assumption plus residual decoherence. That is genuine, useful experimental evidence for the quantum thermodynamics of small systems.

What is new is limited but real: earlier NMR work measured irreversible entropy and work in similar driven qubits; this paper explicitly separates the coherence contribution and reports it as a function of driving time for two final gaps. The theory is not new (Deffner-Lutz, Francica et al., Santos et al.), and Eq. (3) is an identity for any state relative to a diagonal reference, so matching the two sides is mostly a consistency check. The nontrivial comparison is the one against the unitary model.

Soft spots, in proportion: - The stress-test concern is legitimate. Eq. (13) is a single-qubit Hamiltonian, but the experiment uses two coupled 13C spins with J-coupling. The paper never states whether the drive is applied to both spins, whether the coupling is refocused, or whether Eq. (13) is an effective single-qubit description. Without that, the unitary in Eq. (14) and the solid curves in Figs. 4–5 are not uniquely defined. This is the main reason the verification is incomplete as written. - Error bars in Figs. 4–5 have no stated methodology. Not fatal, but a referee should ask. - The symbol τ is used both for the fixed 1/(2J) delay in the pulse sequence and for the variable driving time. Confusing, likely a typo. - “Verified a generalized Clausius inequality” is an overclaim. Better: observed data consistent with the bound for two protocols.

The citation pattern is fine: prior theory and experiments are credited, and the tomography refs are legitimate. The central claim is plausible and probably correct. It needs the Hamiltonian specification and error analysis before publication.

This is a paper for experimentalists in quantum thermodynamics and NMR quantum information. It deserves a serious referee; the verdict should be conditional on the drive Hamiltonian being fully specified and the error-bar method stated.

Referee Report

4 major / 4 minor

Summary. The paper reports an NMR experiment on a two-qubit 13C-glycine system in which the system is prepared at a thermal state and then driven out of equilibrium by a time-dependent Hamiltonian. The authors measure the irreversible entropy production ΔS_irr, the relative-entropy coherence C(ρτ), and the Bures-length lower bound 8L²/π² for two final energy gaps (3600 Hz and 5000 Hz) as functions of driving time. They observe that ΔS_irr decreases as driving time increases, that C(ρτ) follows the same trend, and that ΔS_irr ≥ 8L²/π² for the tested times, which they interpret as experimental verification of a generalized Clausius inequality.

Significance. If the central comparison is made fully specified, the experiment would provide a useful demonstration of how coherence contributes to irreversible entropy production and of the Bures-length bound in a controlled quantum platform. The paper benefits from explicit state tomography and from computing the coherence and entropy-production terms from the same reconstructed density operator. However, the current under-specification of the two-qubit drive Hamiltonian, the missing error-bar methodology, and the unquantified unitary assumption prevent the central claims from being fully supported.

major comments (4)

[Section III, Eq. (13)] The theoretical curves in Figs. 4 and 5 are generated from the driving Hamiltonian H(ν(t)) in Eq. (13), which is written as a single-qubit Hamiltonian acting on σ_x and σ_y. The experiment, however, is performed on two coupled 13C spins whose internal Hamiltonian in Eq. (10) includes the scalar-coupling term ℏJij I_z^i I_z^j. The text never specifies how Eq. (13) acts on the two qubits: whether the same drive is applied independently to each spin, whether the J-coupling is included in or refocused out of the GRAPE pulse, or whether Eq. (13) is an effective single-qubit model after tracing out the second spin. Consequently the unitary U_{τ,0} in Eq. (14) is not uniquely defined and the theoretical predictions for ΔS_irr, C(ρτ), and 8L²/π² in Figs. 4 and 5 cannot be reproduced from the information given. This under-specification affects the central claim that the data verify the generalized Clausius inequality, because the comparison is to curves whose generating Hamiltonian is ambiguous. Please specify the full two-qubit drive, including how the J-coupling is treated, and state explicitly how the computed observables are obtained from the two-qubit state.
[Section IV, Figs. 4 and 5] No methodology is given for the experimental error bars shown in Figs. 4 and 5. The reader cannot tell whether the error bars are standard deviations over repeated experiments, uncertainties propagated from quantum state tomography, or instrumental estimates. Because the quantitative agreement between the measured points and the theoretical curves is part of the evidence for the paper's claims, the error-bar definition and propagation procedure must be stated.
[Section III, unitary assumption] The assumption that the driving is effectively unitary is stated in Section III ('This driving time is much less than the decoherence times...') but no numerical values for T1, T2, or T2* are given for the 13C spins under the experimental conditions, and the assumption is not tested, for example by checking the purity of the reconstructed states against the expected unitary evolution. Since decoherence and relaxation would add incoherent contributions to the measured irreversible entropy, the clean comparison to the unitary theoretical model requires either quantitative support for the assumption or a discussion of how such contributions were identified and excluded.
[Abstract and Conclusions] The abstract and conclusions state that the experiment 'verified' the generalized Clausius inequality. Since Eq. (7) is a mathematical inequality derived from the formalism, the experiment cannot verify it in the sense of testing a conjectured physical law; the data can at most show that the reconstructed states satisfy the bound, which is an internal consistency check. The stronger and more informative experimental claims are the τ-dependence of ΔS_irr and C(ρτ) and their comparison to the unitary model. Please reframe the 'verification' language accordingly.

minor comments (4)

[Section IV, text after Figs. 4 and 5] The sentence 'The irreversible entropy produced is more for a final energy gap of 5000 Hz (Figure 4) as compared to a final energy gap of 3600 Hz (Figure 5)' has the figure numbers reversed; Fig. 4 corresponds to 3600 Hz and Fig. 5 to 5000 Hz.
[References, Eq. (7)] The text attributes the inequality in Eq. (7) to 'Bures [21]' and calls it a 'Bures length inequality,' but Eq. (7) as a generalized Clausius inequality is due to Deffner and Lutz (Ref. [3]); Ref. [21] is Braunstein and Caves. Please correct the attribution.
[Equation (3)] The notation D(∆τ[ρτ || ρB]) in Eq. (3) is ambiguous and ρB is undefined. If the intended expression is D(∆τ[ρτ] || ρB), define ρB and the dephasing map Δτ explicitly, since this determines the decomposition.
[Equations (9) and (15)] The symbol F is used both for Uhlmann fidelity in Eq. (9) and for the Hilbert-Schmidt inner-product fidelity in Eq. (15); use distinct notation or state explicitly that Eq. (15) is not the Uhlmann fidelity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central experiment-theory comparison uses independently computed unitary curves with no fitted parameters.

full rationale

The paper's central claim is an experiment-theory comparison: measured irreversible entropy, coherence, and the Bures-length bound from tomographically reconstructed states are compared with theoretical curves computed from the stated driving Hamiltonian in Eq. (13), with the initial temperature and frequency offsets fixed by the experimental setup rather than fitted to the data. The decomposition in Eq. (3) is an exact relative-entropy identity for a dephasing map, and the paper itself says the positivity is 'by construction'; it is used as an analysis tool, not as a falsifiable prediction or as a fitted input. The Bures-length inequality in Eq. (7) is an external mathematical theorem from Ref. [3]; experimentally checking that measured states satisfy it is a consistency demonstration rather than a prediction derived from the data. The only self-citations are Refs. [38,39] for standard quantum state tomography methods, which are methodological and not load-bearing. The under-specification of how Eq. (13) acts on the two coupled 13C qubits, and the role of the J-coupling term in Eq. (10), is a reproducibility concern but does not constitute circularity. Overall, no circular step reduces a claimed prediction to a fit or to a self-citation chain.

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

No fitted parameters were used; all theoretical curves are computed from the stated experimental settings. The main assumptions are standard quantum thermodynamics results plus domain-specific assumptions about unitarity and the NMR pseudopure state representation. No new entities are introduced.

assumptions (6)

standard math The irreversible entropy of a unitary process equals the relative entropy D(rho_tau || rho_f) (Eq. 2).
This is a standard result from Deffner and Lutz (Ref. [3]) and is used as the definition of irreversible entropy.
standard math The decomposition Delta S_irr = C(rho_tau) + D(Delta_tau[rho_tau] || rho_B) (Eq. 3) holds for the unitary driving, with rho_f diagonal in the dephasing basis.
This is from Francica et al. (Ref. [5]) and Santos et al. (Ref. [6]). It requires the dephasing map to be in the eigenbasis of the final thermal state, which the paper assumes is the computational basis.
standard math The Bures length inequality Delta S_irr >= (8/pi^2) L^2 (Eq. 7) holds.
Proven by Deffner and Lutz (Ref. [3]); the paper treats it as a theoretical benchmark to test experimentally.
domain assumption The external driving is effectively unitary over the 100-800 microsecond time scale.
Stated in Section III: the driving time is much less than the decoherence times, so decoherence is neglected. This is load-bearing because any extra incoherent contribution would break the clean unitary model.
domain assumption The NMR pseudopure state with polarization epsilon about 1e-5 faithfully represents the desired Gibbs state for entropy computations.
The paper prepares pseudopure states (Eq. 11) and then applies rotations and gradients to create a Gibbs state. It does not describe how the identity part of the density matrix is subtracted or handled in the relative entropy calculations, which is a standard but nontrivial step in NMR.
domain assumption The final equilibrium state rho_f is the thermal state at inverse temperature beta_i with the final energy gap nu_f.
Used implicitly when computing D(rho_tau || rho_f) and the theoretical curves. The paper does not write out the operator form of H_f, but assumes it is the diagonal Hamiltonian with gap nu_f.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Experimental investigation of coherence contributions to a nonequilibrium thermodynamic process in a driven quantum system." pith.science (2026). https://pith.science/paper/OGP7J43O

@misc{pith2026241117952,
  author       = {Pith},
  title        = {Pith review of: Experimental investigation of coherence contributions to a nonequilibrium thermodynamic process in a driven quantum system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGP7J43O}},
  note         = {Machine review of arXiv:2411.17952}
}

read the original abstract

The work done when a system at thermal equilibrium is externally driven by a unitary control parameter leads to irreversible entropy production. The entropy produced can be thought of as a combination of coherence generation and a population mismatch between the target equilibrium state and the actually achieved final state. We experimentally explored this out-of-equilibrium process in an NMR quantum processor and studied the contribution of coherence to irreversible entropy generation. We verified a generalized Clausius inequality, which affirms that irreversible entropy production is lower-bounded.

Figures

Figures reproduced from arXiv: 2411.17952 by the authors.

**Figure 2.** FIG. 2. (Color online) Molecular structure of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. (Color online) Dynamics of entropy (∆ [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: FIG. 5. (Color online) Same as Figure 4, but with the energy [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 18 canonical work pages

[1]

Kawai, J

R. Kawai, J. M. R. Parrondo, and C. V. den Broeck, Phys. Rev. Lett. 98, 080602 (2007)

work page 2007
[2]

J. M. R. Parrondo, C. V. den Broeck, and R. Kawai, New Journal of Physics 11, 073008 (2009)

work page 2009
[3]

Deffner and E

S. Deffner and E. Lutz, Phys. Rev. Lett. 105, 170402 (2010)

2010
[4]

Plastina, A

F. Plastina, A. Alecce, T. J. G. Apollaro, G. Falcone, G. Francica, F. Galve, N. Lo Gullo, and R. Zambrini, Phys. Rev. Lett. 113, 260601 (2014)

2014
[5]

Francica, J

G. Francica, J. Goold, and F. Plastina, Phys. Rev. E 99, 042105 (2019)

2019
[6]

J. P. Santos, L. C. C´ eleri, G. T. Landi, and M. Pater- nostro, npj Quantum Information 5, 23 (2019)

work page 2019
[7]

J. V. Koski, T. Sagawa, O.-P. Saira, Y. Yoon, A. Kutvo- nen, P. Solinas, M. M¨ ott¨ onen, T. Ala-Nissila, and J. P. Pekola, Nature Physics 9, 644 (2013)

work page 2013
[8]

T. B. Batalh˜ ao, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Phys. Rev. Lett. 113, 140601 (2014)

work page 2014

Show all 40 references

[9]

T. B. Batalh˜ ao, A. M. Souza, R. S. Sarthour, I. S. Oliveira, M. Paternostro, E. Lutz, and R. M. Serra, Phys. Rev. Lett. 115, 190601 (2015)

2015
[10]

An, J.-N

S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.- Q. Yin, H. T. Quan, and K. Kim, Nature Physics 11, 193 (2015)

2015
[11]

Smith, Y

A. Smith, Y. Lu, S. An, X. Zhang, J.-N. Zhang, Z. Gong, H. T. Quan, C. Jarzynski, and K. Kim, New Journal of Physics 20, 013008 (2018)

2018
[12]

J. P. S. Peterson, T. B. Batalh˜ ao, M. Herrera, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra, Phys. Rev. Lett. 123, 240601 (2019)

2019
[13]

R. J. de Assis, T. M. de Mendon¸ ca, C. J. Villas-Boas, A. M. de Souza, R. S. Sarthour, I. S. Oliveira, and N. G. de Almeida, Phys. Rev. Lett. 122, 240602 (2019)

2019
[14]

Solfanelli, A

A. Solfanelli, A. Santini, and M. Campisi, PRX Quan- tum 2, 030353 (2021)

2021
[15]

Klatzow, J

J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl, K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walm- sley, R. Uzdin, and E. Poem, Phys. Rev. Lett. 122, 110601 (2019)

2019
[16]

Roßnagel, S

J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, Science 352, 325 (2016)

2016
[17]

Maslennikov, S

G. Maslennikov, S. Ding, R. Habl¨ utzel, J. Gan, A. Roulet, S. Nimmrichter, J. Dai, V. Scarani, and D. Matsukevich, Nature Communications10, 202 (2019)

2019
[18]

Baumgratz, M

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

2014
[19]

Streltsov, G

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

2017
[20]

Kakutani, Annals of Mathematics 49, 214 (1948)

S. Kakutani, Annals of Mathematics 49, 214 (1948). 6

1948
[21]

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994)

1994
[22]

Deffner and S

S. Deffner and S. Campbell, Quantum Thermodynamics, 2053-2571 (Morgan & Claypool Publishers, 2019)

2019
[23]

Jarzynski, Phys

C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)

1997
[24]

Umegaki, Kodai Mathematical Seminar Reports 14, 59 (1962)

H. Umegaki, Kodai Mathematical Seminar Reports 14, 59 (1962)

1962
[25]

Irreversible thermody- namics for quantum systems weakly coupled to thermal reservoirs,

H. Spohn and J. L. Lebowitz, “Irreversible thermody- namics for quantum systems weakly coupled to thermal reservoirs,” in Advances in Chemical Physics (1978) pp. 109–142

1978
[26]

Breuer, Phys

H.-P. Breuer, Phys. Rev. A 68, 032105 (2003)

2003
[27]

Salamon and R

P. Salamon and R. S. Berry, Phys. Rev. Lett. 51, 1127 (1983)

1983
[28]

Salamon, J

P. Salamon, J. D. Nulton, and R. S. Berry, The Journal of Chemical Physics 82, 2433 (1985)

1985
[29]

Nulton, P

J. Nulton, P. Salamon, B. Andresen, and Q. Anmin, The Journal of Chemical Physics 83, 334 (1985)

1985
[30]

Ruppeiner, Rev

G. Ruppeiner, Rev. Mod. Phys. 67, 605 (1995)

1995
[31]

G. E. Crooks, Phys. Rev. Lett. 99, 100602 (2007)

2007
[32]

W. K. Wootters, Phys. Rev. D 23, 357 (1981)

1981
[33]

Uhlmann, Reports on Mathematical Physics 9, 273 (1976)

A. Uhlmann, Reports on Mathematical Physics 9, 273 (1976)

1976
[34]

Jozsa, Journal of Modern Optics 41, 2315 (1994)

R. Jozsa, Journal of Modern Optics 41, 2315 (1994)

1994
[35]

I. S. Oliveira, T. J. Bonagamba, R. S. Sarthour, J. C. Freitas, and E. R. deAzevedo, in NMR Quantum Information Processing, edited by I. S. Oliveira, T. J. Bonagamba, R. S. Sarthour, J. C. Freitas, and E. R. deAzevedo (Elsevier Science B.V., Amsterdam, 2007) pp. 137–181

2007
[36]

D. G. Cory, M. D. Price, and T. F. Havel, Physica D: Nonlinear Phenomena 120, 82 (1998), proceedings of the Fourth Workshop on Physics and Consumption

1998
[37]

Khaneja, T

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨ uggen, and S. J. Glaser, Journal of Magnetic Resonance 172, 296 (2005)

2005
[38]

Gaikwad, Arvind, and K

A. Gaikwad, Arvind, and K. Dorai, Quantum Informa- tion Processing 20, 19 (2021)

2021
[39]

Gaikwad, Arvind, and K

A. Gaikwad, Arvind, and K. Dorai, Quantum Informa- tion Processing 21, 388 (2022)

2022
[40]

Zhang, A

J. Zhang, A. M. Souza, F. D. Brandao, and D. Suter, Phys. Rev. Lett. 112, 050502 (2014)

2014

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Kawai, J

R. Kawai, J. M. R. Parrondo, and C. V. den Broeck, Phys. Rev. Lett. 98, 080602 (2007)

work page 2007

[2] [2]

J. M. R. Parrondo, C. V. den Broeck, and R. Kawai, New Journal of Physics 11, 073008 (2009)

work page 2009

[3] [3]

Deffner and E

S. Deffner and E. Lutz, Phys. Rev. Lett. 105, 170402 (2010)

2010

[4] [4]

Plastina, A

F. Plastina, A. Alecce, T. J. G. Apollaro, G. Falcone, G. Francica, F. Galve, N. Lo Gullo, and R. Zambrini, Phys. Rev. Lett. 113, 260601 (2014)

2014

[5] [5]

Francica, J

G. Francica, J. Goold, and F. Plastina, Phys. Rev. E 99, 042105 (2019)

2019

[6] [6]

J. P. Santos, L. C. C´ eleri, G. T. Landi, and M. Pater- nostro, npj Quantum Information 5, 23 (2019)

work page 2019

[7] [7]

J. V. Koski, T. Sagawa, O.-P. Saira, Y. Yoon, A. Kutvo- nen, P. Solinas, M. M¨ ott¨ onen, T. Ala-Nissila, and J. P. Pekola, Nature Physics 9, 644 (2013)

work page 2013

[8] [8]

T. B. Batalh˜ ao, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Phys. Rev. Lett. 113, 140601 (2014)

work page 2014