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arxiv: 2508.15540 · v2 · submitted 2025-08-21 · 🪐 quant-ph · cond-mat.stat-mech

Universal Statistics of Charges Exchanges in Non-Abelian Quantum Transport

Matteo Scandi , Gonzalo Manzano This is my paper

Pith reviewed 2026-05-18 21:45 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords non-Abelian quantum transportfluctuation relationsthermodynamic uncertainty relationcharge exchange statisticsquantum thermodynamicsnon-commuting conserved quantitiesfar-from-equilibrium transport
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The pith

Fluctuation relations and a thermodynamic uncertainty relation hold for exchanges of any number of non-commuting conserved quantities in quantum transport far from equilibrium.

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

The paper derives detailed and integral fluctuation relations together with a thermodynamic uncertainty relation that constrain the statistics of charge exchanges between two quantum systems. These relations apply to an arbitrary number of non-commuting conserved quantities and remain valid in setups arbitrarily far from equilibrium without requiring any efficacy parameter. A sympathetic reader would care because the non-commutativity of the charges permits modifications to standard thermodynamic behaviors that are impossible when charges commute, including apparent second-law violations, tighter bounds on fluctuations, and reversal of currents against their driving affinities. The results directly generalize the familiar fluctuation theorems known for energy and particle transport to the non-Abelian setting.

Core claim

We derive detailed and integral fluctuation relations as well as a Thermodynamic Uncertainty Relation constraining the exchange statistics of an arbitrary number of non-commuting conserved quantities among two quantum systems in transport setups arbitrary far from equilibrium. These universal relations, valid without the need of any efficacy parameter, extend the well-known heat exchange fluctuation theorems for energy and particle transport to the case of non-Abelian quantum transport, where the non-commutativity of the charges allows bending standard thermodynamic rules. In particular, we show that this can lead to apparent violations of the second law of thermodynamics, it enhances the 1/

What carries the argument

Universal fluctuation relations and thermodynamic uncertainty relation derived directly from the non-commuting charge operators that map to exchange statistics in the transport Hamiltonian.

If this is right

  • Apparent violations of the second law become possible because non-commutativity relaxes the usual constraints on entropy production.
  • Current fluctuations can achieve higher precision than the standard thermodynamic uncertainty relation would allow for commuting charges.
  • All currents can invert against their affinity biases simultaneously.
  • The relations remain valid for any number of non-commuting conserved quantities without additional assumptions on the interaction form.

Where Pith is reading between the lines

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

  • The same non-commutativity mechanism could relax efficiency bounds in quantum heat engines or refrigerators that carry multiple internal degrees of freedom.
  • Experimental platforms with spin or orbital angular momentum transport offer direct tests because the charge operators are already non-commuting by construction.
  • The framework suggests that fluctuation theorems in systems with non-Abelian symmetries may require only the algebraic structure of the charges rather than their explicit representation.

Load-bearing premise

The interaction Hamiltonians and transport setups permit a direct mapping from the non-commuting charge operators to the observed exchange statistics without model-specific corrections or efficacy parameters.

What would settle it

Measure the full counting statistics of two non-commuting charges exchanged between two qubits or spins in a controlled transport protocol and test whether the integral fluctuation relation holds exactly as stated for arbitrary driving strengths.

Figures

Figures reproduced from arXiv: 2508.15540 by Gonzalo Manzano, Matteo Scandi.

Figure 1
Figure 1. Figure 1: Depiction of the collisional exchange model. Par [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a) Integral XFT in Eq. (11), together with its standard version for commutative charges; b) second-law inequality in Eq. (12) with and without the quantum correction; c) average currents (in the top panel ⟨∆σz⟩ ≡ ⟨∆σx⟩). Top row corresponds to same temperatures in the baths β A = β B, whereas in the bottom row β A < βB. Shaded areas corresponds to double current inversions. In the bottom row, for the param… view at source ↗
read the original abstract

We derive detailed and intergral fluctuation relations as well as a Thermodynamic Uncertainty Relation constraining the exchange statistics of an arbitrary number of non-commuting conserved quantities among two quantum systems in transport setups arbitrary far from equilibrium. These universal relations, valid without the need of any efficacy parameter, extend the well-known heat exchange fluctuation theorems for energy and particle transport to the case of non-Abelian quantum transport, where the non-commutativity of the charges allows bending standard thermodynamic rules. In particular, we show that this can lead to apparent violations of the second law of thermodynamics, it enhances precision in the current fluctuations, and it allows for the inversion of all currents against their affinity biases.

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

1 major / 2 minor

Summary. The paper claims to derive detailed and integral fluctuation relations as well as a Thermodynamic Uncertainty Relation constraining the exchange statistics of an arbitrary number of non-commuting conserved quantities among two quantum systems in transport setups arbitrarily far from equilibrium. These universal relations are presented as valid without any efficacy parameter, extending standard heat exchange fluctuation theorems to non-Abelian quantum transport where non-commutativity permits bending of thermodynamic rules, including apparent second-law violations, enhanced precision in current fluctuations, and inversion of currents against affinity biases.

Significance. If the derivations hold with the claimed universality, the work would be significant for quantum thermodynamics by providing tools to analyze transport involving multiple non-commuting charges, relevant to systems with symmetries such as spin or angular momentum. The absence of efficacy parameters and the explicit handling of non-commutativity effects on thermodynamic relations could offer new insights into quantum modifications of classical rules. Credit is due for targeting a general setup with arbitrary numbers of charges and far-from-equilibrium conditions, though the overall impact depends on confirming the model-independence of the statistics definition.

major comments (1)
  1. [Definition of exchange statistics / derivation of fluctuation relations] The universality without an efficacy parameter is central to the main claim. In the section defining the joint exchange statistics (likely where the characteristic function or probability distribution for non-commuting charges is introduced), the manuscript must explicitly construct this object from total charge conservation and demonstrate that no protocol-dependent ordering, measurement back-action, or interaction-Hamiltonian-specific corrections arise for arbitrary couplings, as non-commutativity precludes a joint eigenbasis.
minor comments (2)
  1. [Abstract] Typo: 'intergral' should be 'integral'.
  2. [Abstract] 'arbitrary far from equilibrium' should be 'arbitrarily far from equilibrium'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for recognizing the potential significance of our results on universal fluctuation relations in non-Abelian quantum transport. We address the major comment below and will revise the manuscript to improve the clarity of the presentation.

read point-by-point responses

  1. Referee: The universality without an efficacy parameter is central to the main claim. In the section defining the joint exchange statistics (likely where the characteristic function or probability distribution for non-commuting charges is introduced), the manuscript must explicitly construct this object from total charge conservation and demonstrate that no protocol-dependent ordering, measurement back-action, or interaction-Hamiltonian-specific corrections arise for arbitrary couplings, as non-commutativity precludes a joint eigenbasis.

    Authors: We agree that an explicit construction of the joint exchange statistics from total charge conservation is essential to substantiate the claimed universality without efficacy parameters. In the manuscript, the joint statistics are introduced via the characteristic function of the charge exchanges, which is obtained from the unitary time evolution of the composite system under the constraint that the total charges are conserved (i.e., they commute with the total Hamiltonian). This global conservation law ensures that the two-point measurement protocol applied to the initial product state yields a well-defined joint distribution without requiring a common eigenbasis for the non-commuting charges. Nevertheless, to directly address the referee's concern regarding potential protocol dependence, ordering ambiguities, or coupling-specific corrections, we will revise the relevant section to include a step-by-step derivation starting from the conservation of the total charges. We will explicitly show that the resulting characteristic function is independent of the interaction Hamiltonian details beyond conservation and that no additional back-action or ordering terms arise for arbitrary couplings. This revision will strengthen the demonstration that the fluctuation relations and TUR hold model-independently. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on standard techniques

full rationale

The abstract and context describe derivations of fluctuation relations and TUR for non-commuting charges via conservation of total charges and open-system mappings, without any quoted equations reducing predictions to fitted inputs, self-definitions, or load-bearing self-citations. No specific reduction (e.g., Eq. X = Eq. Y by construction) is exhibited in the provided material, and the central claims rest on independent quantum transport assumptions that do not collapse to the target results by definition. This is the expected honest non-finding for a paper whose methods appear externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivations rest on standard quantum mechanics and open-system transport assumptions without introducing new free parameters or postulated entities.

axioms (1)
  • domain assumption Standard assumptions of quantum mechanics and Markovian or non-Markovian open quantum system dynamics in transport setups.
    Invoked to justify the existence of exchange statistics for conserved charges.

pith-pipeline@v0.9.0 · 5635 in / 1200 out tokens · 39308 ms · 2026-05-18T21:45:17.819454+00:00 · methodology

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Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    An environmental unit (molecule) is taken from each bath, leading to an initial product stateρ0 = πA λA ⊗ πB λB, and are measured by projectorsΠA and ΠB in the eigenbasis ofHA and HB respectively

    work page

  2. [2]

    The two units interact through the unitary opera- tion U = Te−i R τ 0 dtHint(t), where Hint is a (possibly time-dependent) interaction Hamiltonian that cou- ples the two baths during the interaction timeτ, and T is the time-ordering operator

    work page

  3. [3]

    Depiction of the collisional exchange model

    The units are measured again in the eigenbasis of HA and HB respectively, after which they return to Figure 1. Depiction of the collisional exchange model. Par- ticles from each bath are projectively measured in the eigen- basis of HA (HB) giving outcome n (ν), they interact with each other through the charge-preserving unitary operation U, and, finally, ...

    work page

  4. [4]

    J. S. Bell and A. Aspect,Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Phi- losophy, 2nd ed. (Cambridge University Press, 2004)

    work page 2004

  5. [5]

    Brukner, Wigner’s friend and relational objectivity, Nature Reviews Physics4, 628 (2022)

    Č. Brukner, Wigner’s friend and relational objectivity, Nature Reviews Physics4, 628 (2022)

    work page 2022

  6. [6]

    Rogge, F

    M. Rogge, F. Cavaliere, M. Sassetti, R. J. Haug, and B. Kramer, Negative conductance patterns of quantum dots: experiment and theory, New Journal of Physics8, 298 (2006)

    work page 2006

  7. [7]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information (Cambridge University Press, 2000)

    work page 2000

  8. [8]

    S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Refer- ence frames, superselection rules, and quantum informa- tion, Rev. Mod. Phys.79, 555 (2007)

    work page 2007

  9. [9]

    Deffner and S

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

    work page 2053

  10. [10]

    Yunger Halpern, P

    N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Win- ter, Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommut- ing charges, Nature Communications7, 12051 (2016)

    work page 2016

  11. [11]

    Guryanova, S

    Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Thermodynamics of quantum systems with multiple conserved quantities, Nature Communica- tions 7, 1 (2016)

    work page 2016

  12. [12]

    Lostaglio, D

    M. Lostaglio, D. Jennings, and T. Rudolph, Thermody- namic resource theories, non-commutativity and max- imum entropy principles, New Journal of Physics 19, 043008 (2017)

    work page 2017

  13. [13]

    Manzano, J

    G. Manzano, J. M. Parrondo, and G. T. Landi, Non- abelian quantum transport and thermosqueezing effects, PRX Quantum 3, 010304 (2022)

    work page 2022

  14. [14]

    Majidy, W

    S. Majidy, W. F. Braasch Jr, A. Lasek, T. Upadhyaya, A. Kalev, and N. Yunger Halpern, Noncommuting con- served charges in quantum thermodynamics and beyond, Nature Reviews Physics5, 689 (2023)

    work page 2023

  15. [15]

    Kranzl, A

    F. Kranzl, A. Lasek, M. K. Joshi, A. Kalev, R. Blatt, C. F. Roos, and N. Yunger Halpern, Experimental ob- servation of thermalization with noncommuting charges, PRX Quantum 4, 020318 (2023)

    work page 2023

  16. [16]

    Manzano, Squeezed thermal reservoir as a generalized equilibrium reservoir, Phys

    G. Manzano, Squeezed thermal reservoir as a generalized equilibrium reservoir, Phys. Rev. E98, 042123 (2018)

    work page 2018

  17. [17]

    Manzano, F

    G. Manzano, F. Galve, R. Zambrini, and J. M. R. Par- rondo, Entropy production and thermodynamic power of the squeezed thermal reservoir, Phys. Rev. E93, 052120 (2016). 6

    work page 2016

  18. [18]

    de Oliveira Junior and M

    A. de Oliveira Junior and M. C. de Oliveira, Unravelling the non-classicality role in gaussian heat engines, Scien- tific reports 12, 10412 (2022)

    work page 2022

  19. [19]

    Shahidani, Equilibration to the non-abelian squeezed- thermal reservoir with three charges, Phys

    S. Shahidani, Equilibration to the non-abelian squeezed- thermal reservoir with three charges, Phys. Rev. A110, 052216 (2024)

    work page 2024

  20. [20]

    Klaers, S

    J. Klaers, S. Faelt, A. Imamoglu, and E. Togan, Squeezed thermal reservoirs as a resource for a nanomechanical en- gine beyond the carnot limit, Phys. Rev. X 7, 031044 (2017)

    work page 2017

  21. [21]

    Yunger Halpern, M

    N. Yunger Halpern, M. E. Beverland, and A. Kalev, Non- commuting conserved charges in quantum many-body thermalization, Phys. Rev. E101, 042117 (2020)

    work page 2020

  22. [22]

    Majidy, Noncommuting charges can remove non- stationary quantum many-body dynamics, Nature Com- mun

    S. Majidy, Noncommuting charges can remove non- stationary quantum many-body dynamics, Nature Com- mun. 15, 8246 (2024)

    work page 2024

  23. [23]

    Murthy, A

    C. Murthy, A. Babakhani, F. Iniguez, M. Srednicki, and N. Yunger Halpern, Non-abelian eigenstate thermaliza- tion hypothesis, Phys. Rev. Lett.130, 140402 (2023)

    work page 2023

  24. [24]

    Popescu, A

    S. Popescu, A. B. Sainz, A. J. Short, and A. Winter, Ref- erence frames which separately store noncommuting con- served quantities, Phys. Rev. Lett.125, 090601 (2020)

    work page 2020

  25. [25]

    Onsager, Reciprocal relations in irreversible processes

    L. Onsager, Reciprocal relations in irreversible processes. i., Phys. Rev.37, 405 (1931)

    work page 1931

  26. [26]

    Onsager, Reciprocal relations in irreversible processes

    L. Onsager, Reciprocal relations in irreversible processes. ii., Phys. Rev.38, 2265 (1931)

    work page 1931

  27. [27]

    H. B. Callen and T. A. Welton, Irreversibility and gen- eralized noise, Phys. Rev.83, 34 (1951)

    work page 1951

  28. [28]

    Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics29, 255 (1966)

    R. Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics29, 255 (1966)

    work page 1966

  29. [29]

    Upadhyaya, W

    T. Upadhyaya, W. F. Braasch, G. T. Landi, and N. Y. Halpern, Non-abelian transport distinguishes three usu- ally equivalent notions of entropy production, PRX Quantum 5, 030355 (2024)

    work page 2024

  30. [30]

    Seifert, Stochastic thermodynamics, fluctuation the- orems and molecular machines, Reports on Progress in Physics 75, 126001 (2012)

    U. Seifert, Stochastic thermodynamics, fluctuation the- orems and molecular machines, Reports on Progress in Physics 75, 126001 (2012)

    work page 2012

  31. [31]

    Manzano and R

    G. Manzano and R. Zambrini, Quantum thermodynam- ics under continuous monitoring: A general framework, AVS Quantum Science4, 025302 (2022)

    work page 2022

  32. [32]

    Jarzynski and D

    C. Jarzynski and D. K. Wójcik, Classical and quantum fluctuation theorems for heat exchange, Phys. Rev. Lett. 92, 230602 (2004)

    work page 2004

  33. [33]

    Andrieux, P

    D. Andrieux, P. Gaspard, T. Monnai, and S. Tasaki, The fluctuation theorem for currents in open quantum sys- tems, New Journal of Physics11, 043014 (2009)

    work page 2009

  34. [34]

    Campisi, P

    M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and appli- cations, Rev. Mod. Phys.83, 771 (2011)

    work page 2011

  35. [35]

    Esposito, U

    M. Esposito, U. Harbola, and S. Mukamel, Nonequilib- rium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys.81, 1665 (2009)

    work page 2009

  36. [36]

    Nakamura, Y

    S. Nakamura, Y. Yamauchi, M. Hashisaka, K. Chida, K. Kobayashi, T. Ono, R. Leturcq, K. Ensslin, K. Saito, Y. Utsumi, and A. C. Gossard, Nonequilibrium fluctu- ation relations in a quantum coherent conductor, Phys. Rev. Lett. 104, 080602 (2010)

    work page 2010

  37. [37]

    Schön, Bidirectional single-electron count- ing and the fluctuation theorem, Phys

    Y.Utsumi, D.S.Golubev, M.Marthaler, K.Saito, T.Fu- jisawa, and G. Schön, Bidirectional single-electron count- ing and the fluctuation theorem, Phys. Rev. B81, 125331 (2010)

    work page 2010

  38. [38]

    J. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, Heat fluctuations in a nonequilibrium bath, Phys. Rev. Lett. 106, 200602 (2011)

    work page 2011

  39. [39]

    Hernández-Gómez, S

    S. Hernández-Gómez, S. Gherardini, F. Poggiali, F. S. Cataliotti, A.Trombettoni, P.Cappellaro,andN.Fabbri, Experimental test of exchange fluctuation relations in an open quantum system, Phys. Rev. Res.2, 023327 (2020)

    work page 2020

  40. [40]

    Hernández-Gómez, N

    S. Hernández-Gómez, N. Staudenmaier, M. Campisi, and N. Fabbri, Experimental test of fluctuation relations for driven open quantum systems with an nv center, New Journal of Physics23, 065004 (2021)

    work page 2021

  41. [41]

    Jevtic, T

    S. Jevtic, T. Rudolph, D. Jennings, Y. Hirono, S. Nakayama, and M. Murao, Exchange fluctuation the- orem for correlated quantum systems, Phys. Rev. E92, 042113 (2015)

    work page 2015

  42. [42]

    Levy and M

    A. Levy and M. Lostaglio, Quasiprobability distribution for heat fluctuations in the quantum regime, PRX Quan- tum 1, 010309 (2020)

    work page 2020

  43. [43]

    Hasegawa and T

    Y. Hasegawa and T. Van Vu, Fluctuation theorem un- certainty relation, Phys. Rev. Lett.123, 110602 (2019)

    work page 2019

  44. [44]

    A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Thermodynamic uncertainty relations from ex- change fluctuation theorems, Phys. Rev. Lett. 123, 090604 (2019)

    work page 2019

  45. [45]

    Lostaglio, A

    M. Lostaglio, A. Belenchia, A. Levy, S. Hernández- Gómez, N. Fabbri, and S. Gherardini, Kirkwood-Dirac quasiprobability approach to the statistics of incompati- ble observables, Quantum7, 1128 (2023)

    work page 2023

  46. [46]

    Micadei, G

    K. Micadei, G. T. Landi, and E. Lutz, Quantum fluctu- ation theorems beyond two-point measurements, Phys. Rev. Lett. 124, 090602 (2020)

    work page 2020

  47. [47]

    E. T. Jaynes, Information theory and statistical mechan- ics, Phys. Rev.106, 620 (1957)

    work page 1957

  48. [48]

    E. T. Jaynes, Information theory and statistical mechan- ics. ii, Phys. Rev.108, 171 (1957)

    work page 1957

  49. [49]

    Rigol, V

    M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Re- laxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1d lattice hard-core bosons, Phys. Rev. Lett. 98, 050405 (2007)

    work page 2007

  50. [50]

    Rigol, Breakdown of thermalization in finite one- dimensional systems, Phys

    M. Rigol, Breakdown of thermalization in finite one- dimensional systems, Phys. Rev. Lett. 103, 100403 (2009)

    work page 2009

  51. [51]

    Langen, S

    T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a generalized gibbs ensemble, Science 348, 207 (2015), https://www.science.org/doi/pdf/10.1126/science.1257026

    work page doi:10.1126/science.1257026 2015

  52. [52]

    Vaccaro and S

    J. Vaccaro and S. Barnett, Information erasure without an energy cost, Proceedings A: Mathematical, Physical and Engineering Sciences467, 1770 (2011)

    work page 2011

  53. [53]

    Croucher, S

    T. Croucher, S. Bedkihal, and J. A. Vaccaro, Discrete fluctuations in memory erasure without energy cost, Phys. Rev. Lett.118, 060602 (2017)

    work page 2017

  54. [54]

    Mur-Petit, A

    J. Mur-Petit, A. Relaño, R. Molina, and D. Jaksch, Re- vealing missing charges with generalised quantum fluctu- ation relations, Nature communications9, 2006 (2018)

    work page 2006

  55. [55]

    F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Resource theory of quantum states out of thermal equilibrium, Phys. Rev. Lett.111, 250404 (2013)

    work page 2013

  56. [56]

    Horodecki and J

    M. Horodecki and J. Oppenheim, Fundamental limita- tions for quantum and nanoscale thermodynamics, Na- ture communications 4, 2059 (2013)

    work page 2059

  57. [57]

    Korzekwa, M

    K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jen- nings, The extraction of work from quantum coherence, 7 New Journal of Physics18, 023045 (2016)

    work page 2016

  58. [58]

    Murashita, K

    Y. Murashita, K. Funo, and M. Ueda, Nonequilibrium equalities in absolutely irreversible processes, Phys. Rev. E 90, 042110 (2014)

    work page 2014

  59. [59]

    Manzano, J

    G. Manzano, J. M. Horowitz, and J. M. R. Parrondo, Quantum fluctuation theorems for arbitrary environ- ments: Adiabatic and nonadiabatic entropy production, Phys. Rev. X8, 031037 (2018)

    work page 2018

  60. [60]

    Luposchainsky, A

    D. Luposchainsky, A. C. Barato, and H. Hinrichsen, Strong fluctuation theorem for nonstationary nonequi- librium systems, Phys. Rev. E87, 042108 (2013)

    work page 2013

  61. [61]

    Scandi, P

    M. Scandi, P. Abiuso, J. Surace, and D. De Santis, Quan- tum fisher information and its dynamical nature, Reports on Progress in Physics88, 076001 (2025)

    work page 2025

  62. [62]

    Jarzynski, Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale, Annual Review of Condensed Matter Physics 2, 329 (2011)

    C. Jarzynski, Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale, Annual Review of Condensed Matter Physics 2, 329 (2011)

    work page 2011

  63. [63]

    Exchange Fluctuation Theorem for Non-Commuting Charges

    N. Merhav and Y. Kafri, Statistical properties of entropy production derived from fluctuation theorems, Journal of Statistical Mechanics: Theory and Experiment2010, P12022 (2010). 1 Supplemental Material to “Exchange Fluctuation Theorem for Non-Commuting Charges" The Supplemental Material contains detailed proofs and further details on the derivation of t...

    work page 2010

  64. [64]

    An environmental unit (molecule) is taken from each bath, leading to an initial product state ˜ρ0 = Θ πA λA ⊗ πB λB Θ−1, and is measured by projectorsΘΠAΘ−1 and ΘΠBΘ−1 in the eigenbasis ofΘHAΘ−1 and ΘHBΘ−1 respectively

    work page

  65. [65]

    The two units interact through the (backward) unitary operation˜U = Te−i R τ 0 dtΘHint(τ −t)Θ−1 , where Hint is the same interaction Hamiltonian as in the original (forward) process that couples the two baths during the interaction time τ

    work page

  66. [66]

    The units are measured again in the eigenbasis ofΘHAΘ−1 and ΘHBΘ−1 respectively, after which they return to their respective baths. It is important to notice that in the backward process the equivalent charge-preserving condition for the interaction between the bath units for all chargesi is verified, as we have that: h ˜U ,Θ QA i + QB i Θ−1 i = Θ U, QA i...

    work page