Decomposable coherence and quantum fluctuation relations

Erick Hinds Mingo1 and David Jennings1,2,3

1Controlled Quantum Dynamics Theory Group, Imperial College London, Prince Consort Road, London SW7 2BW, UK
2Department of Physics, University of Oxford, Oxford, OX1 3PU, UK
3School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK.

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


In Newtonian mechanics, any closed-system dynamics of a composite system in a microstate will leave all its individual subsystems in distinct microstates, however this fails dramatically in quantum mechanics due to the existence of quantum entanglement. Here we introduce the notion of a `coherent work process', and show that it is the direct extension of a work process in classical mechanics into quantum theory. This leads to the notion of `decomposable' and `non-decomposable' quantum coherence and gives a new perspective on recent results in the theory of asymmetry as well as early analysis in the theory of classical random variables. Within the context of recent fluctuation relations, originally framed in terms of quantum channels, we show that coherent work processes play the same role as their classical counterparts, and so provide a simple physical primitive for quantum coherence in such systems. We also introduce a pure state effective potential as a tool with which to analyze the coherent component of these fluctuation relations, and which leads to a notion of temperature-dependent mean coherence, provides connections with multi-partite entanglement, and gives a hierarchy of quantum corrections to the classical Crooks relation in powers of inverse temperature.

How to define work in quantum thermodynamics has been the centre of much debate and received renewed interest after recent no-go results restricting the consistency of quantum and classical definitions. Here we circumvent this issue by first developing a framework for purely deterministic quantum work processes that naturally extend the Newtonian framework into quantum mechanics. We show that this coherent form of work becomes the familiar Newtonian work in the semi-classical limit, however the coherent work output is no longer simply described by a single value but rather it is encoded into a quantum state. Embedding this into recent results from studies of quantum fluctuation theorems, we find that this coherent work naturally arises within recent symmetry-based fluctuation theorems in a way that parallels the way Newtonian work appears in the classical fluctuation theorem settings, and so supports the claim that the framework is an appropriate generalisation into the quantum mechanical regime.

► BibTeX data

► References

[1] P.A.M. Dirac. The Principles of Quantum Mechanics. Comparative Pathobiology - Studies in the Postmodern Theory of Education. Clarendon Press, 1981. ISBN 9780198520115. URL https:/​/​​books?id=XehUpGiM6FIC.

[2] A. Peres. Quantum Theory: Concepts and Methods. Fundamental Theories of Physics. Springer Netherlands, 2006. ISBN 9780306471209. URL https:/​/​​books?id=pQXSBwAAQBAJ.

[3] Anton Zeilinger. Experiment and the foundations of quantum physics. Rev. Mod. Phys., 71: S288–S297, Mar 1999. 10.1103/​RevModPhys.71.S288. URL https:/​/​​doi/​10.1103/​RevModPhys.71.S288.

[4] C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland. A "Schrödinger cat" superposition state of an atom. Science, 272 (5265): 1131–1136, 1996. ISSN 0036-8075. 10.1126/​science.272.5265.1131. URL http:/​/​​content/​272/​5265/​1131.

[5] Giulio Chiribella and Robert W. Spekkens, editors. Quantum Theory: Informational Foundations and Foils, volume 181 of Fundamental Theories of Physics. Springer Netherlands, 1st edition, 2016. 10.1007/​978-94-017-7303-4.

[6] R.P. Feynman, A.R. Hibbs, and D.F. Styer. Quantum Mechanics and Path Integrals. Dover Books on Physics. Dover Publications, 2010. ISBN 9780486477220. URL https:/​/​​books?id=JkMuDAAAQBAJ.

[7] H. Goldstein, C.P. Poole, and J.L. Safko. Classical Mechanics: Pearson New International Edition. Pearson Education Limited, 2014. ISBN 9781292038933. URL https:/​/​​books?id=Xr-pBwAAQBAJ.

[8] Iman Marvian and Robert W. Spekkens. Modes of asymmetry: The application of harmonic analysis to symmetric quantum dynamics and quantum reference frames. Phys. Rev. A, 90: 062110, Dec 2014a. 10.1103/​PhysRevA.90.062110. URL https:/​/​​doi/​10.1103/​PhysRevA.90.062110.

[9] Michael Skotiniotis and Gilad Gour. Alignment of reference frames and an operational interpretation for theG-asymmetry. New Journal of Physics, 14 (7): 073022, jul 2012. 10.1088/​1367-2630/​14/​7/​073022. URL https:/​/​​10.1088.

[10] Gilad Gour and Robert W Spekkens. The resource theory of quantum reference frames: manipulations and monotones. New Journal of Physics, 10 (3): 033023, mar 2008. 10.1088/​1367-2630/​10/​3/​033023. URL https:/​/​​10.1088.

[11] J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Jacobs. Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules. Phys. Rev. A, 77: 032114, Mar 2008. 10.1103/​PhysRevA.77.032114. URL https:/​/​​doi/​10.1103/​PhysRevA.77.032114.

[12] Cristina Cirstoiu and David Jennings. Irreversibility and quantum information flow under global and local gauge symmetries. arXiv:1707.09826, 2017.

[13] E. Lukacs. Characteristic Functions. Griffin books of cognate interest. Hafner Publishing Company, 1970. URL https:/​/​​books?id=uGEPAQAAMAAJ.

[14] Eugene Lukacs. A survey of the theory of characteristic functions. Advances in Applied Probability, 4 (1): 1–38, 1972. ISSN 00018678. URL http:/​/​​stable/​1425805.

[15] D. Raikov. On the decomposition of Gauss and Poisson laws. Izv. Akad. Nauk SSSR Ser. Mat., 2 (1): 91–124, 1938.

[16] I. Z. RUZSA. Arithmetic of probability distributions. Séminaire de Théorie des Nombres de Bordeaux, pages 1–12, 1982. ISSN 09895558. URL http:/​/​​stable/​44166410.

[17] Andreas Winter and Dong Yang. Operational resource theory of coherence. Phys. Rev. Lett., 116: 120404, Mar 2016. 10.1103/​PhysRevLett.116.120404. URL https:/​/​​doi/​10.1103/​PhysRevLett.116.120404.

[18] Benjamin Morris and Gerardo Adesso. Quantum coherence fluctuation relations. Journal of Physics A: Mathematical and Theoretical, 51 (41): 414007, sep 2018. 10.1088/​1751-8121/​aac115. URL https:/​/​​10.1088.

[19] G. E. Crooks. Nonequilibrium measurements of free energy differences for microscopically reversible markovian systems. J. Stat. Phys., 90, 1998. 10.1023/​A:1023208217925. URL https:/​/​​10.1023/​A:1023208217925.

[20] Denis J. Evans and Debra J. Searles. Equilibrium microstates which generate second law violating steady states. Phys. Rev. E, 50: 1645–1648, Aug 1994. 10.1103/​PhysRevE.50.1645. URL https:/​/​​doi/​10.1103/​PhysRevE.50.1645.

[21] Christopher Jarzynski. Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale. Annual Review of Condensed Matter Physics, 2 (1): 329–351, 2011. 10.1146/​annurev-conmatphys-062910-140506. URL https:/​/​​10.1146/​annurev-conmatphys-062910-140506.

[22] Michele Campisi, Peter Hänggi, and Peter Talkner. Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys., 83: 771–791, Jul 2011. 10.1103/​RevModPhys.83.771. URL https:/​/​​doi/​10.1103/​RevModPhys.83.771.

[23] Johan Åberg. Fully quantum fluctuation theorems. Phys. Rev. X, 8: 011019, Feb 2018. 10.1103/​PhysRevX.8.011019. URL https:/​/​​doi/​10.1103/​PhysRevX.8.011019.

[24] Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry. Fluctuating work: From quantum thermodynamical identities to a second law equality. Phys. Rev. X, 6: 041017, Oct 2016. 10.1103/​PhysRevX.6.041017. URL https:/​/​​doi/​10.1103/​PhysRevX.6.041017.

[25] Zoë Holmes, Sebastian Weidt, David Jennings, Janet Anders, and Florian Mintert. Coherent fluctuation relations: from the abstract to the concrete. Quantum, 3: 124, February 2019. ISSN 2521-327X. 10.22331/​q-2019-02-25-124. URL https:/​/​​10.22331/​q-2019-02-25-124.

[26] Hyukjoon Kwon and M. S. Kim. Fluctuation theorems for a quantum channel. Phys. Rev. X, 9: 031029, Aug 2019. 10.1103/​PhysRevX.9.031029. URL https:/​/​​doi/​10.1103/​PhysRevX.9.031029.

[27] Martí Perarnau-Llobet, Elisa Bäumer, Karen V. Hovhannisyan, Marcus Huber, and Antonio Acin. No-go theorem for the characterization of work fluctuations in coherent quantum systems. Phys. Rev. Lett., 118: 070601, Feb 2017. 10.1103/​PhysRevLett.118.070601. URL https:/​/​​doi/​10.1103/​PhysRevLett.118.070601.

[28] A. E. Allahverdyan. Nonequilibrium quantum fluctuations of work. Phys. Rev. E, 90: 032137, Sep 2014. 10.1103/​PhysRevE.90.032137. URL https:/​/​​doi/​10.1103/​PhysRevE.90.032137.

[29] Peter Talkner and Peter Hänggi. Aspects of quantum work. Phys. Rev. E, 93: 022131, Feb 2016. 10.1103/​PhysRevE.93.022131. URL https:/​/​​doi/​10.1103/​PhysRevE.93.022131.

[30] Peter Talkner, Eric Lutz, and Peter Hänggi. Fluctuation theorems: Work is not an observable. Phys. Rev. E, 75: 050102, May 2007. 10.1103/​PhysRevE.75.050102. URL https:/​/​​doi/​10.1103/​PhysRevE.75.050102.

[31] Leonard Susskind and Jonathan Glogower. Quantum mechanical phase and time operator. Physics Physique Fizika, 1: 49–61, Jul 1964. 10.1103/​PhysicsPhysiqueFizika.1.49. URL https:/​/​​doi/​10.1103/​PhysicsPhysiqueFizika.1.49.

[32] S Sivakumar. Studies on nonlinear coherent states. Journal of Optics B: Quantum and Semiclassical Optics, 2 (6): R61–R75, nov 2000. 10.1088/​1464-4266/​2/​6/​02. URL https:/​/​​10.1088.

[33] G. S. Agarwal and K. Tara. Nonclassical properties of states generated by the excitations on a coherent state. Phys. Rev. A, 43: 492–497, Jan 1991. 10.1103/​PhysRevA.43.492. URL https:/​/​​doi/​10.1103/​PhysRevA.43.492.

[34] A. Mahdifar, E. Amooghorban, and M. Jafari. Photon-added and photon-subtracted coherent states on a sphere. Journal of Mathematical Physics, 59 (7): 072109, 2018. 10.1063/​1.5031036. URL https:/​/​​10.1063/​1.5031036.

[35] Luke C. G. Govia, Emily J. Pritchett, Seth T. Merkel, Deanna Pineau, and Frank K. Wilhelm. Theory of Josephson photomultipliers: Optimal working conditions and back action. Phys. Rev. A, 86: 032311, Sep 2012. 10.1103/​PhysRevA.86.032311. URL https:/​/​​doi/​10.1103/​PhysRevA.86.032311.

[36] Luke C G Govia, Emily J Pritchett, and Frank K Wilhelm. Generating nonclassical states from classical radiation by subtraction measurements. New Journal of Physics, 16 (4): 045011, apr 2014. 10.1088/​1367-2630/​16/​4/​045011. URL https:/​/​​10.1088.

[37] David G. Kendall. On infinite doubly-stochastic matrices and birkhoffs problem 111. Journal of the London Mathematical Society, s1-35 (1): 81–84, 1960. 10.1112/​jlms/​s1-35.1.81.

[38] Alexander Barvinok. A course in convexity, volume 54 of Graduate Studies in Mathematics. American Mathematical Society, 2002.

[39] P. Solinas, H. J. D. Miller, and J. Anders. Measurement-dependent corrections to work distributions arising from quantum coherences. Phys. Rev. A, 96: 052115, Nov 2017. 10.1103/​PhysRevA.96.052115. URL https:/​/​​doi/​10.1103/​PhysRevA.96.052115.

[40] Juliette Monsel, Cyril Elouard, and Alexia Auffèves. An autonomous quantum machine to measure the thermodynamic arrow of time. npj Quantum Information, 4 (1): 59, 2018. 10.1038/​s41534-018-0109-8. URL https:/​/​​10.1038/​s41534-018-0109-8.

[41] Iman Marvian Mashhad. Symmetry, Asymmetry and Quantum Information. PhD thesis, 2012. URL http:/​/​​10012/​7088.

[42] Matteo Lostaglio, Kamil Korzekwa, David Jennings, and Terry Rudolph. Quantum coherence, time-translation symmetry, and thermodynamics. Phys. Rev. X, 5: 021001, Apr 2015a. 10.1103/​PhysRevX.5.021001. URL https:/​/​​doi/​10.1103/​PhysRevX.5.021001.

[43] David Jennings and Matthew Leifer. No return to classical reality. Contemporary Physics, 57 (1): 60–82, 2016. 10.1080/​00107514.2015.1063233. URL https:/​/​​10.1080/​00107514.2015.1063233.

[44] Harry J D Miller and Janet Anders. Time-reversal symmetric work distributions for closed quantum dynamics in the histories framework. New Journal of Physics, 19 (6): 062001, jun 2017. 10.1088/​1367-2630/​aa703f. URL https:/​/​​10.1088.

[45] Nicole Yunger Halpern. Jarzynski-like equality for the out-of-time-ordered correlator. Phys. Rev. A, 95: 012120, Jan 2017. 10.1103/​PhysRevA.95.012120. URL https:/​/​​doi/​10.1103/​PhysRevA.95.012120.

[46] Tameem Albash, Daniel A. Lidar, Milad Marvian, and Paolo Zanardi. Fluctuation theorems for quantum processes. Phys. Rev. E, 88: 032146, Sep 2013. 10.1103/​PhysRevE.88.032146. URL https:/​/​​doi/​10.1103/​PhysRevE.88.032146.

[47] Matteo Lostaglio. Quantum fluctuation theorems, contextuality, and work quasiprobabilities. Phys. Rev. Lett., 120: 040602, Jan 2018. 10.1103/​PhysRevLett.120.040602. URL https:/​/​​doi/​10.1103/​PhysRevLett.120.040602.

[48] Simon Kochen and E. P. Specker. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17 (1): 59–87, 1967. ISSN 00959057, 19435274. URL http:/​/​​stable/​24902153.

[49] Ernesto F. Galvao. Foundations of quantum theory and quantum information applications. arXiv:quant-ph/​0212124, 2002.

[50] Robert Raussendorf. Contextuality in measurement-based quantum computation. Phys. Rev. A, 88: 022322, Aug 2013. 10.1103/​PhysRevA.88.022322. URL https:/​/​​doi/​10.1103/​PhysRevA.88.022322.

[51] Markus Frembs, Sam Roberts, and Stephen D Bartlett. Contextuality as a resource for measurement-based quantum computation beyond qubits. New Journal of Physics, 20 (10): 103011, oct 2018. 10.1088/​1367-2630/​aae3ad. URL https:/​/​​10.1088.

[52] Sebastian Deffner and Christopher Jarzynski. Information processing and the second law of thermodynamics: An inclusive, hamiltonian approach. Phys. Rev. X, 3: 041003, Oct 2013. 10.1103/​PhysRevX.3.041003. URL https:/​/​​doi/​10.1103/​PhysRevX.3.041003.

[53] W. Forrest Stinespring. Positive functions on c*-algebras. Proceedings of the American Mathematical Society, 6 (2): 211–216, 1955. ISSN 00029939, 10886826. URL http:/​/​​stable/​2032342.

[54] H.B. Callen. Thermodynamics and an Introduction to Thermostatistics. Wiley, 1985. ISBN 9780471610564. URL https:/​/​​books?id=MFutGQAACAAJ.

[55] Run-Qiu Yang. Complexity for quantum field theory states and applications to thermofield double states. Phys. Rev. D, 97: 066004, Mar 2018. 10.1103/​PhysRevD.97.066004. URL https:/​/​​doi/​10.1103/​PhysRevD.97.066004.

[56] M. Costeniuc, R. S. Ellis, H. Touchette, and B. Turkington. Generalized canonical ensembles and ensemble equivalence. Phys. Rev. E, 73: 026105, Feb 2006. 10.1103/​PhysRevE.73.026105. URL https:/​/​​doi/​10.1103/​PhysRevE.73.026105.

[57] Gavin E. Crooks. Quantum operation time reversal. Phys. Rev. A, 77: 034101, Mar 2008. 10.1103/​PhysRevA.77.034101. URL https:/​/​​doi/​10.1103/​PhysRevA.77.034101.

[58] Dénes Petz. Sufficient subalgebras and the relative entropy of states of a von neumann algebra. Communications in Mathematical Physics, 105 (1): 123–131, Mar 1986. 10.1007/​BF01212345. URL https:/​/​​10.1007/​BF01212345.

[59] Philippe Faist. Quantum coarse-graining: An information-theoretic approach to thermodynamics. arXiv:1607.03104, 2016.

[60] W. Feller. An introduction to probability theory and its applications. Number v. 1 in Wiley series in probability and mathematical statistics. Probability and mathematical statistics. Wiley, 1968. ISBN 9780471257080. URL https:/​/​​books?id=mfRQAAAAMAAJ.

[61] Iman Marvian and Robert W. Spekkens. Asymmetry properties of pure quantum states. Phys. Rev. A, 90: 014102, Jul 2014b. 10.1103/​PhysRevA.90.014102. URL https:/​/​​doi/​10.1103/​PhysRevA.90.014102.

[62] Tomohiro Ogawa and Hiroshi Nagaoka. Strong Converse and Stein's Lemma in Quantum Hypothesis Testing, pages 28–42. 10.1142/​9789812563071_0003. URL https:/​/​​doi/​abs/​10.1142/​9789812563071_0003.

[63] Matteo Lostaglio, David Jennings, and Terry Rudolph. Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nature Communications, 6 (1): 6383, 2015b. 10.1038/​ncomms7383. URL https:/​/​​10.1038/​ncomms7383.

[64] Mehdi Ahmadi, David Jennings, and Terry Rudolph. The wigner–araki–yanase theorem and the quantum resource theory of asymmetry. New Journal of Physics, 15 (1): 013057, jan 2013. 10.1088/​1367-2630/​15/​1/​013057. URL https:/​/​​10.1088.

[65] John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. 10.1017/​9781316848142.

[66] Otfried Gühne, Géza Tóth, and Hans J Briegel. Multipartite entanglement in spin chains. New Journal of Physics, 7: 229–229, nov 2005. 10.1088/​1367-2630/​7/​1/​229. URL https:/​/​​10.1088.

[67] Géza Tóth. Multipartite entanglement and high-precision metrology. Phys. Rev. A, 85: 022322, Feb 2012. 10.1103/​PhysRevA.85.022322. URL https:/​/​​doi/​10.1103/​PhysRevA.85.022322.

[68] Zeqian Chen. Wigner-Yanase skew information as tests for quantum entanglement. Phys. Rev. A, 71: 052302, May 2005. 10.1103/​PhysRevA.71.052302. URL https:/​/​​doi/​10.1103/​PhysRevA.71.052302.

[69] John Goold, Francesco Plastina, Andrea Gambassi, and Alessandro Silva. The Role of Quantum Work Statistics in Many-Body Physics, pages 317–336. Springer International Publishing, Cham, 2018. ISBN 978-3-319-99046-0. 10.1007/​978-3-319-99046-0_13. URL https:/​/​​10.1007/​978-3-319-99046-0_13.

[70] A. S. Said. Some properties of the poisson distribution. AIChE Journal, 4 (3): 290–292, 1958. 10.1002/​aic.690040311. URL https:/​/​​doi/​abs/​10.1002/​aic.690040311.

[71] Sudhakar Prasad, Marlan O. Scully, and Werner Martienssen. A quantum description of the beam splitter. Optics Communications, 62 (3): 139 – 145, 1987. ISSN 0030-4018. https:/​/​​10.1016/​0030-4018(87)90015-0. URL http:/​/​​science/​article/​pii/​0030401887900150.

[72] Dénes Petz. A survey of certain trace inequalities. Banach Center Publications, 30 (1): 287–298, 1994. 10.4064/​-30-1-287-298. URL http:/​/​​doc/​262566.

[73] Pascal Massart. Concentration Inequalities and Model Selection: Ecole d'Eté de Probabilités de Saint-Flour XXXIII - 2003. Lecture Notes in Mathematics. Springer, 2007.

[74] P. Billingsley. Probability and Measure. Wiley Series in Probability and Statistics. Wiley, 1995. ISBN 9780471007104.

Cited by

[1] Adam Teixidó-Bonfill, Alvaro Ortega, and Eduardo Martín-Martínez, "First law of quantum field thermodynamics", Physical Review A 102 5, 052219 (2020).

[2] Benjamin Yadin, Hyejung H Jee, Carlo Sparaciari, Gerardo Adesso, and Alessio Serafini, "Catalytic Gaussian thermal operations", Journal of Physics A: Mathematical and Theoretical 55 32, 325301 (2022).

[3] Takahiro Sagawa, Philippe Faist, Kohtaro Kato, Keiji Matsumoto, Hiroshi Nagaoka, and Fernando G S L Brandão, "Asymptotic reversibility of thermal operations for interacting quantum spin systems via generalized quantum Stein’s lemma", Journal of Physics A: Mathematical and Theoretical 54 49, 495303 (2021).

[4] S. Gherardini, A. Belenchia, M. Paternostro, and A. Trombettoni, "End-point measurement approach to assess quantum coherence in energy fluctuations", Physical Review A 104 5, L050203 (2021).

[5] David Edward Bruschi, Benjamin Morris, and Ivette Fuentes, "Thermodynamics of relativistic quantum fields confined in cavities", Physics Letters A 384 25, 126601 (2020).

[6] Matteo Scandi, Harry J. D. Miller, Janet Anders, and Martí Perarnau-Llobet, "Quantum work statistics close to equilibrium", Physical Review Research 2 2, 023377 (2020).

[7] Bao-Ming Xu, "Quantum fluctuation theorem for initial near-equilibrium system", Journal of Statistical Mechanics: Theory and Experiment 2023 5, 053105 (2023).

[8] Zoë Holmes, Erick Hinds Mingo, Calvin Chen, and Florian Mintert, "Quantifying Athermality and Quantum Induced Deviations from Classical Fluctuation Relations", Entropy 22 1, 111 (2020).

[9] K Khan, J Sales Araújo, W F Magalhães, G H Aguilar, and B de Lima Bernardo, "Coherent energy fluctuation theorems: theory and experiment", Quantum Science and Technology 7 4, 045010 (2022).

[10] Hyukjoon Kwon and M. S. Kim, "Fluctuation Theorems for a Quantum Channel", Physical Review X 9 3, 031029 (2019).

[11] Takahiro Sagawa, Philippe Faist, Kohtaro Kato, Keiji Matsumoto, Hiroshi Nagaoka, and Fernando G. S. L. Brandao, "Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein's Lemma", arXiv:1907.05650, (2019).

[12] Zoe Holmes, "The Coherent Crooks Equality", Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions 195, 301 (2018).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 15:54:16) and SAO/NASA ADS (last updated successfully 2024-05-21 15:54:17). The list may be incomplete as not all publishers provide suitable and complete citation data.