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Fluctuating Work in Coherent Quantum Systems: Proposals and Limitations

  • Elisa BäumerEmail author
  • Matteo Lostaglio
  • Martí Perarnau-Llobet
  • Rui Sampaio
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 195)

Abstract

One of the most important goals in quantum thermodynamics is to demonstrate advantages of themodynamic protocols over their classical counterparts. For that, it is necessary to (i) develop theoretical tools and experimental set-ups to deal with quantum coherence in thermodynamic contexts, and to (ii) elucidate which properties are genuinely quantum in a thermodynamic process. In this short review, we discuss proposals to define and measure work fluctuations that allow to capture quantum interference phenomena. We also discuss fundamental limitations arising due to measurement back-action, as well as connections between work distributions and quantum contextuality. We hope the different results summarised here motivate further research on the role of quantum phenomena in thermodynamics.

Notes

Acknowledgements

We thank Johan Åberg, Armen Allahverdyan, Janet Anders, Peter Hänggi, Simone Gasparinetti, Paolo Solinas and Peter Talkner for useful feedback on the manuscript. E.B. acknowledges contributions from the Swiss National Science Foundation via the NCCR QSIT as well as project No. 200020_165843. M.P.-L. acknowledges support from the Alexander von Humboldt Foundation. M.L. acknowledges financial support from the the European Union’s Marie Sklodowska-Curie individual Fellowships (H2020-MSCA-IF-2017, GA794842), Spanish MINECO (Severo OchoaSEV-2015-0522 and project QIBEQI FIS2016-80773-P), Fundacio Cellex and Generalitat de Catalunya (CERCAProgramme and SGR 875). R. S. acknowledges the Magnus Ehrnrooth Foundation and the Academy of Finland through its CoE grants 284621 and 287750. All authors are grateful for support from the EU COST Action MP1209 on Thermodynamics in the Quantum Regime.

References

  1. 1.
    J. Goold, M. Huber, A. Riera, L. d. Rio, P. Skrzypczyk, The role of quantum information in thermodynamics a topical review. J. Phys. A, 49(14):143001, 2016.  https://doi.org/10.1088/1751-8113/49/14/143001
  2. 2.
    S. Vinjanampathy, J. Anders, Quantum thermodynamics. Contemp. Phys. 57(4), 545–579 (2016).  https://doi.org/10.1080/00107514.2016.1201896ADSCrossRefGoogle Scholar
  3. 3.
    J. Åberg, Truly work-like work extraction via a single-shot analysis. Nat. Commun. 4, 1925 (2013).  https://doi.org/10.1038/ncomms2712ADSCrossRefGoogle Scholar
  4. 4.
    M. Campisi, P. Hänggi, P. Talkner, Colloquium. Rev. Mod. Phys. 83(3), 771–791 (2011).  https://doi.org/10.1103/RevModPhys.83.771
  5. 5.
    M. Esposito, U. Harbola, S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81(4), 1665–1702 (2009).  https://doi.org/10.1103/RevModPhys.81.1665
  6. 6.
    M. Esposito, U. Harbola, S. Mukamel, Erratum: Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems [rev. mod. phys. 81, 1665 (2009)]. Rev. Mod. Phys., 86(3):1125–1125, Sep 2014.  https://doi.org/10.1103/RevModPhys.86.1125
  7. 7.
    C. Jarzynski, Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale. Ann. Rev. Condens. Matter Phys. 2(1), 329–351 (2011).  https://doi.org/10.1146/annurev-conmatphys-062910-140506ADSCrossRefGoogle Scholar
  8. 8.
    J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, E. Lutz, Nanoscale heat engine beyond the carnot limit. Phys. Rev. Lett. 112(3), 030602 (2014).  https://doi.org/10.1103/PhysRevLett.112.030602
  9. 9.
    L.A. Correa, J.e.P. Palao, D. Alonso, G. Adesso, Quantum-enhanced absorption refrigerators. Sci. Rep 4, 3949 (2014).  https://doi.org/10.1038/srep03949
  10. 10.
    R. Alicki, D. Gelbwaser-Klimovsky, Non-equilibrium quantum heat machines. New J. Phys. 17(11), 115012 (2015).  https://doi.org/10.1088/1367-2630/17/11/115012
  11. 11.
    J.B. Brask, N. Brunner, Small quantum absorption refrigerator in the transient regime: Time scales, enhanced cooling, and entanglement. Phys. Rev. E 92(6), 062101 (2015).  https://doi.org/10.1103/PhysRevE.92.062101
  12. 12.
    R. Uzdin, A. Levy, R. Kosloff (2015), Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic Signatures. Phys. Rev. X. 5(3), 031044.  https://doi.org/10.1103/PhysRevX.5.031044
  13. 13.
    M.T. Mitchison, M.P. Woods, J. Prior, M. Huber, Coherence-assisted single-shot cooling by quantum absorption refrigerators. New J. Phys. 17(11), 115013 (2015).  https://doi.org/10.1088/1367-2630/17/11/115013
  14. 14.
    P.P. Hofer, M. Perarnau-Llobet, J.B. Brask, R. Silva, M. Huber, N. Brunner, Autonomous quantum refrigerator in a circuit qed architecture based on a josephson junction. Phys. Rev. B 94(23), 235420 (2016).  https://doi.org/10.1103/PhysRevB.94.235420
  15. 15.
    S. Nimmrichter, J. Dai, A. Roulet, V. Scarani, Quantum and classical dynamics of a three-mode absorption refrigerator. Quantum 1, 37 (2017).  https://doi.org/10.22331/q-2017-12-11-37
  16. 16.
    K. Brandner, M. Bauer, U. Seifert, Universal coherence-induced power losses of quantum heat engines in linear response. Phys. Rev. Lett. 119(17), 170602 (2017).  https://doi.org/10.1103/PhysRevLett.119.170602
  17. 17.
    J. Klatzow, C. Weinzetl, P.M. Ledingham, J.N. Becker, D.J. Saunders, J. Nunn, I.A. Walmsley, R. Uzdin, E. Poem, Experimental demonstration of quantum effects in the operation of microscopic heat engines (2017). arXiv:1710.08716
  18. 18.
    A.E. Allahverdyan, R. Balian, T.M. Nieuwenhuizen, Maximal work extraction from finite quantum systems. EPL (Europhysics Letters) 67(4), 565 (2004).  https://doi.org/10.1209/epl/i2004-10101-2
  19. 19.
    K. Funo, Y. Watanabe, M. Ueda, Thermodynamic work gain from entanglement. Phys. Rev. A 88(5), 052319 (2013).  https://doi.org/10.1103/PhysRevA.88.052319
  20. 20.
    M. Perarnau-Llobet, K.V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, A. Acín, Extractable work from correlations. Phys. Rev. X 5(4), 041011 (2015).  https://doi.org/10.1103/PhysRevX.5.041011
  21. 21.
    K. Korzekwa, M. Lostaglio, J. Oppenheim, D. Jennings, The extraction of work from quantum coherence. New J. Phys 18(2), 023045 (2016).  https://doi.org/10.1088/1367-2630/18/2/023045
  22. 22.
    A. Misra, U. Singh, S. Bhattacharya, A.K. Pati, Energy cost of creating quantum coherence. Phys. Rev. A 93(5), 052335 (2016).  https://doi.org/10.1103/PhysRevA.93.052335
  23. 23.
    N. Lörch, C. Bruder, N. Brunner, P.P. Hofer, Optimal work extraction from quantum states by photo-assisted cooper pair tunneling. Quantum Science and Technology 3(3), 035014 (2018).  https://doi.org/10.1088/2058-9565/aacbf3
  24. 24.
    K.V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, A. Acín, Entanglement generation is not necessary for optimal work extraction. Phys. Rev. Lett. 111(24), 240401 (2013).  https://doi.org/10.1103/PhysRevLett.111.240401
  25. 25.
    N. Brunner, M. Huber, N. Linden, S. Popescu, R. Silva, P. Skrzypczyk, Entanglement enhances cooling in microscopic quantum refrigerators. Phys. Rev. E 89(3), 032115 (2014).  https://doi.org/10.1103/PhysRevE.89.032115
  26. 26.
    R. Uzdin, A. Levy, R. Kosloff, Equivalence of quantum heat machines, and quantum-thermodynamic signatures. Phys. Rev. X 5(3), 031044 (2015).  https://doi.org/10.1103/PhysRevX.5.031044
  27. 27.
    F. Campaioli, F.A. Pollock, F.C. Binder, L. Céleri, J. Goold, S. Vinjanampathy, K. Modi, Enhancing the charging power of quantum batteries. Phys. Rev. Lett. 118(15), 150601 (2017).  https://doi.org/10.1103/PhysRevLett.118.150601
  28. 28.
    D. Ferraro, M. Campisi, G.M. Andolina, V. Pellegrini, M. Polini, High-power collective charging of a solid-state quantum battery. Phys. Rev. Lett. 120(11), 117702 (2018).  https://doi.org/10.1103/PhysRevLett.120.117702
  29. 29.
    G. Watanabe, B.P. Venkatesh, P. Talkner, A. del Campo, Quantum performance of thermal machines over many cycles. Phys. Rev. Lett. 118(5), 050601 (2017).  https://doi.org/10.1103/PhysRevLett.118.050601
  30. 30.
    P. Busch, P. Lahti, R.F. Werner, Colloquium: Quantum root-mean-square error and measurement uncertainty relations. Rev. Mod. Phys. 86(4), 1261–1281 (2014).  https://doi.org/10.1103/RevModPhys.86.1261
  31. 31.
    C. Jarzynski, Nonequilibrium work relations: foundations and applications. J. Euro. Phys. B 64(3), 331–340 (2008).  https://doi.org/10.1140/epjb/e2008-00254-2ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    G.N. Bochkov, I.E. Kuzovlev, General theory of thermal fluctuations in nonlinear systems. Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 72, 238–247 (1977)ADSGoogle Scholar
  33. 33.
    G. Bochkov, Y. Kuzovlev, Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. generalized fluctuation-dissipation theorem. Physica A: Statistical Mechanics and its Applications, 106(3):443 – 479, 1981.  https://doi.org/10.1016/0378-4371(81)90122-9
  34. 34.
    S. Yukawa, A quantum analogue of the jarzynski equality. J. Phys. Soc. Japan. 69(8), 2367–2370 (2000).  https://doi.org/10.1143/JPSJ.69.2367ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    T. Monnai, S. Tasaki, Quantum correction of fluctuation theorem (2003). arXiv:1707.04930
  36. 36.
    V. Chernyak, S. Mukamel, Effect of quantum collapse on the distribution of work in driven single molecules. Phys. Rev. Lett. 93(4), 048302 (2004).  https://doi.org/10.1103/PhysRevLett.93.048302
  37. 37.
    A.E. Allahverdyan, T.M. Nieuwenhuizen, Fluctuations of work from quantum subensembles: The case against quantum work-fluctuation theorems. Phys. Rev. E 71(6), 066102 (2005).  https://doi.org/10.1103/PhysRevE.71.066102
  38. 38.
    A. Engel, R. Nolte, Jarzynski equation for a simple quantum system: Comparing two definitions of work. EPL (Europhysics Letters) 79(1), 10003 (2007).  https://doi.org/10.1209/0295-5075/79/10003
  39. 39.
    M.F. Gelin, D.S. Kosov, Unified approach to the derivation of work theorems for equilibrium and steady-state, classical and quantum hamiltonian systems. Phys. Rev. E 78(1), 011116 (2008).  https://doi.org/10.1103/PhysRevE.78.011116
  40. 40.
    J. Kurchan. A quantum fluctuation theorem (2000). arXiv:cond-mat/0007360
  41. 41.
    H. Tasaki. Jarzynski relations for quantum systems and some applications (2000). arXiv:cond-mat/0009244
  42. 42.
    P. Talkner, E. Lutz, P. Hänggi, Fluctuation theorems: Work is not an observable. Phys. Rev. E 75(5), 050102 (2007).  https://doi.org/10.1103/PhysRevE.75.050102
  43. 43.
    P. Hänggi, P. Talkner, The other qft. Nature Physics 11(2), 108 (2017).  https://doi.org/10.1038/nphys3167
  44. 44.
    M. Campisi, P. Hänggi, P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys. 83(3), 771–791 (2011).  https://doi.org/10.1103/RevModPhys.83.771
  45. 45.
    M. Campisi, P. Hänggi, P. Talkner, Erratum: Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys., 83(4):1653–1653, (2011).  https://doi.org/10.1103/RevModPhys.83.1653
  46. 46.
    M. Campisi, P. Talkner, P. Hänggi, Fluctuation theorem for arbitrary open quantum systems. Phys. Rev. Lett. 102(21), 210401 (2009).  https://doi.org/10.1103/PhysRevLett.102.210401
  47. 47.
    G. Huber, F. Schmidt-Kaler, S. Deffner, E. Lutz, Employing trapped cold ions to verify the quantum jarzynski equality. Phys. Rev. Lett. 101(7), 070403 (2008).  https://doi.org/10.1103/PhysRevLett.101.070403
  48. 48.
    T.B. Batalhão, A.M. Souza, L. Mazzola, R. Auccaise, R.S. Sarthour, I.S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, R.M. Serra, Experimental reconstruction of work distribution and study of fluctuation relations in a closed quantum system. Phys. Rev. Lett. 113(14), 140601 (2014).  https://doi.org/10.1103/PhysRevLett.113.140601
  49. 49.
    S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H.T. Quan, K. Kim, Experimental test of the quantum jarzynski equality with a trapped-ion system. Nature Physics 11(2), 193 (2014).  https://doi.org/10.1038/nphys3197
  50. 50.
    F. Cerisola, Y. Margalit, S. Machluf, A.J. Roncaglia, J.P. Paz, R. Folman, Using a quantum work meter to test non-equilibrium fluctuation theorems. Nat. Commun. 8(1), 1241 (2017).  https://doi.org/10.1038/s41467-017-01308-7
  51. 51.
    C. Jarzynski, H.T. Quan, S. Rahav, Quantum-classical correspondence principle for work distributions. Phys. Rev. X 5(3), 031038 (2015).  https://doi.org/10.1103/PhysRevX.5.031038
  52. 52.
    L. Zhu, Z. Gong, B. Wu, H.T. Quan, Quantum-classical correspondence principle for work distributions in a chaotic system. Phys. Rev. E 93(6), 062108 (2016).  https://doi.org/10.1103/PhysRevE.93.062108
  53. 53.
    A.E. Allahverdyan, Nonequilibrium quantum fluctuations of work. Phys. Rev. E 90(3), 032137 (2014).  https://doi.org/10.1103/PhysRevE.90.032137
  54. 54.
    P. Solinas, S. Gasparinetti, Full distribution of work done on a quantum system for arbitrary initial states. Phys. Rev. E 92(4), 042150 (2015).  https://doi.org/10.1103/PhysRevE.92.042150
  55. 55.
    P. Kammerlander, J. Anders, Coherence and measurement in quantum thermodynamics. Scientific reports 6, 22174 (2016).  https://doi.org/10.1038/srep22174ADSCrossRefGoogle Scholar
  56. 56.
    S. Deffner, J.P. Paz, W.H. Zurek, Quantum work and the thermodynamic cost of quantum measurements. Phys. Rev. E 94(1), 010103 (2016).  https://doi.org/10.1103/PhysRevE.94.010103
  57. 57.
    M. Perarnau-Llobet, E. Bäumer, K.V. Hovhannisyan, M. Huber, A. Acin, No-go theorem for the characterization of work fluctuations in coherent quantum systems. Phys. Rev. Lett. 118(7), 070601 (2017).  https://doi.org/10.1103/PhysRevLett.118.070601
  58. 58.
    G. Watanabe, B.P. Venkatesh, P. Talkner, Generalized energy measurements and modified transient quantum fluctuation theorems. Phys. Rev. E 89(5), 052116 (2014).  https://doi.org/10.1103/PhysRevE.89.052116
  59. 59.
    B.P. Venkatesh, G. Watanabe, P. Talkner, Quantum fluctuation theorems and power measurements. New J. Phys. 17(7), 075018 (2015).  https://doi.org/10.1088/1367-2630/17/7/075018
  60. 60.
    M. Esposito, C.V. den Broeck, Second law and landauer principle far from equilibrium. EPL (Europhysics Letters) 95(4), 40004 (2011).  https://doi.org/10.1209/0295-5075/95/40004
  61. 61.
    D. Janzing, Quantum thermodynamics with missing reference frames: Decompositions of free energy into non-increasing components. J. Stat. Phys. 125(3), 761–776 (2006).  https://doi.org/10.1007/s10955-006-9220-xADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    M. Lostaglio, D. Jennings, T. Rudolph, Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun. 6, 6383 (2015).  https://doi.org/10.1038/ncomms7383ADSCrossRefGoogle Scholar
  63. 63.
    J.J. Alonso, E. Lutz, A. Romito, Thermodynamics of weakly measured quantum systems. Phys. Rev. Lett. 116(8), 080403 (2016).  https://doi.org/10.1103/PhysRevLett.116.080403
  64. 64.
    C. Elouard, D.A. Herrera-Martí, M. Clusel, A. Auffeves, The role of quantum measurement in stochastic thermodynamics. npj Quantum Information, 3(1), 9 (2017).  https://doi.org/10.1038/s41534-017-0008-4
  65. 65.
    A.J. Roncaglia, F. Cerisola, J.P. Paz, Work measurement as a generalized quantum measurement. Phys. Rev. Lett. 113(25), 250601 (2014)  https://doi.org/10.1103/PhysRevLett.113.250601
  66. 66.
    G.D. Chiara, A.J. Roncaglia, J.P. Paz, Measuring work and heat in ultracold quantum gases. New J. Phys. 17(3), 035004 (2015).  https://doi.org/10.1088/1367-2630/17/3/035004
  67. 67.
    P. Talkner, P. Hänggi, Aspects of quantum work. Phys. Rev. E 93(2), 022131 (2016).  https://doi.org/10.1103/PhysRevE.93.022131
  68. 68.
    P. Solinas, H.J.D. Miller, J. Anders, Measurement-dependent corrections to work distributions arising from quantum coherences. Phys. Rev. A 96(5), 052115 (2017).  https://doi.org/10.1103/PhysRevA.96.052115
  69. 69.
    P.P. Hofer, Quasi-probability distributions for observables in dynamic systems. Quantum 1, 32 (2017).  https://doi.org/10.1103/PhysRevA.96.052115CrossRefGoogle Scholar
  70. 70.
    R. Sampaio, S. Suomela, T. Ala-Nissila, J. Anders, T.G. Philbin, Quantum work in the bohmian framework. Phys. Rev. A 97(1), 012131 (2018).  https://doi.org/10.1103/PhysRevA.97.012131
  71. 71.
    M. Lostaglio, Quantum fluctuation theorems, contextuality, and work quasiprobabilities. Phys. Rev. Lett. 120(4), 040602 (2018).  https://doi.org/10.1103/PhysRevLett.120.040602
  72. 72.
    A.M. Alhambra, L. Masanes, J. Oppenheim, C. Perry, Fluctuating work: From quantum thermodynamical identities to a second law equality. Phys. Rev. X 6(4), 041017 (2016).  https://doi.org/10.1103/PhysRevX.6.041017
  73. 73.
    J.G. Richens, L. Masanes, Work extraction from quantum systems with bounded fluctuations in work. Nat. Commun. 7, 13511 (2016)  https://doi.org/10.1038/ncomms13511
  74. 74.
    J. Åberg, Fully quantum fluctuation theorems. Phys. Rev. X 8(1), 011019 (2018).  https://doi.org/10.1103/PhysRevX.8.011019
  75. 75.
    J.S. Bell, On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38(3), 447 (1966).  https://doi.org/10.1103/RevModPhys.38.447
  76. 76.
    S. Kochen, E.P. Specker, The problem of hidden variables in quantum mechanics. In The Logico-Algebraic Approach to Quantum Mechanics, p. 293–328. Springer, 1975.  https://doi.org/10.1007/978-94-010-1795-4_17
  77. 77.
    M. Howard, J. Wallman, V. Veitch, J. Emerson, Contextuality supplies the/magic/’for quantum computation. Nature 510(7505), 351–355 (2014).  https://doi.org/10.1038/nature13460ADSCrossRefGoogle Scholar
  78. 78.
    N. Delfosse, P.A. Guerin, J. Bian, R. Raussendorf, Wigner function negativity and contextuality in quantum computation on rebits. Phys. Rev. X 5(2), 021003 (2015).  https://doi.org/10.1103/PhysRevX.5.021003CrossRefGoogle Scholar
  79. 79.
    J. Bermejo-Vega, N. Delfosse, D.E. Browne, C. Okay, R. Raussendorf, Contextuality as a resource for models of quantum computation with qubits. Phys. Rev. Lett. 119(12), 120505 (2017).  https://doi.org/10.1103/PhysRevLett.119.120505
  80. 80.
    R.W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements. Phys. Rev. A 71(5), 052108 (2005).  https://doi.org/10.1103/PhysRevA.71.052108ADSMathSciNetCrossRefGoogle Scholar
  81. 81.
    N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner, Bell nonlocality. Rev. Mod. Phys. 86(2), 419 (2014).  https://doi.org/10.1103/RevModPhys.86.419
  82. 82.
    P. Solinas, S. Gasparinetti, Probing quantum interference effects in the work distribution. Phys. Rev. A 94(5), 052103 (2016).  https://doi.org/10.1103/PhysRevA.94.052103
  83. 83.
    B.-M. Xu, J. Zou, L.-S. Guo, X.-M. Kong, Effects of quantum coherence on work statistics. Phys. Rev. A 97(5), 052122 (2018).  https://doi.org/10.1103/PhysRevA.97.052122
  84. 84.
    Y. Aharonov, D.Z. Albert, L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60(14), 1351 (1988).  https://doi.org/10.1103/PhysRevLett.60.1351
  85. 85.
    H.M. Wiseman, Weak values, quantum trajectories, and the cavity-qed experiment on wave-particle correlation. Phys. Rev. A 65(3), 032111 (2002).  https://doi.org/10.1103/PhysRevA.65.032111
  86. 86.
    H.J.D. Miller, J. Anders, Time-reversal symmetric work distributions for closed quantum dynamics in the histories framework. New J. Phys. 19(6), 062001 (2017).  https://doi.org/10.1088/1367-2630/aa703f
  87. 87.
    J.W. Hall, Prior information: How to circumvent the standard joint-measurement uncertainty relation. Phys. Rev. A 69(5), 052113 (2004).  https://doi.org/10.1103/PhysRevA.69.052113
  88. 88.
    R.B. Griffiths, Consistent histories and the interpretation of quantum mechanics. J. Stat. Phy. 36(1–2), 219–272 (1984).  https://doi.org/10.1007/BF01015734
  89. 89.
    S. Goldstein, D.N. Page, Linearly positive histories: probabilities for a robust family of sequences of quantum events. Phys. Rev. Lett, 74(19):3715, 1995.  https://doi.org/10.1103/PhysRevLett.74.3715
  90. 90.
    T. Sagawa, Second law-like inequalities with quantum, relative entropy: An introduction, pages 125–190. World Scientific, 2012.  https://doi.org/10.1142/9789814425193_0003
  91. 91.
    M.F. Pusey, Anomalous weak values are proofs of contextuality. Phys. Rev. Lett. 113(20), 200401 (2014).  https://doi.org/10.1103/PhysRevLett.113.200401
  92. 92.
    N.S. Williams, A.N. Jordan, Weak values and the leggett-garg inequality in solid-state qubits. Phys. Rev. Lett. 100(2), 026804 (2008).  https://doi.org/10.1103/PhysRevLett.100.026804
  93. 93.
    R. Blattmann, K. Mølmer, Macroscopic realism of quantum work fluctuations. Physical Review A 96(1), 012115 (2017).  https://doi.org/10.1103/PhysRevA.96.012115ADSCrossRefGoogle Scholar
  94. 94.
    H.J.D. Miller, J. Anders, Leggett-garg inequalities for quantum fluctuating work. Entropy, 20(3), 2018.  https://doi.org/10.3390/e20030200
  95. 95.
    D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" variables. Phys. Rev. I 85(2), 166–179 (1952). https://doi.org/10.1103/PhysRev.85.166
  96. 96.
    D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" variables. Phys. Rev. II 85(2), 180–193 (1952).  https://doi.org/10.1103/PhysRev.85.180
  97. 97.
    L. De Broglie, L’interprétation de la mécanique ondulatoire. J. Phys. Radium. 20(12), 963–979 (1959).  https://doi.org/10.1051/jphysrad:019590020012096300
  98. 98.
    P.R. Holland, The quantum theory of motion: an account of the de Broglie-Bohm causal interpretation of quantum mechanics. Cambridge university press, 1995Google Scholar
  99. 99.
    D. Dürr, S. Goldstein, N. Zanghì, Quantum physics without quantum philosophy. Springer Science & Business Media, 2013.  https://doi.org/10.1007/978-3-642-30690-7
  100. 100.
    Y. Aharonov, D.Z. Albert, L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60(14), 1351–1354 (1988).  https://doi.org/10.1103/PhysRevLett.60.1351
  101. 101.
    I.M. Duck, P.M. Stevenson, E.C.G. Sudarshan, The sense in which a "weak measurement" of a spin particle’s spin component yields a value 100. Phys. Rev. D 40(6), 2112–2117 (1989).  https://doi.org/10.1103/PhysRevD.40.2112
  102. 102.
    S. Kocsis, B. Braverman, S. Ravets, M.J. Stevens, R.P. Mirin, L.K. Shalm, A.M. Steinberg, Observing the average trajectories of single photons in a two-slit interferometer. Science, 332(6034):1170–1173, (2011).  https://doi.org/10.1126/science.1202218
  103. 103.
    D.H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K.J. Resch, H.M. Wiseman, A. Steinberg, Experimental nonlocal and surreal bohmian trajectories. Science Advances 2(2), e1501466 (2016).  https://doi.org/10.1126/sciadv.1501466ADSCrossRefGoogle Scholar
  104. 104.
    Y. Xiao, Y. Kedem, J.-S. Xu, C.-F. Li, G.-C. Guo, Experimental nonlocal steering of bohmian trajectories. Opt. Express 25(13), 14463–14472 (2017).  https://doi.org/10.1364/OE.25.014463ADSCrossRefGoogle Scholar
  105. 105.
    H. Pashayan, J.J. Wallman, S.D. Bartlett, Estimating outcome probabilities of quantum circuits using quasiprobabilities. Physical review letters 115(7), 070501 (2015).  https://doi.org/10.1103/PhysRevLett.115.070501
  106. 106.
    H. Pashayan, J.J. Wallman, S.D. Bartlett, Quantum features and signatures of quantum-thermal machines (2018). arXiv:1803.05586
  107. 107.
    S. Pironio, A. Acín, S. Massar, A.B. de La Giroday, D.N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T.A. Manning et al., Random numbers certified by bell theorem. Nature 464(7291), 1021 (2010).  https://doi.org/10.1038/nature09008ADSCrossRefGoogle Scholar
  108. 108.
    M.D. Mazurek, M.F. Pusey, R. Kunjwal, K.J. Resch, R.W. Spekkens, An experimental test of noncontextuality without unphysical idealizations. Nat. Commun. 7:ncomms11780, (2016).  https://doi.org/10.1038/ncomms11780
  109. 109.
    P. Skrzypczyk, A.J. Short, S. Popescu, Work extraction and thermodynamics for individual quantum systems. Nat. Commun. 5, 4185 (2014).  https://doi.org/10.1038/ncomms5185
  110. 110.
    J. Åberg, Catalytic coherence. Phys. Rev. Lett. 113, 150402 (2014).  https://doi.org/10.1103/PhysRevLett.113.150402
  111. 111.
    D. Reeb, M.M. Wolf, An improved landauer principle with finite-size corrections. New J. Phys. 16(10), 103011 (2014).  https://doi.org/10.1088/1367-2630/16/10/103011

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Elisa Bäumer
    • 1
    Email author
  • Matteo Lostaglio
    • 2
  • Martí Perarnau-Llobet
    • 3
  • Rui Sampaio
    • 4
  1. 1.Institute for Theoretical PhysicsETH ZurichZürichSwitzerland
  2. 2.ICFO-Institut de Ciencies FotoniquesThe Barcelona Institute of Science and TechnologyCastelldefels (Barcelona)Spain
  3. 3.Max-Planck-Institut für QuantenoptikGarchingGermany
  4. 4.QTF Center of Excellence, Department of Applied PhysicsAalto UniversityAaltoFinland

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