Skip to main content
SpringerLink
Log in

Resource Theory of Quantum Thermodynamics: Thermal Operations and Second Laws

  • Chapter
  • First Online:
Thermodynamics in the Quantum Regime

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 195))

Abstract

Resource theories are a generic approach used to manage any valuable resource, such as entanglement, purity, and asymmetry. Such frameworks are characterized by two main elements: a set of predefined (free) operations and states, that one assumes to be easily obtained at no cost. Given these ground rules, one can ask: what is achievable by using such free operations and states? This usually results in a set of state transition conditions, that tell us if a particular state \( \rho \) may evolve into another state \( \rho ' \) via the usage of free operations and states. We shall see in this chapter that thermal interactions can be modelled as a resource theory. The state transition conditions arising out of such a framework, are then referred to as “second laws”. We shall also see how such state transition conditions recover classical thermodynamics in the i.i.d. limit. Finally, we discuss how these laws are applied to study fundamental limitations to the performance of quantum heat engines.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Any state which is not a free state is then called a resource state.

  2. 2.

    Up to an arbitrarily good approximation to the final state \(\rho _S'\) for fixed dimensional \(\mathcal {H}_S\), in operator norm.

References

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

    Google Scholar 

  2. C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53, 2046 (1996a). https://doi.org/10.1103/PhysRevA.53.2046

  3. P.G. Kwiat, S. Barraza-Lopez, A. Stefanov, N. Gisin, Nature 409, 1014 (2001). https://doi.org/10.1038/35059017

  4. H. Takahashi, J.S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, M. Sasaki, Nat. Photonics 4, 178 (2010). https://doi.org/10.1038/nphoton.2010.1

  5. K.G.H. Vollbrecht, C.A. Muschik, J.I. Cirac, Phys. Rev. Lett. 107, 120502 (2011). https://doi.org/10.1103/PhysRevLett.107.120502

  6. F.G. Brandão, G. Gour, Phys. Rev. Lett. 115, 070503 (2015). https://doi.org/10.1103/PhysRevLett.115.070503

  7. G. Gour, R.W. Spekkens, New J. Phys. 10, 033023 (2008). https://doi.org/10.1088/1367-2630/10/3/033023

  8. M.B. Plenio, S. Virmani, Quantum Inf. Comput. 7, 1 (2007). arXiv:quant-ph/0504163

  9. R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865

  10. T. Baumgratz, M. Cramer, M. Plenio, Phys. Rev. Lett. 113, 140401 (2014). https://doi.org/10.1103/PhysRevLett.113.140401

  11. A. Winter, D. Yang, Phys. Rev. Lett. 116, 120404 (2016). https://doi.org/10.1103/PhysRevLett.116.120404

  12. M. Horodecki, P. Horodecki, J. Oppenheim, Phys. Rev. A 67, 062104 (2003). https://doi.org/10.1103/PhysRevA.67.062104

  13. F.G.S.L. Brandão, M. Horodecki, J. Oppenheim, J.M. Renes, R.W. Spekkens, Phys. Rev. Lett. 111, 250404 (2013). https://doi.org/10.1103/PhysRevLett.111.250404

  14. I. Marvian, R.W. Spekkens, New J. Phys. 15, 033001 (2013). https://doi.org/10.1088/1367-2630/15/3/033001

  15. G. Gour, M.P. Müller, V. Narasimhachar, R.W. Spekkens, N. Yunger Halpern, Physics Reports 583, 1 (2015). https://doi.org/10.1016/j.physrep.2015.04.003

  16. C.H. Bennett, Int. J. Theor. Phys. 21, 905 (1982). https://doi.org/10.1007/BF02084158

  17. R. Landauer, IBM J. Res. Dev. 5, 183 (1961). https://doi.org/10.1147/rd.53.0183

  18. C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W.K. Wootters, Phys. Rev. Lett. 76, 722 (1996b). https://doi.org/10.1103/PhysRevLett.76.722

  19. M. Horodecki, J. Oppenheim, R. Horodecki, Phys. Rev. Lett. 89, 240403 (2002). https://doi.org/10.1103/PhysRevLett.89.240403

  20. I. Devetak, A. Harrow, A. Winter, IEEE Trans. Inf. Theory 54, 4587 (2008). https://doi.org/10.1109/TIT.2008.928980

  21. R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki, Open Syst. Inf. Dyn. 11, 205 (2004). arXiv:quant-ph/0402012

  22. R. Bhatia, Matrix Analysis, Graduate texts in mathematics (Springer, Berlin, 1997). https://doi.org/10.1007/978-1-4612-0653-8

  23. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1952)

    Google Scholar 

  24. M.A. Nielsen, Lect. Notes, Department of physics (University of Queensland, Australia, 2002)

    Google Scholar 

  25. J. Scharlau, M.P. Mueller, Quantum 2, 54 (2018). https://doi.org/10.22331/q-2018-02-22-54

  26. W. Pusz, S.L. Woronowicz, Commun. Math. Phys. 58, 273 (1978). https://doi.org/10.1007/BF01614224

  27. D. Janzing, P. Wocjan, R. Zeier, R. Geiss, T. Beth, Int. J. Theor. Phys. 39, 2717 (2000). arXiv:quant-ph/0002048

  28. F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, S. Wehner, Proc. Natl. Acad. Sci. 112, 3275 (2015a). https://doi.org/10.1073/pnas.1411728112

  29. M. Horodecki, J. Oppenheim. Nat. Commun. 4 (2013). https://doi.org/10.1038/ncomms3059

  30. M. Lostaglio, K. Korzekwa, D. Jennings, T. Rudolph, Phys. Rev. X 5, 021001 (2015a). https://doi.org/10.1103/PhysRevX.5.021001

  31. P. Faist, R. Renner, Phys. Rev. X 8, 021011 (2018). https://doi.org/10.1103/PhysRevX.8.021011

  32. N. Yunger Halpern, J.M. Renes, Phys. Rev. E 93, 022126 (2016). https://doi.org/10.1103/PhysRevE.93.022126

  33. N. Yunger Halpern, J. Phys. A: Math. Theor. 51, 094001 (2018). https://doi.org/10.1088/1751-8121/aaa62f

  34. N.Yunger Halpern, P. Faist, J. Oppenheim, A. Winter, Nat. Commun. 7 (2016). https://doi.org/10.1038/ncomms12051

  35. Y. Guryanova, S. Popescu, A.J. Short, R. Silva, P. Skrzypczyk, Nat. Comm. 7 (2016). https://doi.org/10.1038/ncomms12049

  36. M. Lostaglio, D. Jennings, T. Rudolph, New J. Phys. 19, 043008 (2017). https://doi.org/10.1088/1367-2630/aa617f

  37. S. Popescu, A.B. Sainz, A.J. Short, A. Winter, Phil. Trans. Roy. Soc. A 376(2123), 20180111 (2018). https://doi.org/10.1098/rsta.2018.0111

  38. G. Gour, D. Jennings, F. Buscemi, R. Duan, I. Marvian, Nat. Commun. 9(5352) (2018). https://doi.org/10.1038/s41467-018-06261-7

  39. F. Reif, Fundamentals of Statistical and Thermal Physics: (Waveland Press, 2009)

    Google Scholar 

  40. C. Adkins, Equilibrium Thermodynamics (Cambridge University Press, Cambridge, 1983)

    Google Scholar 

  41. D. Schroeder, An Introduction to Thermal Physics (Addison Wesley, USA, 2000). https://doi.org/10.1119/1.19116

  42. K. Huang, Statistical Mechanics (Wiley, New York, 1987)

    Google Scholar 

  43. D. Egloff, O.C.O. Dahlsten, R. Renner, V. Vedral, New J. Phys. 17, 073001 (2015). https://doi.org/10.1088/1367-2630/17/7/073001

  44. L. Del Rio, J. Åberg, R. Renner, O. Dahlsten, V. Vedral, Nature 474, 61 (2011). https://doi.org/10.1038/nature10123

  45. J. Åberg, Nat. Commun. 4 (2013). https://doi.org/10.1038/ncomms2712

  46. G.E. Crooks, J. Stat. Phys. 90, 1481 (1998). https://doi.org/10.1023/A:1023208217925

  47. B. Piechocinska, Phys. Rev. A 61, 062314 (2000). https://doi.org/10.1103/PhysRevA.61.062314

  48. M. Esposito, C.V. den Broeck, EPL (Europhys. Lett.) 95, 40004 (2011). https://doi.org/10.1209/0295-5075/95/40004

  49. H. Hossein-Nejad, E.J. O’Reilly, A. Olaya-Castro, New J. Phys. 17, 075014 (2015). https://doi.org/10.1088/1367-2630/17/7/075014

  50. R. Gallego, J. Eisert, H. Wilming, New J. Phys. 18, 103017 (2016). https://doi.org/10.1088/1367-2630/18/10/103017

  51. J. Gemmer, J. Anders, New J. Phys. 17, 085006 (2015). https://doi.org/10.1088/1367-2630/17/8/085006

  52. M. Perarnau-Llobet, K.V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, A. Acín, Phys. Rev. X 5, 041011 (2015). https://doi.org/10.1103/PhysRevX.5.041011

  53. P. Skrzypczyk, A.J. Short, S. Popescu, Nat. Commun. 5 (2014). https://doi.org/10.1038/ncomms5185

  54. M.O. Scully, Phys. Rev. Lett. 88, 050602 (2002). https://doi.org/10.1103/PhysRevLett.88.050602

  55. J.G. Richens, L. Masanes, Nat. Commun. 7, 13511 (2016). https://doi.org/10.1038/ncomms13511

  56. J. Åberg, Phys. Rev. X 8, 011019 (2018). https://doi.org/10.1103/PhysRevX.8.011019

  57. Á. M. Alhambra, L. Masanes, J. Oppenheim, C. Perry, Phys. Rev. X 6, 041017 (2016). https://doi.org/10.1103/PhysRevX.6.041017

  58. M.P. Woods, N. Ng, S. Wehner (2015). arXiv: 1506.02322

  59. D. Jonathan, M.B. Plenio, Phys. Rev. Lett. 83, 3566 (1999). https://doi.org/10.1103/PhysRevLett.83.3566

  60. M. Lostaglio, D. Jennings, T. Rudolph, Nat. Commun. 6, 6383 (2015b). https://doi.org/10.1038/ncomms7383

  61. F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, S. Wehner, The second laws of quantum thermodynamics (2015b). https://doi.org/10.1073/pnas.1411728112

  62. K.M. Audenaert, J. Phys. A: Math. Theor. 40, 8127 (2007). https://doi.org/10.1088/1751-8113/40/28/S18

  63. M.P. Mueller, Phys. Rev. X 8, 041051 (2018). https://doi.org/10.1103/PhysRevX.8.041051

  64. Á.M. Alhambra, J. Oppenheim, C. Perry, Phys. Rev. X 6, 041016 (2016). https://doi.org/10.1103/PhysRevX.6.041016

  65. S. Carnot, Reflections on the Motive Power of Fire (1824)

    Google Scholar 

  66. M. Scully, M. Zubairy, G. Agarwal, H. Walther, Science 299, 862 (2003). https://doi.org/10.1126/science.1078955

  67. J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, E. Lutz, Phys. Rev. Lett. 112, 030602 (2014). https://doi.org/10.1103/PhysRevLett.112.030602

  68. N.H.Y. Ng, M.P. Woods, S. Wehner, New J. Phys. 19, 113005 (2017). https://doi.org/10.1088/1367-2630/aa8ced

  69. J. Gemmer, M. Michel, G. Mahler, Quantum thermodynamics (Springer, Berlin, 2009). https://doi.org/10.1007/978-3-540-70510-9

  70. R. Kosloff, Entropy 15, 2100 (2013). https://doi.org/10.3390/e15062100

  71. R. Alicki, J. Phys. A: Math. Gen. 12, L103 (1979). https://doi.org/10.1088/0305-4470/12/5/007

  72. N. Brunner, M. Huber, N. Linden, S. Popescu, R. Silva, P. Skrzypczyk, Phys. Rev. E 89, 032115 (2014). https://doi.org/10.1103/PhysRevE.89.032115

  73. N. Linden, S. Popescu, P. Skrzypczyk, Phys. Rev. Lett. 105, 130401 (2010). https://doi.org/10.1103/PhysRevLett.105.130401

  74. K. Szczygielski, D. Gelbwaser-Klimovsky, R. Alicki, Phys. Rev. E 87, 012120 (2013). https://doi.org/10.1103/PhysRevE.87.012120

  75. R. Uzdin, A. Levy, R. Kosloff, Phys. Rev. X 5, 031044 (2015). https://doi.org/10.1103/PhysRevX.5.031044

  76. S. Hilt, E. Lutz, Phys. Rev. A 79, 010101 (2009). https://doi.org/10.1103/PhysRevA.79.010101

  77. H. Peter, G.-L. Ingold, P. Talkner et al., New J. Phys. 10, 115008 (2008). https://doi.org/10.1088/1367-2630/10/11/115008

  78. G.-L. Ingold, P. Hänggi, P. Talkner, Phys. Rev. E 79, 061105 (2009). https://doi.org/10.1103/PhysRevE.79.061105

  79. P. Calabrese, J. Cardy, Phys. Rev. Lett. 96, 136801 (2006). https://doi.org/10.1103/PhysRevLett.96.136801

  80. M. Cramer, C.M. Dawson, J. Eisert, T.J. Osborne, Phys. Rev. Lett. 100, 030602 (2008). https://doi.org/10.1103/PhysRevLett.100.030602

  81. J. Åberg, Phys. Rev. Lett. 113, 150402 (2014). https://doi.org/10.1103/PhysRevLett.113.150402

  82. A.S. Malabarba, A.J. Short, P. Kammerlander, New J. Phys. 17, 045027 (2015). https://doi.org/10.1088/1367-2630/17/4/045027

  83. M.P. Woods, R. Silva, J. Oppenheim (2016). https://doi.org/10.1007/s00023-018-0736-9

  84. C. Perry, P. Ćwikliński, J. Anders, M. Horodecki, J. Oppenheim, Phys. Rev. X 8, 041049 (2018). https://doi.org/10.1103/PhysRevX.8.041049

  85. M. Lostaglio, Á.M. Alhambra, C. Perry, Quantum 2, 52 (2018). https://doi.org/10.22331/q-2018-02-08-52

  86. N. Yunger Halpern, Toward physical realizations of thermodynamic resource theories, in Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge, ed. by I.T. Durham, D. Rickles (Springer International Publishing, Cham, 2017), pp. 135–166. https://doi.org/10.1007/978-3-319-43760-6

  87. C. Jarzynski, Ann. Rev. Condens. Matter Phys. 2, 329 (2011). https://doi.org/10.1146/annurev-conmatphys-062910-140506

  88. M. Esposito, U. Harbola, S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009). https://doi.org/10.1103/RevModPhys.81.1665

  89. M. Campisi, P. Hänggi, P. Talkner, Rev. Mod. Phys. 83, 771 (2011). https://doi.org/10.1103/RevModPhys.83.771

  90. U. Seifert, Rep. Prog. Phys. 75 (2012). https://doi.org/10.1088/0034-4885/75/12/126001

  91. G.M. Wang, E.M. Sevick, E. Mittag, D.J. Searles, D.J. Evans, Phys. Rev. Lett. 89, 050601 (2002). https://doi.org/10.1103/PhysRevLett.89.050601

  92. Y. Utsumi, D.S. Golubev, M. Marthaler, K. Saito, T. Fujisawa, G. Schön, Phys. Rev. B 81, 125331 (2010). https://doi.org/10.1103/PhysRevB.81.125331

  93. 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, Phys. Rev. Lett. 113, 140601 (2014). https://doi.org/10.1103/PhysRevLett.113.140601

  94. S. Salek, K. Wiesner, Phys. Rev. A 96, 052114 (2017). https://doi.org/10.1103/PhysRevA.96.052114

  95. A.M. Alhambra, S. Wehner, M.M. Wilde, M.P. Woods, Phys. Rev. A 97(6), 062114 (2018). https://doi.org/10.1103/PhysRevA.97.062114

Download references

Acknowledgements

N.N. acknowledges funding from the Alexander von Humboldt foundation. M.W. acknowledges the Swiss National Science Foundation (SNSF) via the NCCR QSIT. M.W. would like to acknowledge the COST MP1209 network “Thermodynamics in the quantum regime”, to which he was a member and from which part of the research reviewed in this chapter was made possible.

Author information

Authors and Affiliations

  1. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195, Berlin, Germany

    Nelly Huei Ying Ng

  2. Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093, Zürich, Switzerland

    Mischa Prebin Woods

Authors
  1. Nelly Huei Ying Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mischa Prebin Woods
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mischa Prebin Woods .

Editor information

Editors and Affiliations

  1. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Felix Binder

  2. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Luis A. Correa

  3. ICFO: Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology, Castelldefels, Spain

    Christian Gogolin

  4. CEMPS, Physics and Astronomy, University of Exeter, Exeter, UK

    Janet Anders

  5. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Gerardo Adesso

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ng, N.H.Y., Woods, M.P. (2018). Resource Theory of Quantum Thermodynamics: Thermal Operations and Second Laws. In: Binder, F., Correa, L., Gogolin, C., Anders, J., Adesso, G. (eds) Thermodynamics in the Quantum Regime. Fundamental Theories of Physics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-99046-0_26

Download citation

Publish with us

Policies and ethics

Search

Navigation