Elementary Thermal Operations

Matteo Lostaglio1, Álvaro M. Alhambra2, and Christopher Perry3

1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain
2Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
3QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

full text pdf

To what extent do thermodynamic resource theories capture physically relevant constraints? Inspired by quantum computation, we define a set of elementary thermodynamic gates that only act on 2 energy levels of a system at a time. We show that this theory is well reproduced by a Jaynes-Cummings interaction in rotating wave approximation and draw a connection to standard descriptions of thermalisation. We then prove that elementary thermal operations present tighter constraints on the allowed transformations than thermal operations. Mathematically, this illustrates the failure at finite temperature of fundamental theorems by Birkhoff and Muirhead-Hardy-Littlewood-Polya concerning stochastic maps. Physically, this implies that stronger constraints than those imposed by single-shot quantities can be given if we tailor a thermodynamic resource theory to the relevant experimental scenario. We provide new tools to do so, including necessary and sufficient conditions for a given change of the population to be possible. As an example, we describe the resource theory of the Jaynes-Cummings model. Finally, we initiate an investigation into how our resource theories can be applied to Heat Bath Algorithmic Cooling protocols.

Recent work using tools from quantum information theory has investigated the laws of thermodynamics at the quantum and nanoscale. In deriving these ultimate limits however, it is assumed that the system under consideration can be manipulated arbitrarily precisely, subject only to constraints such as energy conservation. As such, it is unclear how the newly derived laws relate to real experimental setups. In this work, we provide tools for incorporating physically relevant constraints into the previous analysis, limiting the interactions to those that only act on a reduced amount of energy levels, or/and that can be generated via simple light-matter interaction models. This enables the potential for fundamental limits to be investigated for real physical systems.

► BibTeX data

► References

[1] Ernst Ruch. The diagram lattice as structural principle A. New aspects for representations and group algebra of the symmetric group B. Definition of classification character, mixing character, statistical order, statistical disorder; a general principle for the time evolution of irreversible processes. Theoretica Chimica Acta, 38 (3): 167-183, 1975. 10.1007/​BF01125896.

[2] Ernst Ruch and Alden Mead. The principle of increasing mixing character and some of its consequences. Theoretica chimica acta, 41 (2): 95-117, 1976. ISSN 1432-2234. 10.1007/​BF01178071.

[3] C Alden Mead. Mixing character and its application to irreversible processes in macroscopic systems. The Journal of Chemical Physics, 66 (2): 459-467, 1977. 10.1063/​1.433963.

[4] Ernst Ruch, Rudolf Schranner, and Thomas H. Seligman. The mixing distance. J. Chem. Phys., 69 (1): 386-392, 1978. http:/​/​dx.doi.org/​10.1063/​1.436364.

[5] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and Th. Beth. Thermodynamic cost of reliability and low temperatures: Tightening Landauer's principle and the second law. Int. J. Theor. Phys., 39 (12): 2717-2753, 2000. 10.1023/​A:1026422630734.

[6] Fernando G. S. L. Brandão, Michał Horodecki, Jonathan Oppenheim, Joseph M. Renes, and Robert W. Spekkens. Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett., 111: 250404, Dec 2013. 10.1103/​PhysRevLett.111.250404.

[7] Johan Åberg. Truly work-like work extraction via a single-shot analysis. Nat. Commun., 4: 1925, 2013. 10.1038/​ncomms2712.

[8] M. Horodecki and J. Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun., 4: 2059, June 2013. 10.1038/​ncomms3059.

[9] F. G. S. L. Brandão, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner. The second laws of quantum thermodynamics. Proc. Natl. Acad. Sci. U.S.A., 112: 3275, 2015. 10.1073/​pnas.1411728112.

[10] Paul Skrzypczyk, Anthony J Short, and Sandu Popescu. Work extraction and thermodynamics for individual quantum systems. Nat. Commun., 5: 4185, 2014. 10.1038/​ncomms5185.

[11] D Egloff, O C O Dahlsten, R Renner, and V Vedral. A measure of majorization emerging from single-shot statistical mechanics. New Journal of Physics, 17 (7): 073001, 2015. 10.1088/​1367-2630/​17/​7/​073001.

[12] Matteo Lostaglio, David Jennings, and Terry Rudolph. Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun., 6: 6383, 2015a. 10.1038/​ncomms7383.

[13] Matteo Lostaglio, Markus P. Müller, and Michele Pastena. Stochastic independence as a resource in small-scale thermodynamics. Phys. Rev. Lett., 115: 150402, Oct 2015b. 10.1103/​PhysRevLett.115.150402.

[14] V. Narasimhachar and G. Gour. Low-temperature thermodynamics with quantum coherence. Nature Communications, 6: 7689, July 2015. 10.1038/​ncomms8689.

[15] J. Gemmer and J. Anders. From single-shot towards general work extraction in a quantum thermodynamic framework. New Journal of Physics, 17 (8): 085006, 2015. 10.1088/​1367-2630/​17/​8/​085006.

[16] Jonathan G Richens and Lluis Masanes. Work extraction from quantum systems with bounded fluctuations in work. Nat. Commun., 7: 13511, 2016. 10.1038/​ncomms13511.

[17] Lluis Masanes and Jonathan Oppenheim. A general derivation and quantification of the third law of thermodynamics. Nat. Commun., 8: 14538, 2017. 10.1038/​ncomms14538.

[18] Jakob Scharlau and Markus P Mueller. Quantum Horn's lemma, finite heat baths, and the third law of thermodynamics. arXiv:1605.06092, 2016. URL https:/​/​arxiv.org/​abs/​1605.06092.

[19] Henrik Wilming and Rodrigo Gallego. Third law of thermodynamics as a single inequality. Phys. Rev. X, 7: 041033, Nov 2017. 10.1103/​PhysRevX.7.041033.

[20] Nicole Yunger Halpern, Andrew JP Garner, Oscar CO Dahlsten, and Vlatko Vedral. Introducing one-shot work into fluctuation relations. New Journal of Physics, 17 (9): 095003, 2015. 10.1088/​1367-2630/​17/​9/​095003.

[21] Johan Aberg. Fully quantum fluctuation theorems. arXiv:1601.01302, 2016. URL https:/​/​arxiv.org/​abs/​1601.01302.

[22] Álvaro M Alhambra, Jonathan Oppenheim, and Christopher Perry. Fluctuating states: What is the probability of a thermodynamical transition? Phys. Rev. X, 6 (4): 041016, 2016a. 10.1103/​PhysRevX.6.041016.

[23] 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 (4): 041017, 2016b. 10.1103/​PhysRevX.6.041017.

[24] John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, and Paul Skrzypczyk. The role of quantum information in thermodynamics-a topical review. Journal of Physics A: Mathematical and Theoretical, 49 (14): 143001, 2016. 10.1088/​1751-8113/​49/​14/​143001.

[25] Sai Vinjanampathy and Janet Anders. Quantum thermodynamics. Contemporary Physics, 57 (4): 545-579, 2016. 10.1080/​00107514.2016.1201896.

[26] Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information. Cambridge university press, 2010. 10.1017/​CBO9780511976667.

[27] Nicole Yunger Halpern. Toward physical realizations of thermodynamic resource theories. In Information and Interaction, pages 135-166. Springer, 2017. 10.1007/​978-3-319-43760-6.

[28] Michael Reck, Anton Zeilinger, Herbert J Bernstein, and Philip Bertani. Experimental realization of any discrete unitary operator. Phys. Rev. Lett., 73 (1): 58, 1994. 10.1103/​PhysRevLett.73.58.

[29] Robert Franklin Muirhead. Some methods applicable to identities and inequalities of symmetric algebraic functions of $n$ letters. Proceedings of the Edinburgh Mathematical Society, 21: 144-162, 1902. 10.1017/​S001309150003460X.

[30] Godfrey Harold Hardy, John Edensor Littlewood, and George Pólya. Inequalities. Cambridge University Press, 1952. 10.1007/​978-3-319-44299-0_1.

[31] Garrett Birkhoff. Tres observaciones sobre el algebra lineal. Univ. Nac. Tucumán Rev. Ser. A, 5: 147-151, 1946.

[32] Albert W Marshall, Ingram Olkin, and Barry C Arnold. Inequalities: Theory of Majorization and Its Applications. Springer, 2010. 10.1007/​978-0-387-68276-1.

[33] C. Perry, P. Ć wikliński, J. Anders, M. Horodecki, and J. Oppenheim. A sufficient set of experimentally implementable thermal operations. arXiv 1511.06553, November 2015. URL https:/​/​arxiv.org/​abs/​1511.06553.

[34] H. Wilming, R. Gallego, and J. Eisert. Second law of thermodynamics under control restrictions. Phys. Rev. E, 93: 042126, Apr 2016. 10.1103/​PhysRevE.93.042126.

[35] J Lekscha, H Wilming, J Eisert, and R Gallego. Quantum thermodynamics with local control. arXiv:1612.00029, 2016. URL https:/​/​arxiv.org/​abs/​1612.00029.

[36] Paweł Mazurek and Michał Horodecki. Decomposability and convex structure of thermal processes. arXiv preprint arXiv:1707.06869, 2017. URL https:/​/​arxiv.org/​abs/​1707.06869.

[37] Kamil Korzekwa. Coherence, thermodynamics and uncertainty relations. PhD thesis, Imperial College London, 2016. URL https:/​/​spiral.imperial.ac.uk/​handle/​10044/​1/​43343.

[38] Gilad Gour, Markus P Müller, Varun Narasimhachar, Robert W Spekkens, and Nicole Yunger Halpern. The resource theory of informational nonequilibrium in thermodynamics. Physics Reports, 583: 1-58, 2015. http:/​/​dx.doi.org/​10.1016/​j.physrep.2015.04.003.

[39] Edwin T Jaynes and Frederick W Cummings. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proceedings of the IEEE, 51 (1): 89-109, 1963. 10.1109/​PROC.1963.1664.

[40] Bruce W Shore and Peter L Knight. The Jaynes-Cummings model. Journal of Modern Optics, 40 (7): 1195-1238, 1993. 10.1080/​09500349314551321.

[41] Johan Åberg. Catalytic coherence. Phys. Rev. Lett., 113: 150402, Oct 2014. 10.1103/​PhysRevLett.113.150402.

[42] R. Kosloff. Quantum thermodynamics: A dynamical viewpoint. Entropy, 15: 2100-2128, May 2013. 10.3390/​e15062100.

[43] Mário Ziman, Peter Stelmachovic, and Vladimír Buzek. Description of quantum dynamics of open systems based on collision-like models. Open systems & information dynamics, 12 (01): 81-91, 2005. 10.1007/​s11080-005-0488-0.

[44] Valerio Scarani, Mário Ziman, Peter Stelmachovic, Nicolas Gisin, and Vladimír Buzek. Thermalizing quantum machines: Dissipation and entanglement. Phys. Rev. Lett., 88: 097905, Feb 2002. 10.1103/​PhysRevLett.88.097905.

[45] H.-P. Breuer and F. Petruccione. The theory of open quantum systems. Oxford University Press, 2002. 10.1093/​acprof:oso/​9780199213900.001.0001.

[46] E. B. Davies. Markovian master equations. Comm. Math. Phys., 39 (2): 91-110, 1974. ISSN 1432-0916. 10.1007/​BF01608389.

[47] R. Dümcke. The low density limit for anN-level system interacting with a free Bose or Fermi gas. Comm. Math. Phys., 97 (3): 331-359, 1985. ISSN 1432-0916. 10.1007/​BF01213401.

[48] Wojciech Roga, Mark Fannes, and Karol Życzkowski. Davies maps for qubits and qutrits. Rep. Math. Phys., 66 (3): 311-329, 2010. 10.1016/​S0034-4877(11)00003-6.

[49] EB Davies. Embeddable Markov matrices. Electron. J. Probab., 15: 1474-1486, 2010. 10.1214/​EJP.v15-733.

[50] E Brian Davies. Linear operators and their spectra, volume 106. Cambridge University Press, 2007. 10.1017/​CBO9780511618864.

[51] A. F. Veinott. Least d-majorized network flows with inventory and statistical applications. Management Science, 17 (9): 547-567, 1971. 10.1287/​mnsc.17.9.547.

[52] Ernst Ruch, Rudolf Schranner, and Thomas H Seligman. Generalization of a theorem by Hardy, Littlewood, and Pólya. J. Math. Analysis and Applications, 76 (1): 222 - 229, 1980. ISSN 0022-247X. http:/​/​dx.doi.org/​10.1016/​0022-247X(80)90075-X.

[53] A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen. Maximal work extraction from finite quantum systems. Europhys. Lett., 67: 565-571, August 2004. 10.1209/​epl/​i2004-10101-2.

[54] P. Faist, J. Oppenheim, and R. Renner. Gibbs-preserving maps outperform thermal operations in the quantum regime. New Journal of Physics, 17 (4): 043003, April 2015. 10.1088/​1367-2630/​17/​4/​043003.

[55] 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 2014. 10.1103/​PhysRevA.90.062110. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.90.062110.

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

[57] Matteo Lostaglio, Kamil Korzekwa, and Antony Milne. Markovian evolution of quantum coherence under symmetric dynamics. Phys. Rev. A, 96: 032109, Sep 2017. 10.1103/​PhysRevA.96.032109.

[58] Piotr Ć wikliński, Michał Studziński, Michał Horodecki, and Jonathan Oppenheim. Limitations on the evolution of quantum coherences: Towards fully quantum second laws of thermodynamics. Phys. Rev. Lett., 115: 210403, Nov 2015. 10.1103/​PhysRevLett.115.210403.

[59] David P DiVincenzo et al. The physical implementation of quantum computation. arXiv preprint quant-ph/​0002077, 2000. 10.1002/​1521-3978(200009)48:9/​11<771::AID-PROP771>3.0.CO;2-E.

[60] P Oscar Boykin, Tal Mor, Vwani Roychowdhury, Farrokh Vatan, and Rutger Vrijen. Algorithmic cooling and scalable NMR quantum computers. Proceedings of the National Academy of Sciences, 99 (6): 3388-3393, 2002. 10.1073/​pnas.241641898.

[61] Daniel K Park, Nayeli A Rodriguez-Briones, Guanru Feng, Robabeh Rahimi, Jonathan Baugh, and Raymond Laflamme. Heat bath algorithmic cooling with spins: review and prospects. In Electron Spin Resonance (ESR) Based Quantum Computing, pages 227-255. Springer, 2016. 10.1007/​978-1-4939-3658-8_8.

[62] Nayeli A Rodriguez-Briones, Jun Li, Xinhua Peng, Tal Mor, Yossi Weinstein, and Raymond Laflamme. Heat-bath algorithmic cooling with correlated qubit-environment interactions. New Journal of Physics, 19 (11): 113047, 2017. 10.1088/​1367-2630/​aa8fe0.

[63] W. Pusz and S.L. Woronowicz. Passive states and KMS states for general quantum systems. Comm. Math. Phys., 58 (3): 273-290, 1978. ISSN 0010-3616. 10.1007/​BF01614224.

[64] A. Lenard. Thermodynamical proof of the Gibbs formula for elementary quantum systems. Journal of Statistical Physics, 19 (6): 575-586, 1978. ISSN 0022-4715. 10.1007/​BF01011769.

► Cited by (beta)

[1] J. Lekscha, H. Wilming, J. Eisert, R. Gallego, "Quantum thermodynamics with local control", Physical Review E 97, 022142 (2018).

(The above data is from Crossref's cited-by service. Unfortunately not all publishers provide suitable and complete citation data so that some citing works or bibliographic details may be missing.)