Beyond the thermodynamic limit: finite-size corrections to state interconversion rates

Christopher T. Chubb1, Marco Tomamichel2, and Kamil Korzekwa1

1Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney NSW 2006, Australia.
2Centre for Quantum Software and Information, School of Software, University of Technology Sydney, Sydney NSW 2007, Australia.

Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when one of the heat baths it operates between is finite. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.

Thermodynamics is one of the most versatile physical theories, finding applications in almost all fields of science, from cosmology and astrophysics to chemistry and the theory of computation. Its strength comes from the fact that it provides a universal framework that uses statistical tools to study physical phenomena in the so-called thermodynamic limit, i.e., when the number of involved systems is very large. However, our increasing ability to manipulate and control systems at smaller and smaller scales allows us to build novel nanodevices operating well beyond the thermodynamic limit. Therefore, in order to understand the thermodynamic properties of such devices, we need to formulate a theory that is not constrained to the study of macroscopic systems. In this paper we achieve this by developing an information-theoretic framework describing thermodynamic transformations of finite-size systems.

One immediate application of our theoretical results is to the study of irreversible processes in the nanoscale regime. In particular, we show how the amount of ordered energy needed to drive a small system out of equilibrium is larger than the amount one could obtain in a reverse process. This affects reversibility of thermodynamic cycles and, in turn, deteriorates performance of nanoengines. Despite these negative finite-size effects, we find that in specially engineered conditions nanoscale engines can still achieve the ultimate limit of efficiency.

Our results expand the realm of applicability of thermodynamics beyond the constraint of macroscopic systems, and thus provide new tools to study the universe at the smallest scale.

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[1] Robin Giles, Mathematical Foundations of Thermodynamics (Pergamon Press, 1964).

[2] Bob Coecke, Tobias Fritz, and Robert W Spekkens, ``A mathematical theory of resources,'' Inform. Comput. 250, 59-86 (2016).
https://doi.org/10.1016/j.ic.2016.02.008

[3] M. Horodecki and J. Oppenheim, ``(Quantumness in the context of) resource theories,'' Int. J. Mod. Phys. B 27, 1345019 (2013a).
https://doi.org/10.1142/S0217979213450197

[4] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki, ``Quantum entanglement,'' Rev. Mod. Phys. 81, 865-942 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[5] T Baumgratz, M Cramer, and MB Plenio, ``Quantifying coherence,'' Phys. Rev. Lett. 113, 140401 (2014).
https://doi.org/10.1103/PhysRevLett.113.140401

[6] Iman Marvian and Robert W Spekkens, ``How to quantify coherence: distinguishing speakable and unspeakable notions,'' Phys. Rev. A 94, 052324 (2016).
https://doi.org/10.1103/PhysRevA.94.052324

[7] Andreas Winter and Dong Yang, ``Operational resource theory of coherence,'' Phys. Rev. Lett. 116, 120404 (2016).
https://doi.org/10.1103/PhysRevLett.116.120404

[8] I. Marvian and R. W. Spekkens, ``The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations,'' New J. Phys. 15, 033001 (2013).
https://doi.org/10.1088/1367-2630/15/3/033001

[9] Iman Marvian, Symmetry, Asymmetry and Quantum Information, Ph.D. thesis, University of Waterloo (2012).
https:/​/​uwspace.uwaterloo.ca/​handle/​10012/​7088

[10] 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, 2717-2753 (2000).
https://doi.org/10.1023/A:1026422630734

[11] M. Horodecki and J. Oppenheim, ``Fundamental limitations for quantum and nanoscale thermodynamics,'' Nat. Commun. 4, 2059 (2013b).
https://doi.org/10.1038/ncomms3059

[12] 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 (2013).
https://doi.org/10.1103/PhysRevLett.111.250404

[13] Matteo Lostaglio, Kamil Korzekwa, David Jennings, and Terry Rudolph, ``Quantum coherence, time-translation symmetry, and thermodynamics,'' Phys. Rev. X 5, 021001 (2015a).
https://doi.org/10.1103/PhysRevX.5.021001

[14] Piotr Ć wikli ński, 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 (2015).
https://doi.org/10.1103/PhysRevLett.115.210403

[15] V. Narasimhachar and G. Gour, ``Low-temperature thermodynamics with quantum coherence,'' Nat. Commun. 6, 7689 (2015).
https://doi.org/10.1038/ncomms8689

[16] Gilad Gour, David Jennings, Francesco Buscemi, Runyao Duan, and Iman Marvian, ``Quantum majorization and a complete set of entropic conditions for quantum thermodynamics,'' arXiv:1708.04302 (2017).
arXiv:1708.04302

[17] Michael A Nielsen, ``Conditions for a class of entanglement transformations,'' Phys. Rev. Lett. 83, 436 (1999).
https://doi.org/10.1103/PhysRevLett.83.436

[18] Charles H Bennett, Herbert J Bernstein, Sandu Popescu, and Benjamin Schumacher, ``Concentrating partial entanglement by local operations,'' Phys. Rev. A 53, 2046 (1996).
https://doi.org/10.1103/PhysRevA.53.2046

[19] Michał Horodecki, Jonathan Oppenheim, and Carlo Sparaciari, ``Approximate majorization,'' arXiv:1706.05264 (2017).
arXiv:1706.05264

[20] Wataru Kumagai and Masahito Hayashi, ``Second-order asymptotics of conversions of distributions and entangled states based on Rayleigh-normal probability distributions,'' IEEE Trans. Inf. Theory 63, 1829-1857 (2017).
https://doi.org/10.1109/TIT.2016.2645223

[21] Marco Tomamichel and Masahito Hayashi, ``A hierarchy of information quantities for finite block length analysis of quantum tasks,'' IEEE Trans. Inf. Theory 59, 7693-7710 (2013).
https://doi.org/10.1109/TIT.2013.2276628

[22] Ke Li, ``Second-order asymptotics for quantum hypothesis testing,'' Ann. Stat. 42, 171-189 (2014).
https://doi.org/10.1214/13-AOS1185

[23] Marco Tomamichel and Vincent Y. F. Tan, ``Second-order asymptotics for the classical capacity of image-additive quantum channels,'' Commun. Math. Phys. 338, 103-137 (2015).
https://doi.org/10.1007/s00220-015-2382-0

[24] Nilanjana Datta, Marco Tomamichel, and Mark M. Wilde, ``On the second-order asymptotics for entanglement-assisted communication,'' Quantum Inf. Process. 15, 2569-2591 (2016).
https://doi.org/10.1007/s11128-016-1272-5

[25] Nilanjana Datta and Felix Leditzky, ``Second-order asymptotics for source coding, dense coding, and pure-state entanglement conversions,'' IEEE Trans. Inf. Theory 61, 582-608 (2015).
https://doi.org/10.1109/TIT.2014.2366994

[26] Mark M. Wilde, Joseph M. Renes, and Saikat Guha, ``Second-order coding rates for pure-loss bosonic channels,'' Quantum Inf. Process. 15, 1289-1308 (2016).
https://doi.org/10.1007/s11128-015-0997-x

[27] Marco Tomamichel, Mario Berta, and Joseph M. Renes, ``Quantum coding with finite resources,'' Nat. Commun. 7, 11419 (2016).
https://doi.org/10.1038/ncomms11419

[28] Mark M. Wilde, Marco Tomamichel, Seth Lloyd, and Mario Berta, ``Gaussian hypothesis testing and quantum illumination,'' Phys. Rev. Lett. 119, 120501 (2017).
https://doi.org/10.1103/PhysRevLett.119.120501

[29] John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, and Paul Skrzypczyk, ``The role of quantum information in thermodynamics - a topical review,'' J. Phys. A 49, 143001 (2016).
https://doi.org/10.1088/1751-8113/49/14/143001

[30] Johan Åberg, ``Catalytic coherence,'' Phys. Rev. Lett. 113, 150402 (2014).
https://doi.org/10.1103/PhysRevLett.113.150402

[31] Matteo Lostaglio, David Jennings, and Terry Rudolph, ``Description of quantum coherence in thermodynamic processes requires constraints beyond free energy,'' Nat. Commun. 6, 6383 (2015b).
https://doi.org/10.1038/ncomms7383

[32] Raam Uzdin, Amikam Levy, and Ronnie Kosloff, ``Equivalence of quantum heat machines, and quantum-thermodynamic signatures,'' Phys. Rev. X 5, 031044 (2015).
https://doi.org/10.1103/PhysRevX.5.031044

[33] Kamil Korzekwa, Matteo Lostaglio, Jonathan Oppenheim, and David Jennings, ``The extraction of work from quantum coherence,'' New J. Phys. 18, 023045 (2016).
https://doi.org/10.1088/1367-2630/18/2/023045

[34] Robert Alicki and Mark Fannes, ``Entanglement boost for extractable work from ensembles of quantum batteries,'' Phys. Rev. E 87, 042123 (2013).
https://doi.org/10.1103/PhysRevE.87.042123

[35] Martí Perarnau-Llobet, Karen V Hovhannisyan, Marcus Huber, Paul Skrzypczyk, Nicolas Brunner, and Antonio Acín, ``Extractable work from correlations,'' Phys. Rev. X 5, 041011 (2015).
https://doi.org/10.1103/PhysRevX.5.041011

[36] Jonatan Bohr Brask, Géraldine Haack, Nicolas Brunner, and Marcus Huber, ``Autonomous quantum thermal machine for generating steady-state entanglement,'' New J. Phys. 17, 113029 (2015).
https://doi.org/10.1088/1367-2630/17/11/113029

[37] H. Umegaki, ``Conditional expectation in an operator algebra,'' Kodai Math. Sem. Rep. 14, 59-85 (1962).
https://doi.org/10.2996/kmj/1138844604

[38] A. Uhlmann, ``The transition probability for states of star-algebras,'' Ann. Phys. (N. Y). 497, 524-532 (1985).
https://doi.org/10.1002/andp.19854970419

[39] Kosuke Ito, Wataru Kumagai, and Masahito Hayashi, ``Asymptotic compatibility between local-operations-and-classical-communication conversion and recovery,'' Phys. Rev. A 92, 052308 (2015).
https://doi.org/10.1103/PhysRevA.92.052308

[40] Nelly Huei Ying Ng, Mischa Prebin Woods, and Stephanie Wehner, ``Surpassing the Carnot efficiency by extracting imperfect work,'' New J. Phys. 19, 113005 (2017).
https://doi.org/10.1088/1367-2630/aa8ced

[41] 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).
https://doi.org/10.1073/pnas.1411728112

[42] Kamil Korzekwa, Coherence, thermodynamics and uncertainty relations, Ph.D. thesis, Imperial College London (2016).
https:/​/​spiral.imperial.ac.uk/​handle/​10044/​1/​43343

[43] Rajendra Bhatia, Matrix analysis (Springer Science & Business Media, 2013).

[44] Ernst Ruch, Rudolf Schranner, and Thomas H. Seligman, ``The mixing distance,'' J. Chem. Phys. 69 (1978).
https://doi.org/10.1063/1.436364

[45] Joseph M Renes, ``Relative submajorization and its use in quantum resource theories,'' J. Math. Phys. 57, 122202 (2016).
https://doi.org/10.1063/1.4972295

[46] Remco Van Der Meer, Nelly Huei Ying Ng, and Stephanie Wehner, ``Smoothed generalized free energies for thermodynamics,'' Phys. Rev. A 96, 062135 (2017).
https://doi.org/10.1103/PhysRevA.96.062135

[47] Michał Horodecki, Karol Horodecki, Paweł Horodecki, Ryszard Horodecki, Jonathan Oppenheim, Aditi Sen(De), and Ujjwal Sen, ``Local information as a resource in distributed quantum systems,'' Phys. Rev. Lett. 90, 100402 (2003).
https://doi.org/10.1103/PhysRevLett.90.100402

[48] Guifré Vidal, Daniel Jonathan, and MA Nielsen, ``Approximate transformations and robust manipulation of bipartite pure-state entanglement,'' Phys. Rev. A 62, 012304 (2000).
https://doi.org/10.1103/PhysRevA.62.012304

[49] Marco Tomamichel, Quantum information processing with finite resources: mathematical foundations, Vol. 5 (Springer, 2015).

[50] Johan Åberg, ``Truly work-like work extraction via a single-shot analysis,'' Nat. Commun. 4, 1925 (2013).
https://doi.org/10.1038/ncomms2712

[51] Dario Egloff, Oscar CO Dahlsten, Renato Renner, and Vlatko Vedral, ``A measure of majorization emerging from single-shot statistical mechanics,'' New J. Phys. 17, 073001 (2015).
https://doi.org/10.1088/1367-2630/17/7/073001

[52] Chris Perry, Piotr Ć wikliński, Janet Anders, Michał Horodecki, and Jonathan Oppenheim, ``A sufficient set of experimentally implementable thermal operations,'' arXiv:1511.06553 (2015).
arXiv:1511.06553

[53] Michele Campisi, Peter Talkner, and Peter Hänggi, ``Finite bath fluctuation theorem,'' Phys. Rev. E 80, 031145 (2009).
https://doi.org/10.1103/PhysRevE.80.031145

[54] David Reeb and Michael M Wolf, ``An improved Landauer principle with finite-size corrections,'' New J. Phys. 16, 103011 (2014).
https://doi.org/10.1088/1367-2630/16/10/103011

[55] Jonathan G Richens, Álvaro M Alhambra, and Lluis Masanes, ``Finite-bath corrections to the second law of thermodynamics,'' Phys. Rev. E 97, 062132 (2018).
https://doi.org/10.1103/PhysRevE.97.062132

[56] Jakob Scharlau and Markus P Mueller, ``Quantum Horn's lemma, finite heat baths, and the third law of thermodynamics,'' Quantum 2, 54 (2018).
https://doi.org/10.22331/q-2018-02-22-54

[57] Hiroyasu Tajima and Masahito Hayashi, ``Finite-size effect on optimal efficiency of heat engines,'' Phys. Rev. E 96, 012128 (2017).
https://doi.org/10.1103/PhysRevE.96.012128

[58] Kosuke Ito and Masahito Hayashi, ``Optimal performance of generalized heat engines with finite-size baths of arbitrary multiple conserved quantities based on non-iid scaling,'' arXiv:1612.04047 (2016).
arXiv:1612.04047

[59] Mischa P Woods, Nelly Ng, and Stephanie Wehner, ``The maximum efficiency of nano heat engines depends on more than temperature,'' arXiv:1506.02322 (2015).
arXiv:1506.02322

[60] P. Faist, J. Oppenheim, and R. Renner, ``Gibbs-preserving maps outperform thermal operations in the quantum regime,'' New J. of Phys. 17, 043003 (2015).
https://doi.org/10.1088/1367-2630/17/4/043003

[61] Kamil Korzekwa, ``Structure of the thermodynamic arrow of time in classical and quantum theories,'' Phys. Rev. A 95, 052318 (2017).
https://doi.org/10.1103/PhysRevA.95.052318

[62] Masahito Hayashi, Quantum Information - An Introduction (Springer, 2006).

[63] C.T. Chubb, V.Y.F. Tan, and M. Tomamichel, ``Moderate Deviation Analysis for Classical Communication over Quantum Channels,'' Commun. Math. Phys. 355 (2017).
https://doi.org/10.1007/s00220-017-2971-1

[64] Hao-Chung Cheng and Min-Hsiu Hsieh, ``Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing,'' (2017), arXiv:1701.03195.
arXiv:1701.03195

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