Interactions of quantum systems with their environment play a crucial role in resource-theoretic approaches to thermodynamics in the microscopic regime. Here, we analyze the possible state transitions in the presence of "small" heat baths of bounded dimension and energy. We show that for operations on quantum systems with fully degenerate Hamiltonian (noisy operations), all possible state transitions can be realized exactly with a bath that is of the same size as the system or smaller, which proves a quantum version of Horn's lemma as conjectured by Bengtsson and Zyczkowski. On the other hand, if the system's Hamiltonian is not fully degenerate (thermal operations), we show that some possible transitions can only be performed with a heat bath that is unbounded in size and energy, which is an instance of the third law of thermodynamics. In both cases, we prove that quantum operations yield an advantage over classical ones for any given finite heat bath, by allowing a larger and more physically realistic set of state transitions.
 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).
 M. Horodecki, P. Horodecki, and J. Oppenheim, Reversible transformations from pure to mixed states and the unique measure of information, Phys. Rev. A 67, 062104 (2003).
 O. C. O. Dahlsten, R. Renner, E. Rieper, and V. Vedral, Inadequacy of von Neumann entropy for characterizing extractable work, New J. Phys. 13, 053015 (2011).
 F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Resource Theory of Quantum States Out of Thermal Equilibrium, Phys. Rev. Lett. 111, 250404 (2013).
 F. G. S. L. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, Proc. Natl. Acad. Sci. USA 112, 3275 (2015).
 N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, Introducing one-shot work into fluctuation relations, New J. Phys. 17, 095003 (2015).
 D. Egloff, O. C. O. Dahlsten, R. Renner, and V. Vedral, A measure of majorization emerging from single-shot statistical mechanics, New J. Phys. 17, 073001 (2015).
 M. Lostaglio, M. P. Müller, and M. Pastena, Stochastic independence as a resource in small-scale thermodynamics, Phys. Rev. Lett. 115, 150402 (2015).
 W. Nernst, Sitzungsbericht der Königlich Preussischen Akademie der Wissenschaften, p. 134 (1912).
 R. Fowler and E. A. Guggenheim, Statistical Thermodynamics, Cambridge University Press, Cambridge, 1949.
 Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Thermodynamics of quantum systems with multiple conserved quantities, Nat. Comm. 7, 12049 (2016).
 M. Lostaglio, D. Jennings, and T. Rudolph, Description of quantum coherence in thermodynamic processes requires constraints beyond free energy, Nat. Comm. 6, 6383 (2015).
 P. Ć wikliński, M. Studziński, M. Horodecki, and J. Oppenheim, Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics, Phys. Rev. Lett. 115, 210403 (2015).
 R. Webster, Convexity, Oxford University Press, Oxford, 1994.
 G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, The resource theory of informational nonequilibrium in thermodynamics, Phys. Rep. 583, 1-58 (2015).
 C. Browne, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, Guaranteed Energy-Efficient Bit Reset in Finite Time, Phys. Rev. Lett. 113, 100603 (2014).
 K. Życzkowski and I. Bengtsson, On Duality between Quantum Maps and Quantum States, Open Syst. Inf. Dyn. 11, 3 (2004).
 P. Shor, Structure of Unital Maps and the Asymptotic Quantum Birkhoff Conjecture, presentation, slides at http://science-visits.mccme.ru/doc/steklov-talk.pdf, 2010.
 U. Haagerup and M. Musat, Factorization and dilation problems for completely positive maps on von Neumann algebras, Commun. Math. Phys. 303(2), 555-594 (2011).
 J. Scharlau, The resource theories of non-uniformity and athermality: State transitions with finite size environments, Master thesis, Heidelberg University, 2015.
 Alexander Müller-Hermes, Christopher Perry, "All unital qubit channels are 4-noisy operations", Letters in Mathematical Physics (2018).
 Alejandro Pozas-Kerstjens, Eric G Brown, Karen V Hovhannisyan, "A quantum Otto engine with finite heat baths: energy, correlations, and degradation", New Journal of Physics 20 4, 043034 (2018).
 P. Boes, H. Wilming, R. Gallego, J. Eisert, "Catalytic Quantum Randomness", Physical Review X 8 4, 041016 (2018).
 Christopher T. Chubb, Marco Tomamichel, Kamil Korzekwa, "Beyond the thermodynamic limit: finite-size corrections to state interconversion rates", Quantum 2, 108 (2018).
 Jonathan G. Richens, Álvaro M. Alhambra, Lluis Masanes, "Finite-bath corrections to the second law of thermodynamics", Physical Review E 97 6, 062132 (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.)
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.