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).
 J. Gemmer and J. Anders, From single-shot towards general work extraction in a quantum thermodynamic framework, New J. Phys. 17, 085006 (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).
 M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Quantum Coherence, Time-Translation Symmetry, and Thermodynamics, Phys. Rev. X 5, 021001 (2015).
 K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings, The extraction of work from quantum coherence, New J. Phys. 18, 023045 (2016).
 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).
 F. G. S. L. Brandão and G. Gour, Reversible Framework for Quantum Resource Theories, Phys. Rev. Lett. 115, 070503 (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).
 P. Faist, J. Oppenheim, and R. Renner, Gibbs-preserving maps outperform thermal operations in the quantum regime, New J. Phys. 17, 043003 (2015).
 N. H. Y. Ng, M. Mančinska, C. Cirstoiu, J. Eisert, and S. Wehner, Limits to catalysis in quantum thermodynamics, New J. Phys. 17, 085004 (2015).
 J. Scharlau, The resource theories of non-uniformity and athermality: State transitions with finite size environments, Master thesis, Heidelberg University, 2015.
 M. Vergne and M. Walter, Inequalities for Moment Cones of Finite-Dimensional Representations, Journal of Symplectic Geometry 15(4), 1209-1250 (2017).
 D. Reeb and M. M. Wolf, An improved Landauer Principle with finite-size corrections, New J. Phys. 16, 103011 (2014).
 Alexander Müller-Hermes and Christopher Perry, "All unital qubit channels are 4-noisy operations", Letters in Mathematical Physics 109 1, 1 (2019).
 Georgios Styliaris, Álvaro M. Alhambra, and Paolo Zanardi, "Mixing of quantum states under Markovian dissipation and coherent control", Physical Review A 99 4, 042333 (2019).
 Christopher T. Chubb, Marco Tomamichel, and Kamil Korzekwa, "Beyond the thermodynamic limit: finite-size corrections to state interconversion rates", Quantum 2, 108 (2018).
 Martí Perarnau-Llobet and Raam Uzdin, "Collective operations can extremely reduce work fluctuations", New Journal of Physics 21 8, 083023 (2019).
 M. Hamed Mohammady and Alessandro Romito, "Efficiency of a cyclic quantum heat engine with finite-size baths", Physical Review E 100 1, 012122 (2019).
 Markus P. Müller, "Correlating Thermal Machines and the Second Law at the Nanoscale", Physical Review X 8 4, 041051 (2018).
 Sadegh Raeisi, Mária Kieferová, and Michele Mosca, "Novel Technique for Robust Optimal Algorithmic Cooling", Physical Review Letters 122 22, 220501 (2019).
 Paul Boes, Jens Eisert, Rodrigo Gallego, Markus P. Müller, and Henrik Wilming, "Von Neumann Entropy from Unitarity", Physical Review Letters 122 21, 210402 (2019).
 Alejandro Pozas-Kerstjens, Eric G Brown, and 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, and J. Eisert, "Catalytic Quantum Randomness", Physical Review X 8 4, 041016 (2018).
 Nahuel Freitas, Rodrigo Gallego, Lluís Masanes, and Juan Pablo Paz, Fundamental Theories of Physics 195, 597 (2018) ISBN:978-3-319-99045-3.
 Nelly Huei Ying Ng and Mischa Prebin Woods, Fundamental Theories of Physics 195, 625 (2018) ISBN:978-3-319-99045-3.
 Christopher Perry, Piotr Ćwikliński, Janet Anders, Michał Horodecki, and Jonathan Oppenheim, "A Sufficient Set of Experimentally Implementable Thermal Operations for Small Systems", Physical Review X 8 4, 041049 (2018).
 Jonathan G. Richens, Álvaro M. Alhambra, and Lluis Masanes, "Finite-bath corrections to the second law of thermodynamics", Physical Review E 97 6, 062132 (2018).
 Yelena Guryanova, Nicolai Friis, and Marcus Huber, "Ideal Projective Measurements Have Infinite Resource Costs", arXiv:1805.11899.
 Matteo Lostaglio, Álvaro M. Alhambra, and Christopher Perry, "Elementary Thermal Operations", arXiv:1607.00394.
 Kosuke Ito and Masahito Hayashi, "Optimal performance of generalized heat engines with finite-size baths of arbitrary multiple conserved quantities beyond i.i.d. scaling", arXiv:1612.04047.
 Henrik Wilming and Rodrigo Gallego, "Third Law of Thermodynamics as a Single Inequality", Physical Review X 7 4, 041033 (2017).
 Kosuke Ito and Masahito Hayashi, "Optimal performance of generalized heat engines with finite-size baths of arbitrary multiple conserved quantities beyond independent-and-identical-distribution scaling", Physical Review E 97 1, 012129 (2018).
 Nelly Huei Ying Ng, Mischa Prebin Woods, and Stephanie Wehner, "Surpassing the Carnot efficiency by extracting imperfect work", New Journal of Physics 19 11, 113005 (2017).
 Tiago Debarba, Gonzalo Manzano, Yelena Guryanova, Marcus Huber, and Nicolai Friis, "Work Estimation and Work Fluctuations in the Presence of Non-Ideal Measurements", arXiv:1902.08568.
The above citations are from Crossref's cited-by service (last updated 2019-08-18 00:19:50) and SAO/NASA ADS (last updated 2019-08-18 00:19:51). The list may be incomplete as not all publishers provide suitable and complete citation data.
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.