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The second laws of quantum thermodynamics

Brandão, Fernando and Horodecki, Michał and Ng, Nelly and Oppenheim, Jonathan and Wehner, Stephanie (2015) The second laws of quantum thermodynamics. Proceedings of the National Academy of Sciences of the United States of America, 112 (11). pp. 3275-3279. ISSN 0027-8424. PMCID PMC4372001. doi:10.1073/pnas.1411728112.

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The second law of thermodynamics places constraints on state transformations. It applies to systems composed of many particles, however, we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath. Is there a second law of thermodynamics in this regime? Here, we find that for processes which are approximately cyclic, the second law for microscopic systems takes on a different form compared to the macroscopic scale, imposing not just one constraint on state transformations, but an entire family of constraints. We find a family of free energies which generalize the traditional one, and show that they can never increase. The ordinary second law relates to one of these, with the remainder imposing additional constraints on thermodynamic transitions. We find three regimes which determine which family of second laws govern state transitions, depending on how cyclic the process is. In one regime one can cause an apparent violation of the usual second law, through a process of embezzling work from a large system which remains arbitrarily close to its original state. These second laws are relevant for small systems, and also apply to individual macroscopic systems interacting via long-range interactions. By making precise the definition of thermal operations, the laws of thermodynamics are unified in this framework, with the first law defining the class of operations, the zeroth law emerging as an equivalence relation between thermal states, and the remaining laws being monotonicity of our generalized free energies.

Item Type:Article
Related URLs:
URLURL TypeDescription Information CentralArticle
Brandão, Fernando0000-0003-3866-9378
Oppenheim, Jonathan0000-0002-8993-412X
Additional Information:© 2015 National Academy of Sciences. Edited by Peter W. Shor, Massachusetts Institute of Technology, Cambridge, MA, and approved January 12, 2015 (received for review June 26, 2014). Published ahead of print February 9, 2015. We thank Robert Alicki, Piotr Cwiklinski, Milan Mosonyi, Sandu Popescu, Joe Renes, Marco Tomamichel, and Andreas Winter for useful discussions, and Max Frenzel for comments on our draft. J.O. is supported by the Royal Society. M.H. is supported by the Foundation for Polish Science TEAM project cofinanced by the European Union European Regional Development Fund. N.N. and S.W. are supported by the National Research Foundation and Ministry of Education (MOE), Singapore as well as MOE Tier 3 Grant “Random numbers from quantum processes” (MOE2012-T3-1-009). Author contributions: F.B., M.H., N.N., J.O., and S.W. performed research and wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. This article contains supporting information online at
Funding AgencyGrant Number
Foundation for Polish ScienceUNSPECIFIED
European Regional Development FundUNSPECIFIED
National Research Foundation of KoreaUNSPECIFIED
Ministry of Education (Singapore)MOE2012-T3-1-009
Subject Keywords:quantum thermodynamics; quantum information theory; statistical physics; resource theory; free energy
Issue or Number:11
PubMed Central ID:PMC4372001
Record Number:CaltechAUTHORS:20160607-074905677
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Official Citation:Fernando Brandão, Michał Horodecki, Nelly Ng, Jonathan Oppenheim, and Stephanie Wehner The second laws of quantum thermodynamics PNAS 2015 112 (11) 3275-3279; published ahead of print February 9, 2015, doi:10.1073/pnas.1411728112
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:67709
Deposited By: Tony Diaz
Deposited On:07 Jun 2016 15:55
Last Modified:11 Nov 2021 03:53

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