Smoothed generalized free energies for thermodynamics

In the study of thermodynamics for nanoscale quantum systems, a family of quantities known as generalized free energies have been derived as necessary and sufficient conditions that govern state transitions. These free energies become important especially in the regime where the system of interest consists of only a few (quantum) particles. In this work, we introduce a family of smoothed generalized free energies, by constructing explicit smoothing procedures that maximize or minimize the free energy over an ball of quantum states. In contrast to previously known smoothed free energies, these quantities now allow us to make an operational statement for approximate thermodynamic state transitions. We show that these smoothed quantities converge to the standard free energy in the thermodynamic limit.

[1]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[2]  Renato Renner,et al.  Smooth Renyi entropy and applications , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[3]  Nilanjana Datta,et al.  Min- and Max-Relative Entropies and a New Entanglement Monotone , 2008, IEEE Transactions on Information Theory.

[4]  Jonathan Oppenheim,et al.  Fluctuating States: What is the Probability of a Thermodynamical Transition? , 2015, 1504.00020.

[5]  M. Horodecki,et al.  Reversible transformations from pure to mixed states and the unique measure of information , 2002, quant-ph/0212019.

[6]  Jingning Zhang,et al.  Experimental test of the quantum Jarzynski equality with a trapped-ion system , 2014, Nature Physics.

[7]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[8]  R. Renner,et al.  Fundamental work cost of quantum processes , 2017, 1709.00506.

[9]  F. Brandão,et al.  Resource theory of quantum states out of thermal equilibrium. , 2011, Physical review letters.

[10]  R. Duan,et al.  Quantum majorization and a complete set of entropic conditions for quantum thermodynamics , 2017, Nature Communications.

[11]  Joseph M. Renes,et al.  Relative submajorization and its use in quantum resource theories , 2015, 1510.03695.

[12]  M. Horodecki,et al.  Fundamental limitations for quantum and nanoscale thermodynamics , 2011, Nature Communications.

[13]  Matteo Lostaglio,et al.  Stochastic Independence as a Resource in Small-Scale Thermodynamics. , 2014, Physical review letters.

[14]  T. Rudolph,et al.  Quantum coherence, time-translation symmetry and thermodynamics , 2014, 1410.4572.

[15]  Jonathan Oppenheim,et al.  N ov 2 01 1 Fundamental limitations for quantum and nano thermodynamics , 2011 .

[16]  Markus P. Mueller.  Correlating Thermal Machines and the Second Law at the Nanoscale , 2017, Physical Review X.

[17]  Serge Fehr,et al.  On quantum Rényi entropies: A new generalization and some properties , 2013, 1306.3142.

[18]  Nicole Yunger Halpern,et al.  The resource theory of informational nonequilibrium in thermodynamics , 2013, 1309.6586.

[19]  Henrik Wilming,et al.  The third law as a single inequality , 2017 .

[20]  Michal Horodecki,et al.  Towards fully quantum second laws of thermodynamics: limitations on the evolution of quantum coherences , 2014, 1405.5029.

[21]  Henrik Wilming,et al.  Third Law of Thermodynamics as a Single Inequality , 2017, 1701.07478.

[22]  M. Horodecki,et al.  Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics. , 2015, Physical review letters.

[23]  J. Eisert,et al.  Limits to catalysis in quantum thermodynamics , 2014, 1405.3039.

[24]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[25]  Michal Horodecki,et al.  The second laws of quantum thermodynamics , 2013, Proceedings of the National Academy of Sciences.

[26]  K. Southwell.  Quantum coherence , 2008, Nature.

[27]  I. S. Oliveira,et al.  Experimental reconstruction of work distribution and study of fluctuation relations in a closed quantum system. , 2013, Physical review letters.

[28]  Jonathan Oppenheim,et al.  A Sufficient Set of Experimentally Implementable Thermal Operations for Small Systems , 2015, Physical Review X.

[29]  Eric P. Hanson,et al.  Maximum and minimum entropy states yielding local continuity bounds , 2017, 1706.02212.

[30]  Christopher T. Chubb,et al.  Beyond the thermodynamic limit: finite-size corrections to state interconversion rates , 2017, Quantum.

[31]  David Jennings,et al.  Description of quantum coherence in thermodynamic processes requires constraints beyond free energy , 2014, Nature Communications.

[32]  Joseph M. Renes,et al.  Work cost of thermal operations in quantum thermodynamics , 2014, 1402.3496.

[33]  Stephanie Wehner,et al.  Surpassing the Carnot efficiency by extracting imperfect work , 2016, 1606.05532.

[34]  Stephanie Wehner,et al.  The maximum efficiency of nano heat engines depends on more than temperature , 2015, Quantum.

[35]  J Eisert,et al.  Second law of thermodynamics under control restrictions. , 2016, Physical review. E.

[36]  R. Renner,et al.  Gibbs-preserving maps outperform thermal operations in the quantum regime , 2014, 1406.3618.

[37]  J. Åberg.  Truly work-like work extraction via a single-shot analysis , 2011, Nature Communications.