Single-shot quantum thermodynamics

(drawing and copyright by Lidia del Rio)

MVSpec (Advanced Lecture on Special Topics, 4 Credit Points).
Lecture (by Markus Müller): Wednesday (Mittwoch), 14.15 - 15.45, Philosophenweg 19, seminar room.
Tutorial (by Jakob Scharlau): Tuesday (Dienstag), 14.15 - 15.45, Philosophenweg 12, room 060 (ground floor)

To get the credit points, you have to solve a certain percentage of exercise sheets (in groups of 2 or 3 people) that will be handed out every week. There will be 10 exercise sheets in total, and 60% of the points will definitely be enough.

Prerequisites: Basic knowledge of statistical physics and of quantum mechanics; interest in mathematical physics.

Contact: Markus Müller, Jakob Scharlau



Single-shot thermodynamics is a new and exciting application of quantum information theory to physics. Traditionally, thermodynamics is a theory that works in the thermodynamic limit where we have a large number of weakly interacting particles. Single-shot thermodynamics, on the other hand, aims at extending the scope of statistical physics to small and strongly correlated (quantum) systems immersed in heat baths.
The single-shot approach to thermodynamics differs from traditional approaches in several important ways. In particular, this approach takes an "agent-centric" point of view by modelling thermodynamics as a resource theory: a physical (quantum) system is characterized by the amount of control that an experimenter is able to exert on the system; limitations (such as energy conservation or lack of knowledge of the microscopic state) lead to restrictions that in turn allow to quantify, for example, the maximal amount of work that is extractable from a (small or strongly correlated) quantum system. Results on single-shot thermodynamics turn to well-known results from standard thermodynamics in the thermodynamic limit, and generalize these results to other situations.

The goal of this lecture is to get a good understanding of the current state of the art of research in this field. In the end, we should arrive at a point where we understand, for example, recent papers like these ones:

M. Horodecki and J. Oppenheim, Fundamental limitations for quantum and nanoscale thermodynamics, Nature Communications 4, 2059 (2013), arXiv:1111.3834.

F. G. S. L. Brandão, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, arXiv:1305.5278.


Lecture notes (handwritten; no guarantee that there are no mistakes):

  • Lecture 1 (Oct. 15) Overview; rigorous derivations of Landauer's Principle. There are many sources that explain the notions of Maxwell's Demon, Landauer Erasure, or the Szilard Engine. As a starting point, enjoy the following article:
    C. H. Bennett, Demons, Engines, and the Second Law, Scientific American, 1987.
  • Lecture 2 (Oct. 22) Von Neumann entropy, mutual information, the resource theory of nonuniformity
  • Lecture 3 (Oct. 29) Noisy classical operations and majorization
  • Lecture 4 (Nov. 12) Lorenz curves, nonuniformity monotones
  • Lecture 5 (Nov. 19) Distillable nonuniformity and nonuniformity of formation, trace distance.
  • Lecture 6 (Nov. 26) Landauer erasure and I_\infty, approximate formation and distillation, thermodynamic limit
  • Lecture 7 (Dec. 3) Asymptotic state conversion, Shannon's noiseless coding theorem, data compression, AEP
  • Lecture 8 (Dec. 10) Catalysis, trumping, embezzling
  • Lecture 9 (Dec. 17) Passive and completely passive states, definition of the resource theory of athermality, warm-up
  • Lecture 10 (Jan. 7) Asymptotic state conversion and free energy, how (not to) reduce to the classical case
  • Lecture 11 (Jan. 14) Blockdiagonal states, d-majorization and thermal operations, thermal Lorenz curves
  • Lecture 12 (Jan. 21) Rényi divergences; extractable work and work of formation; smoothed versions
  • Lecture 13 (Jan. 28) The second laws of quantum thermodynamics, grandcanonical resource theories.
  • Lecture 14 (Feb. 4) Relation to fluctuation-dissipation theorems, entanglement and Nielsen's Theorem, current research.
    Here are links to the papers that I mentioned in the lecture:
    N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, Unification of fluctuation theorems and one-shot statistical mechanics, arXiv:1409.3878.
    M. Lostaglio, D. Jennings, and T. Rudolph, Thermodynamic laws beyond free energy relations, arXiv:1405.2188.
    P. Cwiklinski, M.Studzinski, M. Horodecki, and J. Oppenheim, Towards fully quantum second laws of thermodynamics: limitations on the evolution of quantum coherences, arXiv:1405.5029.
    S. Ryu and T. Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006); arXiv:hep-th/0603001.
    B. Czech, P. Hayden, N. Lashkari, and B. Swingle, The Information Theoretic Interpretation of the Length of a Curve, arXiv:1410.1540.

 

Exercise sheets (PDF):

  • 01 (Optimality in Landauer erasure; von Neumann entropy and thermal states; Subadditivity of Rényi-0 entropy)
  • 02 (Majorization)
  • 03 (Schur convexity, Rényi entropies, trace distance)
  • 04 (Sharp states, extractable nonuniformity and nonuniformity of formation, perfect distinguishability of states)
  • 05 (Smooth entropies, smoothed nonuniformity measures, Landauer erasure reloaded)
  • 06 (Equivalence of ensembles, catalysis, data compression)
  • 07 (Passive and completely passive states, thermal operations: general properties and small environments)
  • 08 (d-majorization, thermal operations and blockdiagonal states, thermal versus Gibbs-preserving maps)
  • 09 (d-majorization for qubits via the harmonic oscillator, Rényi divergence and its monotonicity)
  • 10 (Gibbs-rescaled Lorenz curves, reading a research paper, Rényi divergence continued)