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Thermodynamics in the Quantum Regime pp 651–679Cite as

One-Shot Information-Theoretical Approaches to Fluctuation Theorems

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Part of the book series: Fundamental Theories of Physics ((FTPH,volume 195))

Abstract

Traditional thermodynamics governs the behaviour of large systems that evolve between states of thermal equilibrium. For these large systems, the mean values of thermodynamic quantities (such as work, heat and entropy) provide a good characterisation of the process. Conversely, there is ever-increasing interest in the thermal behaviour of systems that evolve quickly and far from equilibrium, and that are too small for their behaviour to be well-described by mean values. Two major fields of modern thermodynamics seek to tackle such systems: non-equilibrium thermodynamics, and the nascent field of one-shot statistical mechanics. The former provides tools such as fluctuation theorems, whereas the latter applies “one-shot” Rényi entropies to thermal contexts. In this chapter, I provide a gentle introduction to recent research that draws from both fields: the application of one-shot information theory to fluctuation theorems.

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Notes

  1. 1.

    The surprise involves a logarithm rather than just the reciprocal probability to ensure additivity of surprises. For two mutually exclusive events with joint probability \(p\cdot q\), the joint surprise \(-\ln \left( p\cdot q\right) = -\ln \left( p\right) -\ln \left( q\right) \).

  2. 2.

    The base here is e because we have defined H in units of nats. If we had defined H in bits using \(-\sum _i p_x(i) \log _2 p_x(i)\), the base would be 2.

  3. 3.

    In the context of i.i.d. X, a sequence will be typical if and only if each symbol \(x_i\) appears \(p_i N\) times.

  4. 4.

    Or conversely, we encode both \(x_d\) and \(x_e\) to the same physical configuration. Then, \(\frac{1}{8}\) of messages have an indistinguishable \(x_d\)/\(x_e\), and we can guess the correct symbol \(\frac{1}{2}\) of the time: resulting in a \(\frac{1}{16}\) probability that the message is incorrectly decoded.

  5. 5.

    Sometimes the \(H_\frac{1}{2}\) is also referred to as the max-entropy. To avoid ambiguity, here we label the entropy explicitly by the parameter \(\alpha \), i.e. writing \(H_0\) rather than the ambiguous “\(H_\mathrm{max}\)”.

  6. 6.

    A formal discussion on where the information entropy can be considered a thermodynamic entropy is presented by Weilenmann et al. [19].

  7. 7.

    Conversely, this limit can be sharpened by restricting the set of allowed protocols (such as by bounding the effective dimension of the thermal reservoir [44]).

  8. 8.

    The reader should take care that there are varying notions of energy conservation, and in resource-theoretic frameworks, this will alter the set of permitted “thermal operations”. In particular, one may admit any operation that conserves energy on average (as per [45]), or alternatively could place stricter restrictions, such as mandating that the unitary representation of the dynamics on the system-battery commute with the total Hamiltonian (as per [16]). Here, we present a general scheme that can be adapted to either notion.

  9. 9.

    Furthermore, these two pictures can be seen to be equivalent: a time-varying Hamiltonian can be recast as a time-invariant Hamiltonian with the help of an ancillary “clock” system – for details, see Supplementary Material Sect. VIII of Brandão et al. [46], Appendix D2 of Yunger Halpern et al. [29], or Sect. IIA in Alhambra et al. [47].

  10. 10.

    This constraint is closely related to first, see Appendix A of [29].

  11. 11.

    Sometimes Eq. (27.29) is referred to as Crooks’ theorem itself. However, in [11], Crooks proves a more general statement about entropy production systems with microscopically reversible dynamics: \(P_\mathrm{fwd}\left( \omega \right) /P_\mathrm{rev}\left( -\omega \right) = \exp \left( \omega \right) \). The equation pertaining to work exchanges is the most notable example, and was also provided by Crooks’ in the same article.

  12. 12.

    Under the assumptions that allow us to use Crooks’ nonequilibrium work relation, the initial state is in thermal equilibrium and its free energy may be defined in the usual way. At time \(\tau \), the system may not be in thermal equilibrium. However, we can consider the state that would be reached (for the same final Hamiltonian) if that system were allowed to fully thermalize (limit \(t\rightarrow \infty \)), and take the free energy of that thermal state instead. It is the difference in free energy between the initial thermal state and the thermalized version of the final state that determines \(\Delta F\).

  13. 13.

    Quan and Dong [54] provide a derivation of this in a language familiar to quantum information scientists.

  14. 14.

    In addition to this and the work fluctuation equation discussed in Sect. 27.3, Yunger Halpern et al. [57] also demonstrate that this equality holds for classical phase space fluctuations.

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Acknowledgements

The author is grateful for discussions, comments and suggestions from Felix Binder, Oscar Dahlsten, Jayne Thompson, Vlatko Vedral, and Nicole Yunger Halpern. The author is financially supported by the Foundational Questions Institute “Physics of the Observer” large grant FQXi-RFP-1614.

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  1. Andrew J. P. Garner

    Present address: Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090, Vienna, Austria

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  1. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore, 117543, Singapore

    Andrew J. P. Garner

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Editors and Affiliations

  1. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Felix Binder

  2. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Luis A. Correa

  3. ICFO: Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology, Castelldefels, Spain

    Christian Gogolin

  4. CEMPS, Physics and Astronomy, University of Exeter, Exeter, UK

    Janet Anders

  5. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Gerardo Adesso

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Garner, A.J.P. (2018). One-Shot Information-Theoretical Approaches to Fluctuation Theorems. In: Binder, F., Correa, L., Gogolin, C., Anders, J., Adesso, G. (eds) Thermodynamics in the Quantum Regime. Fundamental Theories of Physics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-99046-0_27

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