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Thermodynamics in the Quantum Regime pp 301–316Cite as

The Coherent Crooks Equality

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

Abstract

This chapter reviews an information theoretic approach (Part V) to deriving quantum fluctuation theorems (see Chap. 10) that was developed in [37, 38]. When a thermal system is driven from equilibrium, random quantities of work are required or produced: the Crooks equality is a classical fluctuation theorem that quantifies the probabilities of these work fluctuations. The framework summarised here generalises the Crooks equality to the quantum regime by modeling not only the driven system but also the control system and energy supply that enables the system to be driven. As is reasonably common within the information theoretic approach but high unusual for fluctuation theorems, this framework explicitly accounts for the energy conservation using only time independent Hamiltonians. We focus on explicating a key result of [37]: a Crooks-like equality for when the energy supply is allowed to exist in a superposition of energy eigenstates states.

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Notes

  1. 1.

    We use the term ‘coherence’ in the sense of a ‘superposition of states belonging to different energy eigenspaces’.

  2. 2.

    To avoid a proliferation of notation we here use the symbol \( \mathcal{T} \) to denote both a mapping on the level of Hilbert spaces and a map on the space of operators on the Hilbert space.

  3. 3.

    The global invariance and factorisability conditions can be naturally reformulated in terms of the Gibbs map. If we define the transition probability from a state \(G_{\rho _A}(H, T)\) to a state \(\rho _B\) as \(P(\rho _B|G_{\rho _A}(H, T)) := {{\,\mathrm{Tr}\,}}[ \rho _B V G_{\rho _A}(H, T) V^\dagger ]\), then global invariance can be rewritten as \(P(\rho _f | G_{\rho _i}(H, T))/P( \mathcal{T} (\rho _i)|G_{ \mathcal{T} (\rho _f)}(H, \, T)) = \tilde{Z}_{ \mathcal{T} (\rho _f)}(H, T)/\tilde{Z}_{\rho _i}(H, T)\). The bipartite setup is factorisable if \(G_{\rho _M \otimes \rho _S}(H_{MS}, T) = G_{\rho _M}(H_M, T) \otimes G_{\rho _S}(H_S, T) \) and \(\tilde{Z}_{\rho _M \otimes \rho _S}(H_{MS}, T) = \tilde{Z}_{\rho _M}(H_M, T) \otimes \tilde{Z}_{\rho _S}(H_S, T)\).

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Acknowledgements

The author thanks Johan Åberg, Álvaro Alhambra, Janet Anders, Florian Mintert, Erick Hinds Mingo, Tom Hebdige and Jake Lishman for commenting on drafts and David Jennings for numerous indispensable discussions. The author is supported by the Engineering and Physical Sciences Research Council Centre for Doctoral Training in Controlled Quantum Dynamics.

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Authors and Affiliations

  1. Department of Physics, Imperial College London, London, SW7 2BW, UK

    Zoe Holmes

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  1. Zoe Holmes
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Correspondence to Zoe Holmes .

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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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Holmes, Z. (2018). The Coherent Crooks Equality. 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_12

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