# Decomposable coherence and quantum fluctuation relations

Erick Hinds Mingo1 and David Jennings1,2,3

1Controlled Quantum Dynamics Theory Group, Imperial College London, Prince Consort Road, London SW7 2BW, UK
2Department of Physics, University of Oxford, Oxford, OX1 3PU, UK
3School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK.

### Abstract

In Newtonian mechanics, any closed-system dynamics of a composite system in a microstate will leave all its individual subsystems in distinct microstates, however this fails dramatically in quantum mechanics due to the existence of quantum entanglement. Here we introduce the notion of a coherent work process', and show that it is the direct extension of a work process in classical mechanics into quantum theory. This leads to the notion of decomposable' and `non-decomposable' quantum coherence and gives a new perspective on recent results in the theory of asymmetry as well as early analysis in the theory of classical random variables. Within the context of recent fluctuation relations, originally framed in terms of quantum channels, we show that coherent work processes play the same role as their classical counterparts, and so provide a simple physical primitive for quantum coherence in such systems. We also introduce a pure state effective potential as a tool with which to analyze the coherent component of these fluctuation relations, and which leads to a notion of temperature-dependent mean coherence, provides connections with multi-partite entanglement, and gives a hierarchy of quantum corrections to the classical Crooks relation in powers of inverse temperature.

How to define work in quantum thermodynamics has been the centre of much debate and received renewed interest after recent no-go results restricting the consistency of quantum and classical definitions. Here we circumvent this issue by first developing a framework for purely deterministic quantum work processes that naturally extend the Newtonian framework into quantum mechanics. We show that this coherent form of work becomes the familiar Newtonian work in the semi-classical limit, however the coherent work output is no longer simply described by a single value but rather it is encoded into a quantum state. Embedding this into recent results from studies of quantum fluctuation theorems, we find that this coherent work naturally arises within recent symmetry-based fluctuation theorems in a way that parallels the way Newtonian work appears in the classical fluctuation theorem settings, and so supports the claim that the framework is an appropriate generalisation into the quantum mechanical regime.

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### Cited by

[1] Zoë Holmes, Erick Hinds Mingo, Calvin Y.-R. Chen, and Florian Mintert, "Quantifying Athermality and Quantum Induced Deviations from Classical Fluctuation Relations", arXiv:2001.04926, Entropy 22 1, 111 (2020).

[2] Hyukjoon Kwon and M. S. Kim, "Fluctuation Theorems for a Quantum Channel", Physical Review X 9 3, 031029 (2019).

[3] Takahiro Sagawa, Philippe Faist, Kohtaro Kato, Keiji Matsumoto, Hiroshi Nagaoka, and Fernando G. S. L. Brandao, "Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein's Lemma", arXiv:1907.05650.

[4] Matteo Scandi, Harry J. D. Miller, Janet Anders, and Marti Perarnau-Llobet, "Quantum work statistics close to equilibrium", arXiv:1911.04306.

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