# Elementary Thermal Operations

Matteo Lostaglio1, Álvaro M. Alhambra2, and Christopher Perry3

1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain
2Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
3QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

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To what extent do thermodynamic resource theories capture physically relevant constraints? Inspired by quantum computation, we define a set of elementary thermodynamic gates that only act on 2 energy levels of a system at a time. We show that this theory is well reproduced by a Jaynes-Cummings interaction in rotating wave approximation and draw a connection to standard descriptions of thermalisation. We then prove that elementary thermal operations present tighter constraints on the allowed transformations than thermal operations. Mathematically, this illustrates the failure at finite temperature of fundamental theorems by Birkhoff and Muirhead-Hardy-Littlewood-Polya concerning stochastic maps. Physically, this implies that stronger constraints than those imposed by single-shot quantities can be given if we tailor a thermodynamic resource theory to the relevant experimental scenario. We provide new tools to do so, including necessary and sufficient conditions for a given change of the population to be possible. As an example, we describe the resource theory of the Jaynes-Cummings model. Finally, we initiate an investigation into how our resource theories can be applied to Heat Bath Algorithmic Cooling protocols.

Recent work using tools from quantum information theory has investigated the laws of thermodynamics at the quantum and nanoscale. In deriving these ultimate limits however, it is assumed that the system under consideration can be manipulated arbitrarily precisely, subject only to constraints such as energy conservation. As such, it is unclear how the newly derived laws relate to real experimental setups. In this work, we provide tools for incorporating physically relevant constraints into the previous analysis, limiting the interactions to those that only act on a reduced amount of energy levels, or/and that can be generated via simple light-matter interaction models. This enables the potential for fundamental limits to be investigated for real physical systems.

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