# Thermodynamics as a Consequence of Information Conservation

Manabendra Nath Bera1,2, Arnau Riera1,2, Maciej Lewenstein1,3, Zahra Baghali Khanian1,4, and Andreas Winter3,4

1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, ES-08860 Castelldefels, Spain
2Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany
3ICREA, Pg. Lluis Companys 23, ES-08010 Barcelona, Spain
4Departament de Física: Grup d'Informació Quàntica, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain

### Abstract

Thermodynamics and information have intricate interrelations. Often thermodynamics is considered to be the logical premise to justify that $\textit{information is physical}$ - through Landauer's principle -, thereby also linking information and thermodynamics. This approach towards information has been instrumental to understand thermodynamics of logical and physical processes, both in the classical and quantum domain. In the present work, we formulate thermodynamics as an exclusive consequence of information conservation. The framework can be applied to the most general situations, beyond the traditional assumptions in thermodynamics: we allow systems and thermal baths to be quantum, of arbitrary sizes and even possessing inter-system correlations.

Here, systems and baths are not treated differently, rather both are considered on an equal footing. This leads us to introduce a ''temperature''-independent formulation of thermodynamics. We rely on the fact that, for a fixed amount of information, measured by the von Neumann entropy, any system can be transformed to a state with the same entropy that possesses minimal energy. This state, known as a $\textit{completely passive}$ state, acquires Boltzmann-Gibbs canonical form with an $\textit{intrinsic temperature}$. We introduce the notions of bound and free energy and use them to quantify heat and work, respectively. Guided by the principle of information conservation, we develop universal notions of equilibrium, heat and work, Landauer's principle and universal fundamental laws of thermodynamics. We demonstrate that the maximum efficiency of a quantum engine with a finite bath is in general lower than that of an ideal Carnot engine. We introduce a resource theoretic framework for our $\textit{intrinsic temperature}$ based thermodynamics, within which we address the problem of work extraction and state transformations. Finally, the framework is extended to multiple conserved quantities.

Thermodynamics is a phenomenological theory developed during the 19th Century in the context of the industrial revolution. Steam engines were the engine of the industrial revolution and it was essential to understand its functioning as well as its efficiency. There is no doubt about the success of thermodynamics which has survived to the scientific revolutions of relativity and quantum mechanics. But, how is it possible that so many completely different systems irrespective of being classical or quantum fulfill the laws of thermodynamics? What is the common feature that makes all these different systems to respect thermodynamics?

In this work we show that the essential feature behind all the systems that fulfill thermodynamics is that they are described by underlying physical theories that are microscopically reversible, or more specifically, that conserve the amount of information quantified by the von Neumann entropy. Guided by this principle of information conservation, we develop a formalism of temperature independent thermodynamics and a notion of heat that does not require the environment to be thermal. This notion of temperature independent thermodynamics is expected to be very useful in the current revolution of nano-technology in which the miniaturisation of components and tools to the nano-scale baths will become of a small size and will be brought easily out of equilibrium. We show that the maximum efficiency of a quantum engine with small baths is in general lower than that of an ideal Carnot’s engine. We finally extend our formalism and results to systems with additional conserved quantities apart from the energy.

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The above citations are from Crossref's cited-by service (last updated successfully 2022-07-05 17:06:09) and SAO/NASA ADS (last updated successfully 2022-07-05 17:06:10). The list may be incomplete as not all publishers provide suitable and complete citation data.