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arXiv:1910.00415

Entropy and applications of the area law in open quantum systems

Tarek Khalil, J. Richert

Published Sep 28, 2019• 0 citations
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Abstract

Open quantum systems interact with their environment and their dynamical behaviour depends strongly both on the spectral properties of the environment and the structure of the interaction between the physical system and the environment. We examine the consequences of these spectral and structural properties on simple but general systems in the case of deterministic (non stochastic) interactions of arbitrary strength. The central point of interest concerns the role played by the semi-group property of the interaction in its relation with entropy and area laws properties of the system of interest.

DOI
10.48550/arxiv.1910.00415

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Open quantum systems interact with their environment and their dynamical behaviour depends strongly both on the spectral properties of the environment and the structure of the interaction between the physical system and the environment. We examine the consequences of these spectral and structural properties on simple but general systems in the case of deterministic (non stochastic) interactions of arbitrary strength. The central point of interest concerns the role played by the semi-group property of the interaction in its relation with entropy and area laws properties of the system of interest.
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The state of a system changes as a consequence of its internal dynamics and of the interaction with the environment that leads to a system‐environment correlation and cannot be represented in terms of unitary Hamiltonian dynamics of the system. In order to investigate the role of interaction between the system and its environment, we derive a general formula which relates the Hamiltonian of the system to entropy. This formula enables us to discriminate a type of interactions which causes the change in the state of the system in a spontaneous process. In other words, this formula enables us to know the type of interactions which would induce a new equilibrium state of the system in the process. It is found that the certain type of interactions does not affect the state (dynamics) of the system even if there are interactions between the system and its environment.
Since the days of Boltzmann the foundation of thermodynamics has been subject to an ongoing discussion. Various problems like the precise definition of entropy, the origin of the second law, etc. have always been issues. For a long time the discussion has not been in the center of interest, for statistical physics work extremely well from a practical point of view. But now, as research deals with single small objects like molecules or atoms, the question arises whether or not thermodynamics are applicable to them, which leads back to the old question of the foundation of thermodynamics. We examine whether thermodynamics can be viewed as emerging from Schrödinger-type quantum dynamics. Therefore we partially transfer concepts of thermodynamical behavior in Gamma-space to the Hilbertspace of compound systems. Central roles play the local Von Neumann entropy of a system and the inevitability of entanglement with its environment, causing it to increase.
Entropy, and its temporal evolution, play a central role in the foundations of quantum theory and in modern quantum technologies. Here we study, in particular, the relations between the --- in general, non-Markovian --- evolution of an open quantum system, the notions of divisibility of a dynamical map and of distinguishability of quantum states, and the temporal behaviour of various entropy-related quantities such as the Renyi (and sandwiched Renyi) divergences, and the so-called min- and max- conditional entropies. This, in turn, gives rise to an operational meaning of (non-)Markovianity.
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