Irreversible information processing cannot be carried out without some inevitable thermodynamical work cost. This fundamental restriction, known as Landauer's principle, is increasingly relevant today, as the energy dissipation of computing devices impedes the development of their performance. Here we determine the minimal work required to carry out any logical process, for instance a computation. It is given by the entropy of the discarded information conditional to the output of the computation. Our formula takes precisely into account the statistically fluctuating work requirement of the logical process. It enables the explicit calculation of practical scenarios, such as computational circuits or quantum measurements. On the conceptual level, our result gives a precise and operational connection between thermodynamic and information entropy, and explains the emergence of the entropy state function in macroscopic thermodynamics.
In this brief paper, we compare two frameworks for characterizing possible operations in quantum thermodynamics. One framework considers thermal operations-unitaries which conserve energy. The other framework considers all maps which preserve the Gibbs state at a given temperature. Thermal operations preserve the Gibbs state; hence a natural question which arises is whether the two frameworks are equivalent. Classically, this is true-Gibbs-preserving maps are no more powerful than thermal operations. Here, we show that this no longer holds in the quantum regime: a Gibbspreserving map can generate coherent superpositions of energy levels while thermal operations cannot. This gap has an impact on clarifying a mathematical framework for quantum thermodynamics.
Information-theoretic approaches provide a promising avenue for extending the laws of thermodynamics to the nanoscale. Here, we provide a general fundamental lower limit, valid for systems with an arbitrary Hamiltonian and in contact with any thermodynamic bath, on the work cost for the implementation of any logical process. This limit is given by a new information measure-the coherent relative entropy-which accounts for the Gibbs weight of each microstate. The coherent relative entropy enjoys a collection of natural properties justifying its interpretation as a measure of information, and can be understood as a generalization of a quantum relative entropy difference. As an application, we show that the standard first and second laws of thermodynamics emerge from our microscopic picture in the macroscopic limit. Finally, our results have an impact on understanding the role of the observer in thermodynamics: Our approach may be applied at any level of knowledge-for instance at the microscopic, mesoscopic or macroscopic scales-thus providing a formulation of thermodynamics that is inherently relative to the observer. We obtain a precise criterion for when the laws of thermodynamics can be applied, thus making a step forward in determining the exact extent of the universality of thermodynamics and enabling a systematic treatment of Maxwell-demon-like situations.
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