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arXiv:1509.04253 · Journal of Experimental and Theoretical Physics

Keldysh formalism for multiple parallel worlds

Mohammad H. Ansari, Yuli V. Nazarov

Published Mar 1, 2016• 14 citations
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Abstract

We present a compact and self-contained review of the recently developed Keldysh formalism for multiple parallel worlds. The formalism has been applied to consistent quantum evaluation of the flows of informational quantities, in particular, to the evaluation of Renyi and Shannon entropy flows. We start with the formulation of the standard and extended Keldysh techniques in a single world in a form convenient for our presentation. We explain the use of Keldysh contours encompassing multiple parallel worlds. In the end, we briefly summarize the concrete results obtained with the method.

DOI
10.1134/s1063776116030134

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We present a compact and self-contained review of the recently developed Keldysh formalism for multiple parallel worlds. The formalism has been applied to consistent quantum evaluation of the flows of informational quantities, in particular, to the evaluation of Renyi and Shannon entropy flows. We start with the formulation of the standard and extended Keldysh techniques in a single world in a form convenient for our presentation. We explain the use of Keldysh contours encompassing multiple parallel worlds. In the end, we briefly summarize the concrete results obtained with the method.
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The Renyi entropies for an interacting quantum dot are computed under a multi-contour Keldysh formalism. With this in hand, the self-information generating function is determined and the probability distribution of self-information is computed, including the effects of Coulomb interactions in the Hartree approximation. The author argues through these computations that fluctuations in the self-information and the Renyi entropies are tied together intimately and that these fluctuations thus offer a way to give the Renyi entropies physical meaning.
The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.
The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation. We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.
In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory.