### Quantum Shannon Theory: Rothschild Distinguished Visiting Fellow Lecture

Holevo, A *(Russian Academy of Sciences )*

Monday 14 October 2013, 17:00-18:00

Seminar Room 1, Newton Institute

#### Abstract

The notions of channel and capacity are central to the classical Shannon theory. "Quantum Shannon theory" denotes a subfield of quantum information science which uses operator analysis, convexity and matrix inequalities, asymptotic techniques such as large deviations and measure concentration to study mathematical models of communication channels and their information-processing performance. From the mathematical point of view quantum channels are normalized completely positive maps of operator algebras, the analog of Markov maps in the noncommutative probability theory, while the capacities are related to certain norm-like quantities. In applications noisy quantum channels arise from irreversible evolutions of open quantum systems interacting with environment-a physical counterpart of a mathematical dilation theorem.
It turns out that in the quantum case the notion of channel capacity splits into the whole spectrum of numerical information-processing characteristics depending on the kind of data transmitted (classical or quantum) as well as on the additional communication resources. An outstanding role here is played by quantum correlations - entanglement - inherent in tensor-product structure of composite quantum systems. This talk presents a survey of basic coding theorems providing analytical expressions for the capacities of quantum channels in terms of entropic quantities. We also touch upon some open mathematical problems, such as additivity and Gaussian optimizers, concerning the entropic characteristics of both theoretically and practically important Bosonic Gaussian channels.

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