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      Quantum Information Processing with Finite Resources - Mathematical Foundations

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          Abstract

          One of the predominant challenges when engineering future quantum information processors is that large quantum systems are notoriously hard to maintain and control accurately. It is therefore of immediate practical relevance to investigate quantum information processing with limited physical resources, for example to ask: How well can we perform information processing tasks if we only have access to a small quantum device? Can we beat fundamental limits imposed on information processing with classical resources? This book will introduce the reader to the mathematical framework required to answer such questions. A strong emphasis is given to information measures that are essential for the study of devices of finite size, including R\'enyi entropies and smooth entropies. The presentation is self-contained and includes rigorous and concise proofs of the most important properties of these measures. The first chapters will introduce the formalism of quantum mechanics, with particular emphasis on norms and metrics for quantum states. This is necessary to explore quantum generalizations of R\'enyi divergence and conditional entropy, information measures that lie at the core of information theory. The smooth entropy framework is discussed next and provides a natural means to lift many arguments from information theory to the quantum setting. Finally selected applications of the theory to statistics and cryptography are discussed.

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          Possible generalization of Boltzmann-Gibbs statistics

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            �ber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik

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              Separability Criterion for Density Matrices

              A quantum system consisting of two subsystems is separable if its density matrix can be written as \(\rho=\sum_A w_A\,\rho_A'\otimes\rho_A''\), where \(\rho_A'\) and \(\rho_A''\) are density matrices for the two subsytems. In this Letter, it is shown that a necessary condition for separability is that a matrix, obtained by partial transposition of \(\rho\), has only non-negative eigenvalues. This criterion is stronger than Bell's inequality.
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                Author and article information

                Journal
                2015-04-01
                2015-10-18
                Article
                10.1007/978-3-319-21891-5
                1504.00233
                20ea9358-d915-4582-a413-68b7dca0e5b1

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                135 pages, partly based on arXiv:1203.2142, v3: minor fixes, published version, SpringerBriefs in Mathematical Physics (2016)
                quant-ph cs.IT math-ph math.IT math.MP

                Mathematical physics,Quantum physics & Field theory,Numerical methods,Mathematical & Computational physics,Information systems & theory

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