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Quantum Information Processing with Finite Resources -- Mathematical Foundations
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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.
Forward citations
Cited by 7 Pith papers
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Achievable rates in non-asymptotic bosonic quantum communication
First easily computable lower bounds on non-asymptotic capacities of Gaussian channels, plus a tail bound on Gaussian photon statistics and a fixed-precision trace-distance algorithm.
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Communication Advantages from Quantum Dense Network Coding
Dense network coding computes group operations over multiaccess networks with half the classical communication cost using shared entanglement plus quantum channels, and yields measurement-device-independent quantum ke...
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No off-diagonal quantum focusing for R\'enyi divergences
No Rényi-type divergence obeying DPI, tensor additivity and matched cq conditioning admits a universal off-diagonal quantum focusing inequality.
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Maximum channel entropy principle and microcanonical channels
A maximum-entropy principle for quantum channels yields thermal channels with exponential form, analogous to thermal states.
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Uhlmann's theorem for relative entropies
For all alpha in [1/2, infinity], the regularized alpha-Renyi divergence from a bipartite state to the set of extensions of a fixed marginal equals the marginal divergence, and the measured version lies between the me...
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Erasure cost of a quantum process: A thermodynamic meaning of the dynamical min-entropy
The adversarial erasure cost of a quantum channel equals, in the zero-error limit, the negative of the channel's min-entropy times k_B T ln 2.
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Thermodynamics of quantum processes: An operational framework for free energy and reversible athermality
For quantum channels, athermality distillation and formation under Gibbs-preserving superchannels both converge asymptotically to the channel's relative-entropy free energy, making the resource theory asymptotically r...
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