The Theory of Quantum Information
Cambridge University Press, 26.04.2018
This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.
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2 Basic Notions of Quantum Information
3 Similarity and Distance Among States and Channels
4 Unital Channels and Majorization
5 Quantum Entropy and Source Coding
6 Bipartite Entanglement
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a e XX a€XX aeXX alphabet assume Borel bounded trace norm choice classical state set coding collection completely bounded trace completely positive maps completes the proof complex Euclidean space compound register consider Corollary defined definition denote density operator dim(A ensemble entanglement equation equivalent exist fidelity function given Haar measure Herm(X Hermitian operators holds identity implies statement inequality isometry Lemma linear LOCC channel mixed-unitary channel Neumann entropy nonnegative nonzero notion orthonormal p e D(A Pos(A Pos(X Pos(Y positive integer positive real number positive semidefinite operators probabilistic probability vector Proof Let Proposition proved quantum capacity quantum channels quantum information quantum relative entropy random variable respect satisfying semidefinite program Shannon entropy space and let subset subspace tensor product uniform spherical measure unit vector unital channels unitary operator vec(A von Neumann entropy