Quantum Systems, Channels and Information :

Holevo, Alexander S.

Quantum Systems, Channels and Information : A Mathematical Introduction. - 1 online resource (349 pages) - De Gruyter Studies in Mathematical Physics Ser. ; v.16 . - De Gruyter Studies in Mathematical Physics Ser. .

Intro -- Preface -- I Basic structures -- 1 Vectors and operators -- 1.1 Hilbert space -- 1.2 Operators -- 1.3 Positivity -- 1.4 Trace and duality -- 1.5 Convexity -- 1.6 Notes and references -- 2 States, observables, statistics -- 2.1 Structure of statistical theories -- 2.1.1 Classical systems -- 2.1.2 Axioms of statistical description -- 2.2 Quantum states -- 2.3 Quantum observables -- 2.3.1 Quantum observables from the axioms -- 2.3.2 Compatibility and complementarity -- 2.3.3 The uncertainty relation -- 2.3.4 Convex structure of observables -- 2.4 Statistical discrimination between quantum states -- 2.4.1 Formulation of the problem -- 2.4.2 Optimal observables -- 2.5 Notes and references -- 3 Composite systems and entanglement -- 3.1 Composite systems -- 3.1.1 Tensor products -- 3.1.2 Naimark's dilation -- 3.1.3 Schmidt decomposition and purification -- 3.2 Quantum entanglement vs "local realism" -- 3.2.1 Paradox of Einstein-Podolski-Rosen and Bell's inequalities -- 3.2.2 Mermin-Peres game -- 3.3 Quantum systems as information carriers -- 3.3.1 Transmission of classical information -- 3.3.2 Entanglement and local operations -- 3.3.3 Superdense coding -- 3.3.4 Quantum teleportation -- 3.4 Notes and references -- II The primary coding theorems -- 4 Classical entropy and information -- 4.1 Entropy of a random variable and data compression -- 4.2 Conditional entropy and the Shannon information -- 4.3 The Shannon capacity of the classical noisy channel -- 4.4 The channel coding theorem -- 4.5 Wiretap channel -- 4.6 Gaussian channel -- 4.7 Notes and references -- 5 The classical-quantum channel -- 5.1 Codes and achievable rates -- 5.2 Formulation of the coding theorem -- 5.3 The upper bound -- 5.4 Proof of the weak converse -- 5.5 Typical projectors -- 5.6 Proof of the Direct Coding Theorem -- 5.7 The reliability function for pure-state channel. 5.8 Notes and references -- III Channels and entropies -- 6 Quantum evolutions and channels -- 6.1 Quantum evolutions -- 6.2 Completely positive maps -- 6.3 Definition of the channel -- 6.4 Entanglement-breaking and PPT channels -- 6.5 Quantum measurement processes -- 6.6 Complementary channels -- 6.7 Covariant channels -- 6.8 Qubit channels -- 6.9 Notes and references -- 7 Quantum entropy and information quantities -- 7.1 Quantum relative entropy -- 7.2 Monotonicity of the relative entropy -- 7.3 Strong subadditivity of the quantum entropy -- 7.4 Continuity properties -- 7.5 Information correlation, entanglement of formation and conditional entropy -- 7.6 Entropy exchange -- 7.7 Quantum mutual information -- 7.8 Notes and references -- IV Basic channel capacities -- 8 The classical capacity of quantum channel -- 8.1 The coding theorem -- 8.2 The χ - capacity -- 8.3 The additivity problem -- 8.3.1 The effect of entanglement in encoding and decoding -- 8.3.2 A hierarchy of additivity properties -- 8.3.3 Some entropy inequalities -- 8.3.4 Additivity for complementary channels -- 8.3.5 Nonadditivity of quantum entropy quantities -- 8.4 Notes and references -- 9 Entanglement-assisted classical communication -- 9.1 The gain of entanglement assistance -- 9.2 The classical capacities of quantum observables -- 9.3 Proof of the Converse Coding Theorem -- 9.4 Proof of the Direct Coding Theorem -- 9.5 Notes and references -- 10 Transmission of quantum information -- 10.1 Quantum error-correcting codes -- 10.1.1 Error correction by repetition -- 10.1.2 General formulation -- 10.1.3 Necessary and sufficient conditions for error correction -- 10.1.4 Coherent information and perfect error correction -- 10.2 Fidelities for quantum information -- 10.2.1 Fidelities for pure states -- 10.2.2 Relations between the fidelity measures. 10.2.3 Fidelity and the Bures distance -- 10.3 The quantum capacity -- 10.3.1 Achievable rates -- 10.3.2 The quantum capacity and the coherent information -- 10.3.3 Degradable channels -- 10.4 The private classical capacity and the quantum capacity -- 10.4.1 The quantum wiretap channel -- 10.4.2 Proof of the Private Capacity Theorem -- 10.4.3 Large deviations for random operators -- 10.4.4 The Direct Coding Theorem for the quantum capacity -- 10.5 Notes and references -- V Infinite systems -- 11 Channels with constrained inputs -- 11.1 Convergence of density operators -- 11.2 Quantum entropy and relative entropy -- 11.3 Constrained c-q channel -- 11.4 Classical-quantum channel with continuous alphabet -- 11.5 Constrained quantum channel -- 11.6 Entanglement-assisted capacity of constrained channels -- 11.7 Entanglement-breaking channels in infinite dimensions -- 11.8 Notes and references -- 12 Gaussian systems -- 12.1 Preliminary material -- 12.1.1 Spectral decomposition and Stone's Theorem -- 12.1.2 Operators associated with the Heisenberg commutation relation -- 12.1.3 Classical signal plus quantum noise -- 12.1.4 The classical-quantum Gaussian channel -- 12.2 Canonical commutation relations -- 12.2.1 Weyl-Segal CCR -- 12.2.2 The symplectic space -- 12.2.3 Dynamics, quadratic operators and gauge transformations -- 12.3 Gaussian states -- 12.3.1 Characteristic function -- 12.3.2 Definition and properties of Gaussian states -- 12.3.3 The density operator of Gaussian state -- 12.3.4 Entropy of a Gaussian state -- 12.3.5 Separability and purification -- 12.4 Gaussian channels -- 12.4.1 Open bosonic systems -- 12.4.2 Gaussian channels: basic properties -- 12.4.3 Gaussian observables -- 12.4.4 Gaussian entanglement-breaking channels -- 12.5 The capacities of Gaussian channels -- 12.5.1 Maximization of the mutual information. 12.5.2 Gauge-covariant channels -- 12.5.3 Maximization of the coherent information -- 12.5.4 The classical capacity: conjectures -- 12.6 The case of one mode -- 12.6.1 Classification of Gaussian channels -- 12.6.2 Entanglement-breaking channels -- 12.6.3 Attenuation/amplification/classical noise channel -- 12.6.4 Estimating the quantum capacity -- 12.7 Notes and references -- Bibliography -- Index.

The subject of this book is theory of quantum system presented from information science perspective. The central role is played by the concept of quantum channel and its entropic and information characteristics. Quantum information theory gives a key to understanding elusive phenomena of quantum world and provides a background for development of experimental techniques that enable measuring and manipulation of individual quantum systems. This is important for the new efficient applications such as quantum computing, communication and cryptography. Research in the field of quantum informatics, including quantum information theory, is in progress in leading scientific centers throughout the world. This book gives an accessible, albeit mathematically rigorous and self-contained introduction to quantum information theory, starting from primary structures and leading to fundamental results and to exiting open problems.

9783110273403


Information theory in physics.;Quantum entropy.


Electronic books.

QC28 -- .K53613 2012eb

530.15