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.