1. Finite-dimensional Hilbert spaces, Dirac formalism. Orthonormal bases, trace, Hilbert-Schmidt inner product. Special operators, spectral decomposition, functional calculus. Finite-dimensional operator algebraic models, states, measurements, Born rule. Composite systems, tensor product of Hilbert spaces and observable algebras. 2. Completely positive maps and their representations. State transformation problem for pure states. Reversibility on a subspace, Knill-Laflamme theorem. 3. Separable and entangled states, linear entanglement witnesses, PPT and reduction criteria. Werner states and isotropic states. 4. Single-copy state discrimination, trace norm distance, fidelity, application in state cloning. Single-copy channel discrimination, diamond norm. 5. Asymptotic binary state discrimination. Operator convexity of the power functions, Audenaert’s inequality. Pinching inequality, large deviation bounds. Direct exponent, Chernoff exponent, Stein’s lemma. Petz-type Rényi divergences, monotonicity from the operational interpretation. 6. Quantum source coding in the purified and the ensemble picture. Quantum Rényi entropies, von Neumann entropy, conditional entropy and mutual information. Entropy inequalities (strong subadditivity, weak monotonicity), continuity properties.7. Quantum thermodynamics, free energy, Gibbs states, variational formulas. Majorization. Entropy-based entanglement criteria. 8. Classical-quantum channel coding, random coding exponent, divergence radius representation of the capacity. 9. Unitarily invariant norms, Hölder inequality. Variational representation of the Rényi (alpha-z)-divergences. Monotonicity of the sandwiched Rényi divergences, connection to reversibility. Strong converse exponent of binary state discrimination and classical-quantum channel coding.
Nyomtatóbarát változat
