Open question for deriving Quantum vs Classical Mechanics; 2-level systems ("Quantum Bit") - Quantum vs Classical rotations; State update rule - Luder's Rule (post-selected scenario vs non-post-selected scenario).
Meaning of labels; state up-date rule as a consequence of postulates; state up-date rule classically - Kochen-Specker model of qubit. Some additional references on the Kochen Specker model: Part II; Chap. D part 3.
Composite Quantum Systems - Examples and basic mathematical properties; Entanglement - Schmidt decomposition/Schmidt number; Infinite dimensional Hilbert space and unbounded operators - Dirac's approach vs von Neumann's approach.
Infinite Dimensional Systems - Dirac's approach vs von Neumann's approach; Operational Postulates for open (real-world) quantum systems: probabilistic mixtures of pure states vs Partial description of composite system (partial trace, reduced state) - Operational postulate 1.
Real-world quantum systems description - Pure state vs Mixed states - Purification of Mixed States; Generalized measurement - Mixtures of "ideal measurement" - Ideal measurement of a composite system - POVMs.