PSI 2017/2018 - Quantum Theory (Emerson)

Motivations, Postulate 1 (Ideal preparations), Hilbert space. Articles mentioned by Joseph: L. Hardy's article (2011) on physical principles that imply mathematical axioms; Howard et al (2014) on Contextuality and quantum computing; Schrödinger's article on the cat paradox (not sure about this…

Notion of dual space - Riesz' theorem; Examples of Hilbert spaces; Postulate 2 - Born's rule; Postulate 3 - The von Neumann eq (global phase = gauge symmetry).

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).

Sequential Measurements and state update rule - Mathematical rules - Non-rank 1 projector - Degenerate vs non-degenerate measurement operationally.

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.

Theory of Quantum Measurement - Von Neuman inderect measurement model - Inconsistency btw result obtained from Schrödinger eq and from State update rule - Stern-Gerlach example.

Generalized Measurement: Mathematical structure - POVM - Postulate 2 (Discrete case) - POVM Continuous case - Postulate 2 (Continuous case) - Example: Trine POVM measurement - Naimark/Neumark's dilation theorem - Generalized transformations: Postulate 3.

Generalized transformationsv - Kraus decomposition - Example: CNOT gate - Stinespring dilation theorem - Postulate 3 - Counter example.

Counter example to generalized postulate 3 and theory of entanglement; State-update rule under POVM - Entanglement and non-locality - Entropy.