The Mathematical Language of Quantum Theory
From Uncertainty to Entanglement

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CONTENT

1. Hilbert space refresher
   1.1 Hilbert spaces
   1.2 Operators on Hilbert spaces
   1.3 Additional useful mathematics
2. States and Effects
   2.1 Duality of states and effects
   2.2 Superposition structure of pure states
   2.3 Symmetry
   2.4 Composite systems
3. Observables
   3.1 Observables as positive-operator valued measures
   3.2 Sharp observables
   3.3 Informationally complete observables
   3.4 Testing quantum systems
   3.5 Relations between observables
   3.6 Example: photon-counting observable
4. Operations and Channels
   4.1 Transforming quantum systems
   4.2 Physical model of quantum channels
   4.3 Elementary properties of quantum channels
   4.4 Parametrizations of quantum channels
   4.5 Special classes of channels
   4.6 Example: qubit channels
5. Measurement models and Instruments
   5.1 Three levels of description of measurements
   5.2 Disturbance cause by measurements
   5.3 Luders instruments
   5.4 Repeatable measurements
   5.5 Programmable quantum processors
6. Entanglement
   6.1 Entangled bipartite systems
   6.2 Entanglement and LOCC
   6.3 Entanglement detection
   6.4 Additional topics in entanglement theory
   6.5 Example: Werner states