Elements of Quantum Informatics 0800-ELINKW

1. Theory of convex sets, sets of classical and quantum states:
a) convex sets, convex combinations, extreaml points, Krein-Millmann theorem, Caratheodory theorem
b) setz of classical and quantum states, inscribed and circumscribed ball,
c) qubit, Bloch ball, pure states vs. state vectors, global phase, first Hopf bundle
d) decomposition of mixed states into combinations of pure states
2) Consequences of non-commutativity
a) uncertainty principle, Shannon entropy, and Maasen-Uffink formulation
b) Mutuallu unbiased bases and methods of construction
c) random access codes
3. Quantum mechanics of composed and open systems:
a) tensor product and partial trace
b) separable and entangled states
c) quantum channels
d) generalised measurements
4. Dilation theorems
a) purification of a mixed state
b) realisation of a quantum channel as an effect of unitary evolution on a larger system
c) realisation of a generalised measurement as a projective measurement on a larger system
5. Theory of quantum channels
a) vectorisation and realignment
b) finding a KRauss representation of a given quantum channel
c) qubit channels, bistochastic channels, random unitary, Pauli channels
6. Theory of generalised measurements
a) distinguishing quantum states, Hellstrom theory
b) SIC POVMs
7. Optical systems in polarisation domain
a) quantum gates
b) realisation of POVMs
8. Non-clonning, BB84
9. Hidden variables space, Bell inequalities, non-signaling
10. Teleportation, local filtering.
11. Measures of entanglement
a) distillation of entanglement
b) EOF and EOD, bound entanglement
c) concurrence, negativity
d) relations between properties: PPT, distilability, non-locality
12. Entanlement detection
a) partial transposition criterion
b) positive maps criterion
c) entanglement witnesses and Jamiołkowski isomorphism
d) Choi map
e) realignment criterion
13. Shor algorithm of factorisation.