Mathematical methods of quantum information theory

In 2017 Reinhard Werner gave a series of lectures on the mathematical methods of quantum information theory at the Leibniz Universität Hannover. These lectures were recorded and I have the pleasure of hosting these videos on my youtube channel. Over the coming weeks I’ll be posting these lectures there.

The prerequisites for these lectures are a standard course on quantum mechanics and some familiarity with mathematical analysis, e.g., Hilbert space, operators, etc., although these topics are reviewed in the first lectures.

The material covered in these lectures covered a range of topics in quantum information theory, a partial list is given below:

Lecture 1: Hilbert spaces, scalar product, bra, ket, operators.

Lecture 2: operators, diagonalization, functional calculus, qubit, composite systems, tensor product.

Lecture 3: composition, tensor product, channels, Heisenberg picture, Schrödinger picture, complete positivity, channel examples: unitary, depolarizing, von Neumann measurement.

Lecture 4: state space, probabilites, composition positivity, geometry of cones.

Lecture 5: geometry, extremal points, pure states, POVM, effect operators.

Lecture 6: Choi-Jamiokowski isomorphism, Kraus operators.

Lecture 7: Wigner’s theorem, anti unitary operators, symmetry groups, one-parameter groups, irreducible representations

Lecture 8: How to construct a Hilbert space, positive kernel, kolmogorov dilation, completion, going to the larger Hilbert space.

Lecture 9: Stinespring dilation Theorem and proof, Example: Naimark dilation, GNS representation, comparison theorem.

Lecture 10: Corollary of Stinespring, Kraus Form.

Lecture 11: Instrument, statistical structure; entanglement, Choi isomorphism and channels, classical models, Bell correlation.

Lecture 12: Mixed state entanglement, Bell inequalites, Tsirelsons inequality, pure state entanglement, Schmidt decomposition, maximally entangled states.

Lecture 13: Dispersion-free preparation, Joint measurement, measurement uncertainty relation, copying, transmitting a quantum state via a classical channel, signalling on correlations, teleportation.

Lecture 14: quantum teleportation; dense coding

Lecture 15: teleportation vs. dense coding, star trek

Lecture 16: norms and fidelities, operator norms, Schatten norms, trace norm, diamond norm, cb norm.

Lecture 17: some semidefinite tasks in QI SDPs, examples: unambiguous state discrimination, entanglement detection, code optimization, dual SDP, optimization on a convex cone (interior point method).

Lecture 18: noisy resources and conversion rates classical-quantum information transmission, two-step encoding inequality.

Lecture notes and exercises will not be distributed.

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