Mathematical methods of quantum information theory, Lecture 1

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 here.

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.

Lecture notes and exercises will not be distributed.

In this first lecture, Hilbert spaces, scalar product, bra, ket, operators were introduced.

[xiaoliang]

Let me make a ToC:
Lecture 1: Hilbert spaces, scalar product, bra, ket, operators
Lecture 2: operators, diagonalization, functional calculus, qubit, composite systems, and the tensor product
Lecture 3: composition, tensor product, channels, Heisenberg picture, Schrödinger picture, complete positivity, channel examples: unitary, depolarizing, von Neumann measurement
Lecutre 4: state space, probabilites, composition, the positive cone, positivity, and the geometry of cones
Lecture 5: extremal points, pure states
extremal observables, POVMs, and effect operators
Lecture 6: extremal observables, the Choi-Jamiokowski isomorphism, Kraus operators, and symmetries of the positive cone
Lecture 7: symmetries of the positive cone
Wigner's theorem, anti unitary operators, symmetry groups, one-parameter groups, and irreducible representations
Lecture 8: how to construct a Hilbert space, positive kernel, kolmogorov dilation, completion, Naimark dilation, going to the larger Hilbert space, and Stinespring dilation
Lecture 9: the Stinespring dilation Theorem and proof, Example: Naimark dilation, GNS representation, comparison theorem
Lecture 10: corollaries for the Stinespring dilation Theorem
Lecture 11: instruments, statistical structure, Choi isomorphism and channels, classical models, and Bell correlations
Lecture 12: entanglement
Lecture 13: tasks and resources
Lecture 14: quantum teleportation and dense coding 1
Lecture 15: quantum teleportation and dense coding 2
Lecture 16: norms and fidelities
Lecture 17: some semidefinite tasks in QI
Lecture 18: noisy resources and conversion rates

[emanuelemarconato]

The most amazing part is the cleaning of the blackboard. Absolutely love <3

[jakubjan]

These are grate lectures. The essence of the topic is presented here in very intuitive way yet in a strict mathematical manner.

[terencedeng]

It is a very good lecture for us to study the quantum information theory

[sumitdey]

Thank you Sir. You are doing a great service(at least for us lesser mortals!)

[hakanakgun]

I don't know the last part after concrete hilbert spaces. Can anyone say by which words should I search for this part? Also if you can recommend resources about that part I'd appriciate a lot.

[weishanlei]

So exciting to watch the lecture series you post, Professor Orsborn!

[nassimelaflej]

Thanks a lot, this is exactly what I was looking for. Warm regards from Lille :D.

[arthurzhang]

great thanks for sharing the lectures, that is awesome stuff!

[bzbrz]

Thanks for posting this series, looks promising. Looking forward to next lectures.

[ratulbanerjee]

always wanted a maths introduction qi theory, thanks a lot sir

[richardlidar]

Great lectures! What is the reference textbook?

[garywpearson]

Thanx for posting. Looks like we're off to an interesting start!

[jzcftty]

Osborn is very known in that field, he has a lot of seminal papers.

[anguskan]

it's like christmas came early :D

[AGN]

❤ beautiful. Thank you for this video and this YouTube channel.

[noono-eh]

can anyone share link for homework problems and solutions for these lectures? if there are any

[shawn]

is there a lecture note for this wonderful course?

[sidddddddddddddd]

Guys, relax! This is not Prof. Osborne. Watch his CFT lectures or QFT to know who he is 😅

[soumitrabose]

Some one must henceforth see the output of every such video before publishing, because the board could not be read because of the poor focus by the cameraperson. The content is so important and so appropriate, and we viewers lost half of it because we could not see the board clearly .