Mathematical methods of quantum information theory

September 10, 2018

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.


What is quantum field theory? A quantum information theorist’s perspective

October 19, 2015

Daniel Burgarth and Matt Leifer have really been an inspiration for me as they have successfully embraced and exploited new technologies to improve scientific communication and discourse. One of their innovations, with the help of Ravi Kunjwal, has been the Q+ hangouts talk series. This excellent series of talks is delivered online via google hangouts, and has now attracted a wonderful line up of topics and speakers. It has also inspired the TCS+ hangouts series.

Last week I had the honour and pleasure to participate in the Q+ hangout series. I spoke about a topic very close to my heart, namely, quantum field theory, and would like to share the video with you here today

In this talk I described a way to formalise Wilson’s approach to quantum field theory as an effective theory in quantum information theorist friendly terms. If you are interested in seeing the details behind what I discussed in the talk then please do check out the (as yet still incomplete) paper draft of the content at my github repository. This paper contains very many definitions and worked out examples. The purpose of my talk and the paper is to provide a definition for what a quantum field state is in such a way that it: (a) matches the modern Wilsonian definition used by physicists; (b) allows us to answer quantum information theory style questions about QFT; and (c) is mathematically rigourous (or, at least, rigourisable…)

If you feel like contributing to this project then please don’t hesitate to fork the github repository!