Educational materials

Here we collect some materials that are hopefully useful to students interested in the topic, and the quantum research community overall.


This monographs gives an introduction to measures of quantum information (divergences and entropies) and their properties and applications. It is aimed at graduate students interested in mathematical aspects of quantum information theory.

The published version is out of date, please consider the version on arXiv instead.

Erreta for current version on arXiv:

  • In Section 4.1.1., Axiom I on continuity: Continuity is actually only ensured when both \rho \ll \sigma and \sigma \ll \rho , that is, wherever the supports of the two states coincide. I thank Omar Fawzi for pointing out this mistake.
  • A proof of this continuity is currently missing in the book. Please check out the lecture notes on the Masterclass on Entropy Inequalities instead.

Various Lectures and Tutorials

Tutorial on Quantum Information Theory

I gave a tutorial at QIP 2022 at CalTech titled “From one-shot to asymptotic quantum information theory”. The slides, with some corrections, are available here.

Masterclass on Entropy Inequalities

I taught a short course (two lectures) on “An Axiomatic Approach to Quantum Divergences” at the 2022 QMath Masterclass in Copenhagen. The lecture notes will be published here soon.

Introduction to Quantum Learning Theory

I gave a few introductory lectures on quantum learning theory as part of the course Advanced Topics in Quantum Information at ETH Zurich. Lecture notes are here, provided with the help of Jan Seyfried.

Teaching at NUS

  • EE5139/6139: Information Theory and its Applications
  • EE4032: Blockchain Engineering
  • EE4205: Quantum Communication and Cryptography