This course is a selection of some topics in quantum information theory. The guiding question we consider throughout the course is how to encode information in the presence of noise. In the first part of the course, we consider fundamental limits of what is possible within quantum theory. This area is typically called quantum Shannon theory. The second part will be dedicated to efficient constructions of encoding schemes. This area is called quantum error correcting codes. The mathematical tools you will learn are also used in many other areas of quantum information science.
Prerequisites: Familiarity with the basic formalism of quantum theory (in particular density operators) will be assumed, but a quick review will be provided, and motivated students with no prior knownledge of quantum theory should be able to catch up.
The notes used for the lecture as well as the exercices can be found here.
The homework can be found here.
| Date | Topic | References |
| Jan 18th | Mathematical formalism: density operators, quantum channels, trace distance and interpretation in terms of distinguishability. Start classical information over quantum channels. |
[Ren, Chap 4], Ligong Wang's Thesis, Chap 2 |
| Jan 25th |
Hypothesis testing, quantum relative entropy (and conditional entropy, mutual information), quantum Stein lemma (with partial proof) Proof of one-shot converse for classical quantum channels |
See quant-ph/0307170 for a full proof of quantum Stein lemma (with references) [Wil, Chap 11] for a detailed treatment of the quantum relative entropy and the derived entropies Ligong Wang's Thesis, Chap 2 |
| Feb 1st |
Proof of one-shot achievability for classical quantum channels Memoryless channels: Use Stein lemma. Proof of additivity. General quantum channels. State one-shot theorem and HSW theorem. |
Ligong Wang's Thesis, Chap 2 See [Wil, Chap 13 and Chap 20] for a detailed treatment of Holevo information and the classical capacity See Hasting's paper or the book Alice and Bob meet Banach for counterexamples to additivity |
| Feb 8th |
More on distances: fidelity More on quantum channel: Choi operator, Stinespring dilation, Kraus operators Started quantum information transmission |
[Ren, Chap 4] [Pre, Chap 10] |
| Feb 15th |
Equivalence of average fidelity and fidelity for max entangled state Coherent information of a quantum channel Sketch of proof for converse, and achievability via decoupling theorem |
[Pre, Chap 10] Check this paper for other measures for encoding/decoding Check Frederic Dupuis' thesis for more on decoupling |
| Feb 22nd | Quantum error correction | See lecture notes |
| Mar 8th | Quantum error correction | See lecture notes |
| Mar 15th | Quantum error correction | See lecture notes |
| Mar 22nd | Quantum error correction | See lecture notes |