Quantum Information Theory

Winter 2022 at ICFP

General information

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.

Resources/related courses

Schedule

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

Projects

Send report to all three instructors by April 3rd 23:59. The presentations will be held on April 5th.

Suggestions for projects

Groups of 2 students, choice before Feb 25th. Please send an email to instructors to discuss and confirm your choice.