- Omar Fawzi (Inria, ENS Lyon), Anthony Leverrier (Inria Paris), Lorenzo Piroli (ENS Paris)
- Tuesdays 14:00 to 17:30 at ENS Paris, 24 rue Lhomond, Room L367

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.

- [NC] The standard textbook Nielsen and Chuang: Quantum Computation and Quantum Information
- [Ren] Renato Renner's lecture notes
- [Pre] John Preskill's lecture notes
- [Wat] John Watrous' The theory of quantum information
- [Wil] Mark Wildes's Quantum information theory

The notes used for the lecture as well as the exercices can be found here.

Date | Topic | References |

Jan 17th | 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 24th |
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 |

Jan 31st |
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. More on distances: fidelity More on quantum channel: Choi operator, Stinespring dilation, Kraus operators [3 hours of lecture, i.e., no tutorial and ONLINE] |
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 7th | [Tutorial only] | [Ren, Chap 4] [Pre, Chap 10] |

Feb 14th | Quantum error correction [3 hours of lecture] | See lecture notes |

Feb 21st | Quantum error correction [3 hours of tutorial] | See lecture notes |

Mar 7th | Quantum error correction | See lecture notes |

Mar 14th | Quantum error correction | See lecture notes |

Mar 21st | Quantum error correction | See lecture notes |