Quantum Shannon Theory and Beyond
In this course, we will study the transmission of information over a noisy quantum communication channel. In particular, students will learn about quantum mechanics, entanglement, teleportation, tomography, quantum estimation, hypothesis testing, and various capacity theorems involving classical bits, qubits, and entangled bits. There will be a strong focus on entropy measures and their application to numerous quantum tasks.
Outline.
The contents of the course are initially fixed as follows, although they are subject to modification depending on the interest of the students:
1. Introduction to fundamental concepts and the basic formalism: Pure/mixed states, evolution, completely positive maps, measurements Schmidt decomposition.
2. Quantum channels, Kraus representation, Choi matrices, POVM formalism. Interpretation of channels and their application to coherent communication and purification.
3. Trace distance, fidelity and entropy measures. Quantum relative entropy and quantum entropy inequalities. Hypothesis testing.
4. Monotonicity, recoverability and quantum data compression. Classical and quantum data communication.
5. Entanglement in dense coding, quantum teleportation and quantum cryptography. Use of Bell inequalities to characterize quantum weirdness of entanglement and non-locality.
Prerequisites.
The students should have taken the basic modules on Analysis and Linear Algebra. Probability theory is also desirable. No previous knowledge on quantum mechanics is assumed.
Literature.
- Lecture notes. Updated every week. Link
- Nielsen and Chuang, Quantum Computation and Quantum Information. Link
- Wilde, From Classical to Quantum Shannon Theory. Link
- Watrous, The theory of quantum information. Link
- Carlen, Trace inequalities and quantum entropy. Link
- Wolf, Quantum Channels and Operations Guided Tour. Link
Evaluation.
There will be weekly assignments of exercises to be solved every Friday during the exercises sessions. Active participation in those sessions will be essential for the final grade. Additionally, there will be an oral exam at the end of the course, consisting on the presentation of a short project on a specific topic of interest for each student.
Exercises
29.04.22 - Brief review in Classical Information Theory and basic notions in Quantum Information Theory
06.05.22 - Measurements, quantum circuits and quantum teleportation.
13.05.22 - Quantum nonlocality and nonlocal games.
20.05.22 - Pure states and quantum channels.
27.05.22 - Quantum channels.
03.06.22 - Quantum hypothesis testing.
24.06.22 - Quantum entropies.
01.07.22 - Quantum entropies. Part II.
Graded Exercises
DUE: 03.06.22 - SDPs for quantum hypothesis testing.
DUE: 15.07.22 - Data processing inequality and Petz recovery map.