Summary
Course description: This course gives an introduction to the theory of quantum channels in the finite-dimensional setting of quantum information theory. We discuss the various mathematically equivalent representations of quantum channels, focus on some important subclasses of channels, and make connections to the theory of majorization and covariant channels.
Prerequisites: MATH 415 or MATH 416
Throughout the course I will draw connections to quantum information theory, in particular the subfield of "quantum Shannon theory" that is concerned with the study of capacities of quantum channels amongst other things. However, no prior knowledge in this area is necessary to follow the course.
Grading policy: There will be no homework assignments or written exams for this course. Grading will be based on active class participation. However, I will provide exercise sheets, and it is strongly recommended to attempt to solve them.
Remark: This is a half-course, spanning the first 8 weeks of the term. I will teach a follow-up course Quantum channels II - data-processing, recovery channels, and quantum Markov chains in weeks 9-16. While attendance of both courses is recommended, the second course will be fairly independent from this course.
Table of contents
- Representations of quantum channels:
Choi isomorphism, Isometric picture, Unitary evolution, Kraus representation, Linear operator representation - Classes of quantum channels:
Qubit channels, entanglement-breaking channels, PPT channels, antidegradable and degradable channels - Covariant quantum channels:
Definition, properties, examples, Holevo information and minimum output entropy
Lectures
- Prerequisites: [lecture notes]
- Exercises: [sheet 1] (last update: 07/22/2021) [sheet 2] (last update: 07/22/2021) [sheet 3] (last update: 07/22/2021)
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Tue, Jan 26: Evolution of closed and open quantum systems, Choi-Jamiołkowski isomorphism
[recording] [lecture notes] -
Thu, Jan 28: Properties of linear maps and their Choi states, CP maps, Kraus representation
[recording] [lecture notes] (last update: 02/02/2021) - Tue, Feb 2: Kraus representation, isometric and unitary representation of quantum channels
[recording] [lecture notes] - Thu, Feb 4: Vectorization mapping, linear representation of quantum channels
[recording] [lecture notes] - Tue, Feb 9: Isometric representation and complementary channel, examples of qubit channels (bit- and phase flip channels, depolarizing channel, Pauli channels)
[recording] [lecture notes] - Thu, Feb 11: examples of qubit channels (amplitude damping channel, erasure channel), generalized dephasing channels
[recording] [lecture notes] - Tue, Feb 16: Holevo information, classical capacity, entanglement-breaking channels
[recording] [lecture notes] - Thu, Feb 18: Characterization of entanglement-breaking channels, separability, PPT criterion, PPT channels
[recording] [lecture notes] (last update: 02/22/2021) - Tue, Feb 23: Characterization of PPT channels, distillability of states, antidegradable channels
[recording] [lecture notes] -
Thu, Feb 25: Antidegradability of erasure and depolarizing channel
[recording] [lecture notes] -
Tue, Mar 02: Degradable channels, no-cloning theorem
[recording] [lecture notes] -
Thu, Mar 04: Covariant quantum channels, examples of covariant channels
[recording] [lecture notes] (last update: 03/09/2021) -
Tue, Mar 09: Characterization of depolarizing channel via unitary covariance, structure of Werner states
[recording] [lecture notes] -
Thu, Mar 11: Channel twirling, Haar measure, entanglement fidelity, irreducibly covariant channels
[recording] [lecture notes] -
Tue, Mar 16: Irreducibly covariant channels, Holevo information, minimum output entropy
[recording] [lecture notes] (last update: 03/18/2021) -
Thu, Mar 18: Minimum output entropy, classical capacity, additivity conjecture
[recording] [lecture notes]
Jacob Beckey has generously committed to typing up the lecture notes in LaTeX, which are available on GitHub.
Throughout the lecture I am trying to give some quantum information-theoretic context while discussing quantum channels and their properties. Unfortunately, it is beyond the scope of this lecture to discuss these information-theoretic aspects in more detail. Along with Mark M. Wilde's book on Quantum Information Theory (see the course literature below), the lecture Theory of Quantum Communication (Fall 2020) taught by Debbie Leung at the University of Waterloo is an excellent resource to learn more about this topic.
Literature
- Stephane Attal: Quantum channels, Lecture notes.
- John Watrous: Theory of Quantum Information, Cambridge University Press, 2018.
- Mark M. Wilde: Quantum Information Theory, Cambridge University Press, 2017.
- Michael M. Wolf: Quantum Channels & Operations: Guided Tour, Lecture notes, 2012.
Contact
Email: <mylastname>@illinois.edu
Postal address:
Illini Hall, Office 341B
725 S. Wright Street
Champaign, IL 61820
USA