Summary
Course description: This course gives an introduction to the theory of quantum Markov chains in the finitedimensional setting of quantum information theory. We first discuss the quantum relative entropy and its fundamental property, the dataprocessing inequality, and give a proof of this inequality that naturally leads to equality conditions and the concept of recovery channels. Specializing this analysis to the partial trace, we obtain the strong subadditivity property of the von Neumann entropy, as well as a natural definition of quantum Markov chains. We then review a structure theorem for quantum Markov chains, the fundamental differences to classical Markov chains, and  time permitting  venture into the active research topic of approximate quantum Markov chains.
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 halfcourse, spanning weeks 916 of the term. The first part of this course is Quantum channels I  Representations and properties in weeks 18. While attendance of both courses is recommended, this course will be fairly independent from the first one.
Table of contents
 Quantum relative entropy:
Definition and operational interpretation, Joint convexity and dataprocessing inequality  Equality conditions for dataprocessing:
Petz's proof of the equality condition, Formulation in terms of recovery channels  Quantum Markov chains:
Strong subadditivity of von Neumann entropy, Quantum conditional mutual information and its operational interpretations, Structure theorem for exact quantum Markov chains 
Approximate quantum Markov chains:
Classical vs. quantum setting, Approximate recovery channels
Lectures
 Exercises: [sheet 1]

Tue, Mar 23: Recap (quantum states, quantum channels), quantum state discrimination, trace distance
[recording] [lecture notes] 
Thu, Mar 25: Quantum state discrimination, trace distance, data processing, hypothesis testing, Quantum Stein's Lemma, relative entropy
[recording] [lecture notes] 
Tue, Mar 30: Properties of relative entropy, functions on operators
[recording] [lecture notes] 
Thu, Apr 1: Relative entropy and joint convexity, functions on operators, relative modular operator
[recording] [lecture notes] 
Tue, Apr 6: Relative modular operator, proof of dataprocessing inequality
[recording] [lecture notes] 
Thu, Apr 8: Properties of von Neumann entropy and conditional entropy
[recording] [lecture notes]  Tue, Apr 13: No lecture (noninstructional day at University of Illinois)

Thu, Apr 15: Properties of conditional entropy, coherent information, and mutual information
[recording] [lecture notes] 
Tue, Apr 20: Holevo bound, conditional mutual information, classical and quantum Markov chains
[recording] [lecture notes] 
Thu, Apr 22: Quantum Markov chains, Petz recovery channel, equality in dataprocessing
[recording] [lecture notes] 
Tue, Apr 27: Quantum Markov chains, Petz recovery channel, equality in dataprocessing
[recording] [lecture notes] 
Thu, Apr 29: Quantum Markov chains, Petz recovery channel, equality in dataprocessing
[recording] [lecture notes] 
Tue, May 4: Equality in the Holevo bound
[recording] [lecture notes] 
Mon, May 10 (part 1): Structure of quantum Markov chains
[recording] [lecture notes] 
Mon, May 10 (part 2): Robustness of Markov chains, approximate recovery
[recording] [lecture notes]
Literature
 David Sutter: Approximate quantum Markov chains, Vol. 28. SpringerBriefs in Mathematical Physics, 2018.
 Denes Petz: Quantum Information Theory and Quantum Statistics. Theoretical and Mathematical Physics. Springer, 2008.
 Michael A. Nielsen and Denes Petz: A Simple Proof of the Strong Subadditivity Inequality, Quantum Information & Computation 5.6 (2005), pp. 507–513.
 Denes Petz: Monotonicity of quantum relative entropy revisited, Reviews in Mathematical Physics 15.01 (2003), pp. 79–91.
 Mary Beth Ruskai: Inequalities for quantum entropy: A review with conditions for equality, Journal of Mathematical Physics 43.9 (2002), pp. 4358–4375.
 Ben Ibinson et al.: Robustness of quantum Markov chains, Communications in Mathematical Physics 277.2 (2008), pp. 289–304.
Contact
Email: <mylastname>@illinois.edu
Postal address:
Illini Hall, Office 341B
725 S. Wright Street
Champaign, IL 61820
USA