Functional inequalities for classical and quantum Markov chains [MA5947]

Summer Semester 2020

Dr. Ángela Capel, Dr. Cambyse Rouzé

Dozent:	Dr. Ángela Capel, Dr. Cambyse Rouzé
Übungsleitung:
Mitwirkende:
Vorlesung:		Anmeldung
Zentralübung:		Anmeldung
Tutorübungen:		Anmeldung

News

Due to the coronavirus crisis, this course might be moved to the Winter Semester.

Content (from Module description)

1. Classical Markov chains
1. Review of Markov semigroups and ergodic properties
2. Spectral gap
3. Logarithmic Sobolev inequality and hypercontractivity
4. Modified logarithmic Sobolev inequality and entropic convergence
5. Tensorization
2. Quantum Markov chains
1. Review on quantum Markov chains
2. Ergodicity properties
3. Quantum functional inequalities
4. Quantum Stroock Varopoulos inequality
5. The tensorization problem
3. Spin systems
1. Review on Gibbs states and locality
2. Clustering of correlations at equilibrium
3. Approximate tensorization
4. Rapid mixing
5. Connection via functional inequalities

Exam

The exam will be in oral form (30 minutes).

Exercises

There will be four exercise sessions.

Literature

There will be available lecture notes for the course.

Recommended bibliography:

• L. Saloff-Coste, "Lectures on finite Markov chains", Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1665. Springer, Berlin, Heidelberg, 1997, link .
• N. Berestycki, "Mixing Times of Markov Chains: Techniques and Examples", 2016, link .
• D. Bakry, "L'hypercontractivité et son utilisation en theorie des semi-groupes", Lectures on Probability Theory. Lecture Notes in Mathematics, vol 1581. Springer, Berlin, Heidelberg, 1994, link .
• M. M. Wolf, "Quantum Channels and Operations. Guided Tour", 2012, link.
• F. Martinelli, "Lectures on Glauber Dynamics for Discrete Spin Models", Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1717. Springer, Berlin, Heidelberg, 1999, link .

Complementary reading:

• E. A. Carlen and J. Maas, "Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance", Journal of Functional Analysis, 273, no. 5, (2017) 1810-1869, link .
• P. Dai Pra, A. M. Paganoni, and G. Posta. “Entropy inequalities for unbounded spin systems”, Annals of Probabality, 30 (2002), 1959-1976,link .
• P. Diaconis and L. Saloff-Coste, "Logarithmic Sobolev Inequalities for Finite Markov Chains", The Annals of Applied Probability, 6 (3), (1996), 695-750, link .
• M. J. Kastoryano and F. G. S. L. Brandao. “Quantum Gibbs Samplers: The Commuting Case”, Communications in Mathematical Physics, 344 (2016), 915–957, link .