Markov semigroups with quantum symmetry
Organizers
Speaker
Zishuo Zhao
Time
Wednesday, June 11, 2025 10:30 AM - 12:00 PM
Venue
A3-3-301
Online
Zoom 242 742 6089
(BIMSA)
Abstract
We study bimodule quantum Markov semigroups, which describe the dynamics of quantum systems that preserve a large class of symmetries.
Mathematically, the symmetry is encoded by a finite index inclusion of von Neumann algebras N⊆M, and can be extracted from the standard invariant of the inclusion.
Using the standard invariant, we show how the generator of a bimodule quantum Markov semigroup can be expressed in terms of the Fourier multiplier.
This allows us to extend the notion of equilibrium states to bimodule equilibirum.
A bimodule quantum channel can have no stationary states but still admit a bimodule equilibrium.
Moreover, a bimodule quantum channel admits a bimodule equilibrium, we prove that the fixed points of the channel form a von Neumann subalgebra of M.
Finally, we discuss how well-known functional inequalities, such as the Poincaré inequality, the logarithmic Sobolev inequality and the Talagrand inequality, can be generalized to the bimodule equilibrium setting.
Reference: https://arxiv.org/abs/2504.09576
(by Jinsong Wu and Zishuo Zhao)
Mathematically, the symmetry is encoded by a finite index inclusion of von Neumann algebras N⊆M, and can be extracted from the standard invariant of the inclusion.
Using the standard invariant, we show how the generator of a bimodule quantum Markov semigroup can be expressed in terms of the Fourier multiplier.
This allows us to extend the notion of equilibrium states to bimodule equilibirum.
A bimodule quantum channel can have no stationary states but still admit a bimodule equilibrium.
Moreover, a bimodule quantum channel admits a bimodule equilibrium, we prove that the fixed points of the channel form a von Neumann subalgebra of M.
Finally, we discuss how well-known functional inequalities, such as the Poincaré inequality, the logarithmic Sobolev inequality and the Talagrand inequality, can be generalized to the bimodule equilibrium setting.
Reference: https://arxiv.org/abs/2504.09576
(by Jinsong Wu and Zishuo Zhao)