Gradient flow and quantum Markov semigroups with detailed balance

Li, V.C.Y.

Gradient flow and quantum Markov semigroups with detailed balance

Master thesis (2021)

Authors

V.C.Y. Li Electrical Engineering, Mathematics and Computer Science

Contributors

M. Caspers Analysis - (mentor)

Frank Redig Applied Probability - (graduation committee member)

Faculty

Electrical Engineering, Mathematics and Computer Science, Electrical Engineering, Mathematics and Computer Science

Quantum Markov semigroup Detailed balance Gradient flow

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Published Date

02-07-2021

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

In this thesis we present a study of quantum Markov semigroups. In particular, we mainly consider quantum Markov semigroups with detailed balance that are defined on finite-dimensional C*-algebras. They have an invariant density matrix ρ. Carlen and Maas showed that the evolution on the set of invertible density matrices that is given by such a semigroup is gradient flow for the relative entropy with respect to ρ for some Riemannian metric. This result is a non-commutative analog of certain diffusion equations that are gradient flow in the second order Wasserstein space. We provide a self-contained and accessible account to these issues. Moreover, we give a complete introduction to Tomita-Takesaki theory which has a close relation with quantum Markov semigroups satisfying detailed balance. Finally, we present some examples of these semigroups that arise from quantum theory.

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