Gradient flow and quantum Markov semigroups with detailed balance
V.C.Y. Li (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Martijn Caspers – Mentor (TU Delft - Analysis)
F. REDIG – Graduation committee member (TU Delft - Applied Probability)
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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.