Jakub Stavina: An introduction to quantum information geometry

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Abstract: In this talk, we present the framework of quantum information geometry. We first review key concepts in quantum statistical mechanics, focusing on the Hamiltonian, thermal states, and quantum relative entropy. We then introduce the exponential family as a prototypical quantum statistical manifold and apply Eguchi’s construction to derive its information-geometric structure and associated family of alpha-connections. We next examine the relations between Petz’s classification of monotone quantum metrics and Chentsov’s classical uniqueness theorem. Finally, we outline selected applications of the framework.