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Description
Rooted in quantum information theory, entanglement and state complexity have been extensively applied in contexts ranging from condensed matter physics to quantum gravity. Within the framework of the AdS/CFT correspondence, they shed light on phenomena such as black hole evaporation and the emergence of quantum-gravitational spacetime. Entanglement is also tightly related to von Neumann operator algebras, a tool crucial for a mathematically rigorous formulation of quantum field theories. Motivated by recent developments in holography, we study the connection between the different types of von Neumann algebras locally describing a system and the presence of quantum phase transitions. On the other hand, state complexity quantifies how difficult it is to achieve a quantum state via a sequence of gates or by implementing a specific quantum evolution. As Susskind suggested, when interpreted holographically, complexity is a promising probe of deep regions of the gravitational dual otherwise insensitive to entanglement measures. I will show one of the few instances where a definition of complexity, known as Krylov complexity, can be precisely interpreted in terms of gravitational dynamics in the holographic bulk dual.
