Developments in holographic complexity and quantum information

Holography has unveiled an interrelation between quantum information and gravity. In this framework, complexity is supposed to capture the interior of black holes, overcoming the limitations of entanglement entropy. My PhD research has focused on the interplay between the two quantities, covering complexity aspects in the quantum information and holographic realms.

Complexity quantifies the hardness of implementing an operator or preparing a quantum state through elementary operations. Huge arbitrariness stems from the identification of operations with high computational cost. For an n-qubits system, we have detected a choice compatible with exponential lower bounds and chaotic behavior of operator complexity, as required to mimic black hole interiors. Then, we have analyzed the relation between operator and state complexity using the formalism of Riemannian submersions.

Several candidates have been proposed for the dual of state complexity: the volume, the gravitational action, and the spacetime volume of proper bulk
regions. Specializing to subsystems, we have explored the conjectures in various static settings, finding that subsystem complexity and entanglement entropy contain different information. The same conclusion holds for a holographic global quench, during which subsystem volume complexity evolves non-monotonically in time, contrary to entanglement entropy.

Finally, we have studied an example of local quench in which entanglement entropy suffices to discern between diverse holographic realizations.