Uncomplexity and Black Hole Geometry
Ying Zhao
TL;DR
This work extends the concept of uncomplexity to mixed states by defining Bob’s usable computational power without relying on a particular mixed-state complexity measure. It shows that UC(rho_B) is determined by operations on B that cannot be undone by Alice, and interprets these in holographic terms as the portion of interior spacetime lying inside Bob’s entanglement wedge, thereby integrating subregion duality and black hole interior growth. Through simple black hole examples, perturbation analyses, and an epidemic-model analogy, the paper links circuit-level gate dynamics to bulk geometric growth, clarifying how different kinds of operations populate distinct spacetime regions. It also ties apparent-horizon physics to the degree of active computation, suggesting a coherent framework for understanding how interior spacetime and computational power co-evolve in holographic systems. Overall, the results resolve puzzles about thermofield-double uncomplexity and illuminate how interior geometry encodes usable quantum information for subsystems.
Abstract
We give a definition of uncomplexity of a mixed state without invoking any particular definitions of mixed state complexity, and argue that it gives the amount of computational power Bob has when he only has access to part of a system. We find geometric meanings of our definition in various black hole examples, and make a connection with subregion duality. We show that Bob's uncomplexity is the portion of his accessible interior spacetime inside his entanglement wedge. This solves a puzzle we encountered about the uncomplexity of thermofield double state. In this process, we identify different kinds of operations Bob can do as being responsible for the growth of different parts of spacetime.