Derive holographic complexity of subsystems

Derive a rigorous formulation of holographic complexity for boundary subsystems (for example, using the complexity=volume prescription that measures the size of the entanglement wedge) in AdS/CFT, so that the time-dependent formulas for the complexity of an interval in a 1+1-dimensional holographic CFT undergoing a global quench are established rather than conjectural.

Background

The paper compares rigorous results from random unitary circuits with heuristic holographic predictions for subsystem complexity in 1+1-dimensional systems after a global quench from the B-state. In the holographic setup, the complexity of a subsystem is proposed to correspond to a measure of the entanglement wedge (e.g., its spatial volume in the complexity=volume prescription), yielding a linear growth up to a sharp collapse at time T = ℓ/2 for intervals smaller than half the system.

However, the authors emphasize that these holographic formulas for subsystem complexity have not been derived from first principles in AdS/CFT. As a result, the predicted behavior—particularly the abrupt collapse—rests on conjectural grounds, motivating a need for a rigorous derivation to validate or correct the proposed time dependence.