Spread Complexity Rate as Proper Momentum

TL;DR

This paper establishes a precise link between the rate of quantum state complexity growth in 2D conformal field theories and the radial momentum of particles in 3D Anti-de Sitter space, clarifying the operator growth dynamics.

Contribution

It provides a rigorous relation between quantum complexity rate and bulk radial momentum, refining previous qualitative models with exact definitions and measurements.

Findings

Complexity growth rate matches bulk radial momentum in AdS

Precise definition of spread complexity used for quantum states

Sharp correspondence established between boundary and bulk dynamics

Abstract

We demonstrate a precise relation between the rate of complexity of quantum states excited by local operators in two-dimensional conformal field theories and the radial momentum of particles in 3-dimensional Anti-de Sitter spacetimes. Similar relations have been anticipated based on qualitative models for operator growth. Here, we make this correspondence sharp with two key ingredients: the precise definition of quantum complexity given by the spread complexity of states, and the match of its growth rate to the bulk momentum measured in the proper radial distance coordinate.