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Spread Complexity Rate as Proper Momentum

Pawel Caputa, Bowen Chen, Ross W. McDonald, Joan Simón, Benjamin Strittmatter

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.

Spread Complexity Rate as Proper Momentum

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.

Paper Structure

This paper contains 6 sections, 90 equations, 1 figure.

Table of Contents

  1. A. Spread complexity and $\mathrm{SL}(2,\mathbb{R})$
  2. B. Embeddings and proper distance.
  3. Black Hole
  4. Global coordinates
  5. Poincaré patch
  6. C. Coherent states in a harmonic oscillator

Figures (1)

  • Figure 1: The holographic dual of a state excited by a local operator \ref{['LOES']} corresponds to a massive particle following a timelike geodesic in $AdS_3$. Here we show the example of an excitation to the CFT vacuum on a line and the corresponding geodesic in \ref{['AdSPoincare']}.