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Switchbacks and the Bridge to Nowhere

Leonard Susskind, Ying Zhao

TL;DR

This work extends the ERB concept to one-sided black holes by introducing a bridge-to-nowhere whose interior growth mirrors quantum complexity. It adopts Nielsen’s geometric approach to quantum complexity, defining operator and state complexities as geodesic lengths under a penalty metric that emphasizes low-body-term interactions. Focusing on precursors and the switchback effect, the authors develop a two-dimensional precursor geometry with negative curvature, deriving explicit expressions for precursor complexity and size and clarifying the switchback phenomenon. The study links interior gravitational dynamics to quantum information geometry, offering a framework to explore how GR and complexity co-evolve and outlining avenues to incorporate Hawking radiation and tensor-network descriptions.

Abstract

This paper is in three parts: Part 1 explains the relevance of Einstein-Rosen bridges for one-sided black holes. Like their two-sided counterparts, one-sided black holes are connected to ERBs whose growth tracks the increasing complexity of the quantum state. Quantitative solutions for one-sided ERBs are presented in the appendix. Part 2 calls attention to the work of Nielsen and collaborators on the geometry of quantum complexity. This geometric formulation of complexity provides a valuable tool for studying the evolution of complexity for systems such as black holes. Part 3 applies the Nielsen approach to geometrize two related black hole quantum phenomena: the rapid mixing of information through fast-scrambling; and the time dependence of the complexity of precursors, in particular the switchback effect.

Switchbacks and the Bridge to Nowhere

TL;DR

This work extends the ERB concept to one-sided black holes by introducing a bridge-to-nowhere whose interior growth mirrors quantum complexity. It adopts Nielsen’s geometric approach to quantum complexity, defining operator and state complexities as geodesic lengths under a penalty metric that emphasizes low-body-term interactions. Focusing on precursors and the switchback effect, the authors develop a two-dimensional precursor geometry with negative curvature, deriving explicit expressions for precursor complexity and size and clarifying the switchback phenomenon. The study links interior gravitational dynamics to quantum information geometry, offering a framework to explore how GR and complexity co-evolve and outlining avenues to incorporate Hawking radiation and tensor-network descriptions.

Abstract

This paper is in three parts: Part 1 explains the relevance of Einstein-Rosen bridges for one-sided black holes. Like their two-sided counterparts, one-sided black holes are connected to ERBs whose growth tracks the increasing complexity of the quantum state. Quantitative solutions for one-sided ERBs are presented in the appendix. Part 2 calls attention to the work of Nielsen and collaborators on the geometry of quantum complexity. This geometric formulation of complexity provides a valuable tool for studying the evolution of complexity for systems such as black holes. Part 3 applies the Nielsen approach to geometrize two related black hole quantum phenomena: the rapid mixing of information through fast-scrambling; and the time dependence of the complexity of precursors, in particular the switchback effect.

Paper Structure

This paper contains 17 sections, 90 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The Bridge to Nowhere
  3. The Geometry of Complexity
  4. The Geometry of Complexity
  5. Operator Complexity
  6. State Complexity
  7. The Geometry of Precursors and Switchbacks
  8. Size and Complexity of Precursors
  9. Quantum Circuit Analysis
  10. Switchback Effect
  11. Complexity-geometry of Precursors
  12. Complexity Calculation
  13. Size of Precursor
  14. Questions
  15. Geometry of ERB1
  16. ...and 2 more sections

Figures (14)

  • Figure 1: Penrose diagram for one-sided ADS black hole. The red line is an infalling shell.
  • Figure 2: The blue curves show maximal volume spatial surfaces anchored at the boundary. The green curve is the asymptotic late-time limit of the ERB.
  • Figure 3: Embedding diagrams of one and two-sided ERBs.
  • Figure 4: The shaded area represents the group manifold $SU(2^K).$
  • Figure 5: Graphs of the complexity and size of a precursor.
  • ...and 9 more figures