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Towards the Web of Quantum Chaos Diagnostics

Arpan Bhattacharyya, Wissam Chemissany, S. Shajidul Haque, Bin Yan

TL;DR

This work investigates the interconnected roles of three quantum-chaos diagnostics—the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and circuit complexity—arguing that averaged OTOCs, sub-system LE, and a specific complexity measure reflect the same underlying chaotic dynamics. It extends the OTOC–LE connection to higher-point correlators and multi-fold echoes, including infinite-dimensional and local-subsystem generalizations via Haar and unitary-1-design averaging. A concrete inverted-harmonic-oscillator model is used to relate LE to complexity, highlighting scrambling and intermediate regimes with distinct scaling relations and suggesting universal ties ${ m C}^2 \nsim - obreak\log({\rm LE})$ in the scrambling window. The paper also connects these diagnostics to holographic shockwaves and wormhole growth, offering a geometric interpretation and pointing toward experimental CV implementations as potential tests. Overall, it builds a cohesive

Abstract

We study the connections between three quantities that can be used as diagnostics for quantum chaos, i.e., the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and complexity. We generalize the connection between OTOC and LE for infinite dimensions and extend it for higher-order OTOCs and multi-fold LEs. Novel applications of this intrinsic relation are proposed. We also propose a relationship between a specific circuit complexity and LE by using the inverted oscillator model. These relationships signal a deeper connection between these three probes of quantum chaos.

Towards the Web of Quantum Chaos Diagnostics

TL;DR

This work investigates the interconnected roles of three quantum-chaos diagnostics—the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and circuit complexity—arguing that averaged OTOCs, sub-system LE, and a specific complexity measure reflect the same underlying chaotic dynamics. It extends the OTOC–LE connection to higher-point correlators and multi-fold echoes, including infinite-dimensional and local-subsystem generalizations via Haar and unitary-1-design averaging. A concrete inverted-harmonic-oscillator model is used to relate LE to complexity, highlighting scrambling and intermediate regimes with distinct scaling relations and suggesting universal ties in the scrambling window. The paper also connects these diagnostics to holographic shockwaves and wormhole growth, offering a geometric interpretation and pointing toward experimental CV implementations as potential tests. Overall, it builds a cohesive

Abstract

We study the connections between three quantities that can be used as diagnostics for quantum chaos, i.e., the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and complexity. We generalize the connection between OTOC and LE for infinite dimensions and extend it for higher-order OTOCs and multi-fold LEs. Novel applications of this intrinsic relation are proposed. We also propose a relationship between a specific circuit complexity and LE by using the inverted oscillator model. These relationships signal a deeper connection between these three probes of quantum chaos.

Paper Structure

This paper contains 17 sections, 72 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Loschmidt echo and OTOC
  3. 2k-OTOC and 2(k-1)-fold echo
  4. Local OTOCs
  5. Infinite dimensional generalization
  6. Application I: butterfly velocity
  7. Application II: shockwave and Loschmidt echo
  8. Echo evolution, Precursors and black holes
  9. Single shock
  10. Multiple shocks
  11. The triangle links
  12. Perspectives from infinite dimensional continuous variable systems
  13. Loschmidt echo and complexity
  14. Introducing complexity
  15. LE-complexity connection
  16. ...and 2 more sections

Figures (14)

  • Figure 1: LE as an echo quantity measures how much a quantum state is recovered by an imperfect time reversal.
  • Figure 2: Local structure of the total system and the choice of subsystems in the 4-point OTOC.
  • Figure 3: Local structure of the total system and the choice of subsystems in the 2k-OTOC.
  • Figure 4: Local structure of the total system and the choice of subsystems in the local 4-point OTOC.
  • Figure 5: Comparison between the regular two-loop LE and the four-loop LE. The former shows immediate decay after perturbation, while the latter exhibits an initial plateau regime.
  • ...and 9 more figures