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Isospectral Twirling and Quantum Chaos.

Lorenzo Leone1, Salvatore F E Oliviero1, Alioscia Hamma1

  • 1Physics Department, University of Massachusetts Boston, Boston, MA 02125, USA.

Entropy (Basel, Switzerland)
|August 27, 2021
PubMed
Summary

We introduce isospectral twirling, a unified framework for quantum chaos measures like OTOCs. This method allows smooth interpolation between integrable and chaotic Hamiltonians, distinguishing their spectral properties.

Keywords:
entanglementinformation scramblingquantum chaostwirling

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Area of Science:

  • Quantum Physics
  • Quantum Information Theory

Background:

  • Quantum chaos is typically studied through spectral properties or eigenvectors of Hamiltonians.
  • Existing measures of quantum chaos include frame potentials, scrambling, Loschmidt echo, and out-of-time-order correlators (OTOCs).

Purpose of the Study:

  • To present a unified framework, isospectral twirling, for describing key quantum chaos measures.
  • To demonstrate how this framework allows interpolation between integrable and chaotic Hamiltonians.
  • To show how spectral properties can distinguish between integrable and chaotic systems.

Main Methods:

  • Utilizing the Haar average of a k-fold unitary channel as the core of the isospectral twirling framework.
  • Casting quantum chaos measures as expectation values of the isospectral twirling.
  • Employing random matrix theory to analyze spectral properties (GUE, Poisson, GDE).

Main Results:

  • Isospectral twirling unifies various quantum chaos measures, including OTOCs.
  • The framework enables smooth transitions between integrable and chaotic Hamiltonians.
  • Hamiltonians with eigenvector stabilizer states lack chaotic features, unlike those with Haar measure eigenvectors.
  • OTOCs with Clifford resources decay to higher values than with universal resources.
  • Doping Hamiltonians with non-Clifford resources shows a crossover in OTOC behavior.
  • Spectral measures clearly distinguish chaotic (GUE) from integrable (Poisson, GDE) systems.

Conclusions:

  • Isospectral twirling provides a powerful, unified approach to studying quantum chaos.
  • The framework offers new insights into the relationship between Hamiltonian properties and chaotic behavior.
  • This technique facilitates the analysis and differentiation of quantum systems based on their spectral characteristics.