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Information Scrambling and Loschmidt Echo

Bin Yan, Lukasz Cincio, Wojciech H. Zurek

arXiv:1903.02651v4quant-ph

TL;DR

The precise quantitative equivalence between the OTOC and Loschmidt echo had remained unresolved despite efforts to connect them. This paper derives the OTOC as a thermal average of Loschmidt echo signals and tests the connection in two models. The relation is reported in scrambling and intermediate-decay regimes, with implications for the bound on OTOC decay rate.

  • Problem

    A precise quantitative equivalence between the OTOC and Loschmidt echo was missing despite efforts to connect them, although establishing one could benefit both areas of quantum chaos.

    Source: p. 1

  • Method

    The paper averages over local operators and relates the OTOC to a thermal average of Loschmidt echoes, with coupling to the rest of the system acting as a perturbation.

    Source: pp. 1–2

  • Results

    The OTOC–Loschmidt echo relation is demonstrated in two models in the scrambling and intermediate exponential-decay regimes, and the numerical OTOC and echo match well in the random-matrix model.

    Source: pp. 1, 4

  • Takeaways & Limitations

    The connection links OTOC and Loschmidt echo studies and has implications for the bound on the OTOC decay rate.

    Source: p. 4

    The analysis assumes operators on distinct local subsystems and focuses on β = 0 for simplicity, while stating that finite temperature can be handled by generalization.

    Source: p. 1

Abstract

from arXiv · show

We demonstrate analytically and verify numerically that the out-of-time order correlator is given by the thermal average of Loschmidt echo signals. This provides a direct link between the out-of-time-order correlator -- a recently suggested measure of information scrambling in quantum chaotic systems -- and the Loschmidt echo, a well-appreciated familiar diagnostic that captures the dynamical aspect of chaotic behavior in the time domain, and is accessible to experimental studies.

Appendix A: Haar integral of subsystem operators

Appendix A derives Haar averaging identities for operators on a subsystem and on a composed system, using invariance of the Haar measure and a normalization convention.

Appendix A: Haar integral of subsystem operators

  • The appendix proves the single-system Haar integral formula and identifies dA with integration over the unitary group’s Haar measure.The measure is introduced as dA ≡ dµ(A).Source: Appendix A · p. 7
  • Haar invariance under A → AU makes the averaged operator proportional to the identity, with its coefficient fixed by the trace in finite dimensions.The measure is normalized by convention; the identity is also stated to hold in infinite dimensions, whose formal treatment is deferred to [54].Source: Appendix A · p. 7
  • For an operator on a composed system, averaging over subsystem operators reduces to taking the partial trace over that subsystem.The resulting operator is the identity on the subsystem tensored with the partial trace of the original operator.Source: Appendix A · p. 7

Appendix B: Reduced dynamics for operator B

Appendix B derives reduced dynamics for operator B after tracing out subsystem A, relating weak-coupling evolution to a stochastic-field description under a stated hierarchy of energy scales.

Appendix B: Reduced dynamics for operator B

  • The derivation assumes a small subsystem A, a much larger complementary subsystem B, and coupling strength δ ≪ 1 relative to HB.The stated hierarchy is ĤA ≪ ĤI ≪ ĤB.Source: Appendix B · pp. 7–8
  • For δ ≪ 1, the second-order reduced evolution of B takes Lindblad form and can be simulated under HB without coupling to other systems, using a stochastic field.The appendix interprets this as decoherence: B becomes mixed through coupling to subsystem A.Source: Appendix B · p. 8
  • The approximation treats A’s dynamics as much slower than B’s dynamics, allowing each stochastic noise field to be a random constant ±1 with equal probability.The operators VᵢA are chosen Hermitian and orthonormal; normalization conventions determine factors in the stochastic field.Source: Appendix B · pp. 7–8
  • Each noise realization is a random ±1 linear combination of the coupling operators, and the noisy evolution is averaged over N realizations.The reduced dynamics is obtained after tracing out subsystem A.Source: Appendix B · p. 8

Appendix C: OTOC - LE connection

Appendix C derives the OTOC–Loschmidt echo connection from the reduced dynamics and extends the relation to finite temperature through a complex-time representation.

Appendix C: OTOC - LE connection

  • The appendix derives Eq. (9) by substituting an alternative expression for B’s reduced dynamics and then averaging over unitary operators on subsystem B.The result is identified as Eq. (9a) in the main text.Source: Appendix C · pp. 8–9
  • At finite temperature, distributing operators around the thermal circle maps the averaged OTOC to an infinite-temperature echo average in complex time and then to a finite-temperature regular echo in real time.The effective inverse temperature for the Loschmidt echo is β/2.Source: Appendix C · p. 9
  • The chosen regularization gives the OTOC a non-unit value at t = 0, so the numerical simulations remove its prefactor by convention.The prefactor is Z²β/2/Zβ.Source: Appendix C · p. 9
  • The Haar-averaged OTOC is bounded in the complex-time strip −β/4 ≤ τ ≤ β/4, with the bound applying to a quantity at effective positive temperature β/4 ± τ.The passage identifies this boundedness as crucial to the upper bound on OTOC growth rate.Source: Appendix C · p. 9

Appendix D: OTOC and LE for the inverted harmonic oscillator

Appendix D introduces the derivation of the inverted harmonic oscillator’s OTOC and Loschmidt echo.

Appendix D: OTOC and LE for the inverted harmonic oscillator

  • The appendix presents the derivation of the OTOC and Loschmidt echo for the inverted harmonic oscillator.The passage does not specify the derivation’s steps or results.Source: Appendix D · p. 9

1. Wavefunction of the IHO

A coordinate transformation reduces the coupled inverted harmonic oscillators to uncoupled oscillators. The chosen initial state and propagator construction make the time-dependent wavefunction tractable in the transformed coordinates.

1. Wavefunction of the IHO

  • A coordinate transformation reduces the coupled IHO Hamiltonian to two uncoupled harmonic oscillators.The transformed coordinates have effective masses and frequencies.Source: §1 · p. 10
  • The chosen product state is not physically essential, but its separability in the new coordinates simplifies the time-dependent wavefunction calculation.The state is a pure product of Gaussian wavefunctions, up to a normalization factor.Source: §1 · p. 10
  • The two-IHO wavefunction is built from the single-IHO propagator in transformed coordinates and then transferred back to the original coordinates.The single-IHO time-dependent wavefunction can be expressed by a path integral.Source: §1 · p. 10

2. Haar averaged OTOC for the IHO

For the IHO calculation, the authors replace the regularized thermal state with a pure state and Haar-average unitary operators on the two oscillators. In this setting, the resulting OTOC exactly equals the purity of oscillator 2.

2. Haar averaged OTOC for the IHO

  • The Haar-averaged OTOC calculation uses unitary operators on the two oscillators and replaces the regularized thermal state with a pure state.A thermal state of the IHO is not well defined in this treatment.Source: §2 · pp. 10–11
  • After evaluating the Haar integral and total trace, the OTOC is expressed in terms of the evolved two-oscillator wavefunction.The expression contains products of wavefunction amplitudes at time t.Source: §2 · p. 11
  • In this setting, the OTOC exactly equals the purity of oscillator 2.The reduced density matrix of oscillator 2 determines the purity.Source: §2 · p. 11

3. Baker-Campbell-Hausdorff expansion of the LE

The appendix expands a Loschmidt echo to second order in the perturbation strength using the Baker-Campbell-Hausdorff formula. The derivation uses the Heisenberg evolution of the perturbation operator to determine the expansion terms.

3. Baker-Campbell-Hausdorff expansion of the LE

  • The Loschmidt echo is expanded to second order in the perturbation strength δ.The derivation drops a term that contributes only above second order.Source: §3 · p. 12
  • The echo operator is expanded with the Baker-Campbell-Hausdorff formula by treating H0 ≡ H − δx as the unperturbed Hamiltonian.The expansion terms are determined by the Heisenberg evolution of the perturbation operator.Source: §3 · p. 12
  • A coordinate shift yields the evolved position operator, which is used to evaluate the expansion terms Σ(t) and Γ(t).The appendix then expands the average echo and gives its final expression; for its chosen zero-mean Gaussian state, only the first term in Σ(t) contributes to M(t).Source: §3 · pp. 12–13

Appendix E: Finite size study of the SYK model.

The finite-size SYK study finds Gaussian OTOC decay with weak temperature dependence, while the decay rate is close to the Hamiltonian bandwidth. Exponential decay appears when couplings are manually reduced, with a rate that depends quadratically on the reduction factor.

Appendix E: Finite size study of the SYK model

  • The fermionic SYK OTOC shows clear Gaussian decay with very weak temperature dependence across the displayed N = 15, 16, and 17 systems.The figure inset compares temperatures for 16 fermions.Source: Appendix E · p. 13, Figure E.1 · p. 13
  • The extracted decay rate is close to the SYK Hamiltonian bandwidth, indicating that the finite-system decay is in the Gaussian regime.The bandwidth is the width of the spectral density.Source: Appendix E · p. 14
  • The exponential OTOC decay rate depends quadratically on the coupling factor g, consistent with the Fermi’s golden rule prediction for exponential LE decay.Figure E.2 plots the decay rate against g.Source: Figure E.2 · p. 14, Appendix E · p. 14
  • Exponential decay was not observed up to 17 fermions, but manually reducing couplings involving the target subsystems produced exponential decay.The couplings involving those subsystems were scaled by 0 < g < 1 while the rest were unchanged.Source: Appendix E · p. 14
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