[Paper Review] Operator scrambling and quantum chaos

Xiaohong Chen, Tianci Zhou|arXiv (Cornell University)|Apr 23, 2018

Quantum chaos and dynamical systems42 citations

TL;DR

The paper analyzes operator scrambling in three models (chaotic spin-1/2 chain, 2-local spin model, and a modified quantum linear map) and shows how operator entanglement entropy and spectral correlations signal scrambling and quantum chaos, with differing scrambling times and behavior of C(t).

ABSTRACT

Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this paper, we discuss operator scrambling in three representative models: a chaotic spin-$1/2$ chain with spatially local interactions, a 2-local spin model and the quantum linear map. In the first two examples, although the speeds of scrambling are quite different, a simple Pauli spin operator can eventually approach a "highly entangled" operator with operator entanglement entropy taking a volume law value (close to the Page value). Meanwhile, the spectrum of the operator reduced density matrix develops a universal spectral correlation which can be characterized by the Wishart random matrix ensemble. In the second example, we further connect the 2-local model into a one dimensional chain and briefly discuss the operator scrambling there. In contrast, in the quantum linear map, although the square of commutator can increase exponentially with time, a simple operator does not scramble but performs chaotic motion in the operator basis space determined by the classical linear map. We show that once we modify the quantum linear map such that operator can mix in the operator basis, the operator entanglement entropy can grow and eventually saturate to its Page value, thus making it a truly quantum chaotic model.

Motivation & Objective

• Motivate and define operator scrambling as a diagnostic of quantum chaos.
• Explore scrambling dynamics in three representative models: chaotic local-spin chain, 2-local Hamiltonian, and quantum linear map.
• Characterize scrambling via operator entanglement entropy and reduced-density-matrix spectra.
• Compare Lyapunov-like growth regimes and scrambling times across models.
• Demonstrate when and how universal spectral correlations emerge in operator bases.

Proposed method

• Define operator scrambling through the Heisenberg-evolved operator and its expansion in a Pauli basis.
• Use operator length/h height distributions and a reduced density matrix to quantify scrambling (operator EE and entanglement spectrum).
• Analyze squared commutator C(t) as a chaotic-dynamics diagnostic.
• Compute spectral form factors of operator-reduced density matrices to detect Wishart-random-matrix-like correlations.
• For the quantum linear map, compare chaotic motion in operator space with true scrambling, and show how nonlinear perturbations induce scrambling.

Experimental results

Research questions

• RQ1 How does operator scrambling manifest in different chaotic quantum systems (local vs. nonlocal interactions)?
• RQ2 What roles do operator entanglement entropy and spectral correlations play in diagnosing scrambling and chaos across models?
• RQ3 Can a quantum linear map exhibit true scrambling, and how do perturbations affect this behavior?
• RQ4 What are the characteristic scrambling times and their dependence on system size or map parameters for each model?
• RQ5 How do random-matrix theory signatures (Wishart ensembles) emerge in operator spectra as scrambling proceeds?

Key findings

• In the chaotic spin-1/2 chain, a local Pauli operator grows nonlocally with a scrambling time proportional to system size (no exponential C(t) regime).
• In the 2-local Hamiltonian, scrambling is fast with scrambling time scaling as log N, and C(t) grows exponentially in the early-time regime; operator EE approaches Page value and spectral correlations emerge via the Wishart ensemble.
• In the quantum linear map, C(t) can grow exponentially due to classical chaos, but scrambling in operator space does not occur unless a nonlinear perturbation is added, after which operator EE saturates to Page value and spectral correlations appear; scrambling time scales with a function f(κ) log K.
• The emergence of universal spectral correlations in the operator-reduced density matrix (via spectral form factor) accompanies full scrambling in the subsystems.
• A one-dimensional extension and Fisher/Kolmogorov-Petrovsky-Piskunov dynamics suggest traveling-wave behavior for operator growth in space, paralleling scrambling in extended systems.