Operator scrambling and quantum chaos

TL;DR

This paper investigates operator scrambling in various quantum systems, demonstrating how simple operators evolve into highly entangled states with universal spectral properties, and contrasting chaotic and non-chaotic dynamics.

Contribution

It provides a comparative analysis of operator scrambling across three models, highlighting the conditions under which quantum chaos manifests and how operator entanglement develops.

Findings

Operators reach volume-law entanglement entropy in chaotic models.

Spectral correlations align with Wishart random matrix ensemble.

Modified quantum linear map exhibits true quantum chaos with entanglement saturation.

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.