Quantum walks with quantum chaotic coins: Of the Loschmidt echo, classical limit and thermalization

TL;DR

This paper investigates quantum walks with chaotic coins, revealing how classical limits emerge, the role of fidelity, and the entanglement dynamics, bridging quantum chaos, thermalization, and classical behavior.

Contribution

It demonstrates how chaotic coins induce classical-like distributions and thermalization in quantum walks, connecting quantum chaos with classical diffusion and entanglement growth.

Findings

Classical binomial distribution emerges from chaotic quantum walks.

Entanglement grows logarithmically and saturates at the Page value in coin-dominated regimes.

Fidelity acts as a characteristic function linking quantum walks to classical random walks.

Abstract

Coined discrete-time quantum walks are studied using simple deterministic dynamical systems as coins whose classical limit can range from being integrable to chaotic. It is shown that a Loschmidt echo like fidelity plays a central role and when the coin is chaotic this is approximately the characteristic function of a classical random walker. Thus the classical binomial distribution arises as a limit of the quantum walk and the walker exhibits diffusive growth before eventually becoming ballistic. The coin-walker entanglement growth is shown to be logarithmic in time as in the case of many-body localization and coupled kicked rotors, and saturates to a value that depends on the relative coin and walker space dimensions. In a coin dominated scenario, the chaos can thermalize the quantum walk to typical random states such that the entanglement saturates at the Haar averaged Page value, unlike in a walker dominated case when atypical states seem to be produced.