Connecting velocity and entanglement in quantum walks

TL;DR

This paper explores how the transport properties of quantum walks relate to entanglement between internal and external degrees of freedom, providing formulas for long-term behavior based on initial states and coin parameters.

Contribution

It derives closed-form expressions linking velocity and entanglement in one-dimensional quantum walks for various initial states and coin configurations.

Findings

Asymptotic entanglement depends on the walker's limit velocity and initial qubit orientation.

Knowledge of velocity and initial state parameters predicts long-term entanglement.

Formulas apply to both local and Gaussian initial position distributions.

Abstract

We investigate the relation between transport properties and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following $\delta$-like (local) and Gaussian distributions. We find out that the knowledge of the limit velocity of the walker together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while this velocity with the quantum coin phases give it for highly delocalized states.