Information Dimension of Dissipative Quantum Walks

TL;DR

This paper investigates how the von Neumann entropy grows over time in dissipative quantum walks, revealing a logarithmic growth linked to the information dimension, bridging quantum and classical walk behaviors.

Contribution

It introduces a method to connect entropy growth in dissipative quantum walks with the classical information dimension, extending understanding of quantum stochastic processes.

Findings

Logarithmic entropy growth observed across various networks.

Entropy growth rate correlates with the information dimension.

Results confirmed by numerical simulations.

Abstract

We study the temporal growth of the von Neumann entropy for dissipative quantum walks on networks. By using a phenomenological quantum master equation, the quantum stochastic walk (QSW), we are able to parametrically scan the crossover from purely coherent quantum walks to purely diffusive random walks. In the latter limit the entropy shows a logarithmic growth, which is proportional to the information dimension of the random walk on the network. Here we present results for the von Neumann entropy based on the reduced density operator of the QSW. It shows a similar logarithmic growth for a wide range of parameter values and networks. As a consequence, we propose the logarithmic growth rate of the von Neumann entropy to be a natural extension of the information dimension to dissipative quantum systems. We corroborate our results by comparing to numerically exact simulations.