The Evolution of Network Entropy in Classical and Quantum Consensus Dynamics

TL;DR

This paper explores how network entropy evolves in classical and quantum consensus dynamics, revealing contrasting entropy behaviors and comparing various gossiping algorithms.

Contribution

It introduces a comparative analysis of entropy evolution in classical and quantum networks and links quantum gossiping to classical algorithms with random coefficients.

Findings

Classical network entropy decreases at consensus for Bernoulli initial values.

Gaussian initial values lead to non-increasing differential entropy.

Quantum von Neumann entropy is non-decreasing during consensus.

Abstract

In this paper, we investigate the evolution of the network entropy for consensus dynamics in classical or quantum networks. We show that in the classical case, the network entropy decreases at the consensus limit if the node initial values are i.i.d. Bernoulli random variables, and the network differential entropy is monotonically non-increasing if the node initial values are i.i.d. Gaussian. While for quantum consensus dynamics, the network's von Neumann entropy is in contrast non-decreasing. In light of this inconsistency, we compare several gossiping algorithms with random or deterministic coefficients for classical or quantum networks, and show that quantum gossiping algorithms with deterministic coefficients are physically related to classical gossiping algorithms with random coefficients.