Entropy Evolution in Consensus Networks

Shuangshuang Fu,Guodong Shi,Ian R Petersen,Matthew R James

doi:10.1038/s41598-017-01615-5

Shuangshuang Fu, Guodong Shi + Show 2 more

Open Access

https://doi.org/10.1038/s41598-017-01615-5

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Abstract

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

Highlights

How agreement emerging among a group of agents interacting each other is an intriguing subject in various research disciplines[1,2,3,4,5]
On the other hand, such precise state observation among different components in a quantum network is proven to be impossible[11], and quantum consensus dynamics is realized via nodes interacting not directly, but with the help of local environments which by themselves are quantum subsystems
We investigate the evolution of the network entropy for consensus dynamics in classical and quantum networks

Summary

Introduction

How agreement emerging among a group of agents interacting each other is an intriguing subject in various research disciplines[1,2,3,4,5]. The two categories of dynamics over classical and quantum networks can be put together into a group-theoretic framework[10], and quantum consensus dynamics can be equivalently mapped into certain parallel classical dynamics over disjoint subsets of the entries of the network density matrix[12] This line of research on consensus dynamics is related to the work on quantum walks over complex networks[13, 14], where associated with the network Laplacian there is a Hermitian operator defining the evolution of the network state in a quantum space. The result shows that quantum gossiping algorithms with deterministic coefficients are physically consistent with classical gossiping algorithms with random coefficients

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