Average scattering entropy of quantum graphs

TL;DR

This paper introduces a new measure called average scattering entropy for quantum graphs, quantifying the complexity of scattering amplitudes based on Shannon entropy principles.

Contribution

It proposes a novel methodology to compute scattering entropy in quantum graphs and explores its dependence on graph topology and boundary conditions.

Findings

Scattering entropy varies with graph topology and boundary conditions.

Graphs with similar structural properties can have different entropies.

Entropy decreases and saturates as the number of elementary structures increases.

Abstract

The scattering amplitude in simple quantum graphs is a well-known process which may be highly complex. In this work, motivated by the Shannon entropy, we propose a methodology that associates to a graph a scattering entropy, which we call the average scattering entropy. It is defined by taking into account the period of the scattering amplitude which we calculate using the Green's function procedure. We first describe the methodology on general grounds, and then exemplify our findings considering several distinct groups of graphs. We go on and investigate other possibilities, one that contains groups of graphs with the same number of vertices, with the same degree, and the same number of edges, with the same length, but with distinct topologies and with different entropies. And the other, which contains graphs of the fishbone type, where the scattering entropy depends on the boundary conditions on the vertices of degree $1$, with the corresponding values decreasing and saturating very rapidly, as we increase the number of elementary structures in the graphs.