Entanglement Entropy on the Cayley Tree

TL;DR

This paper investigates how entanglement entropy behaves on a Cayley tree, revealing different growth patterns depending on the partitioning method and state type, with implications for complex systems.

Contribution

It provides a detailed analysis of entanglement entropy in Cayley trees, highlighting how partitioning and state excitation influence entropy growth patterns, a novel insight for graph-based quantum systems.

Findings

Ground-state EE grows logarithmically when pruning a branch.

Ground-state EE grows exponentially when trimming around the root.

Highly excited states' EE grows exponentially regardless of partitioning.

Abstract

The properties of the entanglement entropy (EE) of a clean Cayley tree (CT) are studied. The EE shows a completely different behaviour depending on the way the CT is partitioned into two regions and whether we consider the ground-state or highly excited many-particle wave function. The ground-state EE increases logarithmically as function of number of generation if a single branch is pruned off the tree, while it grows exponentially if the region around the root is trimmed. On the other hand, in both cases the highly excited states' EE grows exponentially. Implications of these results to general graphs and disordered systems are shortly discussed.