Quantum walks do not like bridges

TL;DR

This paper proves that quantum perfect state transfer is impossible between certain vertices in graphs with bridges, highlighting the importance of connectivity in quantum walks and supporting the conjecture about trees.

Contribution

It introduces a novel proof technique showing the absence of perfect state transfer in specific graph structures with bridges.

Findings

Quantum perfect state transfer is impossible in graphs with bridges unless the graph has only one vertex.

Connectivity significantly influences quantum walk behavior.

Supports the conjecture that no tree with four or more vertices admits state transfer.

Abstract

We consider graphs with two cut vertices joined by a path with one or two edges, and prove that there can be no quantum perfect state transfer between these vertices, unless the graph has no other vertex. We achieve this result by applying the 1-sum lemma for the characteristic polynomial of graphs, the neutrino identities that relate entries of eigenprojectors and eigenvalues, and variational principles for eigenvalues (Cauchy interlacing, Weyl inequalities and Wielandt minimax principle). We see our result as an intermediate step to broaden the understanding of how connectivity plays a key role in quantum walks, and as further evidence of the conjecture that no tree on four or more vertices admits state transfer. We conclude with some open problems.