No Laplacian Perfect State Transfer in Trees

TL;DR

This paper proves that perfect quantum state transfer does not occur in trees with more than two vertices under the Heisenberg Hamiltonian, and conjectures similar results for the XY-Hamiltonian case.

Contribution

It establishes a no-go theorem for perfect state transfer in trees with the Heisenberg model and explores related graph classes, proposing conjectures for the XY-Hamiltonian.

Findings

No perfect state transfer in trees with >2 vertices under Heisenberg Hamiltonian.

Exploration of bipartite and odd spanning tree graphs.

Conjecture that no perfect transfer in trees with >3 vertices under XY-Hamiltonian.

Abstract

We consider a system of qubits coupled via nearest-neighbour interaction governed by the Heisenberg Hamiltonian. We further suppose that all coupling constants are equal to $1$. We are interested in determining which graphs allow for a transfer of quantum state with fidelity equal to $1$. To answer this question, it is enough to consider the action of the Laplacian matrix of the graph in a vector space of suitable dimension. Our main result is that if the underlying graph is a tree with more than two vertices, then perfect state transfer does not happen. We also explore related questions, such as what happens in bipartite graphs and graphs with an odd number of spanning trees. Finally, we consider the model based on the $XY$-Hamiltonian, whose action is equivalent to the action of the adjacency matrix of the graph. In this case, we conjecture that perfect state transfer does not happen in trees with more than three vertices.