No Laplacian Perfect State Transfer in Trees
Gabriel Coutinho, Henry Liu · 2014
We consider a system of qubits coupled via nearest-neighbour in-teraction governed by the Heisenberg Hamiltonian. We further sup-pose 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. 1