Laplacian pair state transfer in Q-graph

In 2018, Chen and Godsil proposed the concept of Laplacian perfect pair state transfer which is a brilliant generalization of Laplacian perfect state transfer. Studying Laplacian pair state transfer will provide a theoretical foundation for constructing quantum communication networks capable of quantum state transfer. In this paper, we study the existence of Laplacian perfect pair state transfer in the Q-graph of an r-regular graph for r≥2. By combining the spectral decomposition of the graph with the Laplacian eigenvalue support of pair state, we prove that the Q-graph of an r-regular graph does not have Laplacian perfect pair state transfer when r+1 is prime or a power of 2. By contrast, we also give sufficient conditions for Q-graph to have Laplacian pretty good pair state transfer. The approach used in this paper can effectively verify the existence of Laplacian perfect (or pretty good) pair state transfer in other families of graphs.