Let G be a finite, simple, and undirected graph with the Laplacian matrix L. We study the continuous-time quantum walk on G, governed by the transition matrix UL(t) = eitL, where t ∈ R. In this work, we explore Laplacian state transfer on a double subdivided star Tm,m, constructed by connecting the coalescence vertices of two copies of a subdivided star SK1,m, with an additional edge. We present a complete characterisation for the existence of Laplacian pretty good state transfer and Laplacian pretty good pair state transfer in Tm,m. Furthermore, we demonstrate that an edge perturbation in Tm,m yields infinitely many bicyclic graphs that exhibit Laplacian perfect pair state transfer.