  A subdivided star $SK_{1,l}$ is obtained by identifying exactly one pendant
vertex from $l$ copies of the path $P_3.$ This study is on the existence of
quantum state transfer on double subdivided star $T_{l,m}$ which is a pair of
subdivided stars $SK_{1,l}$ and $SK_{1,m}$ joined by an edge to the respective
coalescence vertices. Using the Galois group of the characteristic polynomial
of $T_{l,m},$ we analyze the linear independence of its eigenvalues which
uncovers no perfect state transfer in double subdivided stars when considering
the adjacency matrix as the Hamiltonian of corresponding quantum system. Then
we establish a complete characterization on double subdivided stars exhibiting
pretty good state transfer.
