Entanglement entropy in two-particle Grover walks on graphs

We define a two-particle quantum walk of identical particles on a graph $G$ via the one-particle Grover walk on the Kronecker product $G \otimes G$, and call it the two-particle Grover walk. In systems of identical particles, quantum mechanics requires that quantum states have a certain invariance with respect to the exchange of particles. Focusing on the symmetry of the Kronecker product $G \otimes G$ as a graph, we show that the time evolution operator of this walk commutes with the swap operator, which ensures that this requirement is satisfied. Furthermore, we study the entanglement entropy of quantum states evolved by this walk. For the complete bipartite graph $K_{n,n}$, we completely determine the values of $n$ for which the quantum states evolved from specific initial states attain the upper bound of the entropy at some time, and prove that they are exactly $1$ and $2$.