A simple coin for a 2d entangled walk

We analyze the effect of a simple coin operator, built out Bell pairs, in $2d$ Discrete Quantum Random Walk (DQRW) problem. The specific form enables us to find analytical and closed solutions recursion relations DQRW. induces entanglement between spin position degrees freedom, which oscillates with time reaches constant value asymptotically. probe entangling properties operator further, by two different measures. First, integrating over space initial tensor product states, we determine {\it Entangling Power} operator. Secondly, compute Generalized Relative R\'{e}nyi Entropy} corresponding density matrices for entangled state pure unentangled state. Both behaves similar time. Finally, continuum limit, reduces DQRW into $1d$ massive fermions coupled synthetic gauge fields, where both mass term fields are parameters.