Itô formula for one-dimensional continuous-time quantum random walk

In the paper, we first extend the discrete time quantum random walk (DQRW) to continuous time quantum random walk (CQRW). Then we establish an Itô formula for the one-dimensional CQRW. As an application, we present Tanaka’s formula with the Itô formula and characterize the relation between quantum local time L t a and Dirac delta function δ . Finally, we discuss quantum stochastic calculus for the CQRW.