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 Lta and Dirac delta function δδ.
• Finally, we discuss quantum stochastic calculus for the CQRW.

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 Lta and Dirac delta function δδ. Finally, we discuss quantum stochastic calculus for the CQRW.