  Since a limit distribution of a discrete-time quantum walk on the line was
derived in 2002, a lot of limit theorems for quantum walks with a localized
initial state have been reported. On the other hand, in quantum probability
theory, there are four notions of independence (free, monotone, commuting, and
boolean independence) and quantum central limit theorems associated to each
independence have been investigated. The relation between quantum walks and
quantum probability theory is still unknown. As random walks are fundamental
models in the Kolmogorov probability theory, can the quantum walks play an
important role in quantum probability theory? To discuss this problem, we focus
on a discrete-time 2-state quantum walk with a non-localized initial state and
present a limit theorem. By using our limit theorem, we generate probability
laws in the quantum central limit theorems from the quantum walk.
