  We study the quantum mechanical evolution of the tensor perturbations during
inflation with non-linear tensor interactions. We first obtain the Lindblad
terms generated by non-linear interactions by tracing out unobservable
sub-horizon modes. Then we calculate explicitly the reduced density matrix for
the super-horizon modes, and show that the probability of maintaining the
unitarity of the squeezed state decreases in time. The decreased probability is
transferred to other elements of the reduced density matrix including
off-diagonal ones, so the evolution of the reduced density matrix describes the
quantum-to-classical transition of the tensor perturbations. This is different
from the classicality accomplished by the squeezed state, the suppression of
the non-commutative effect, which is originated from the quadratic, linear
interaction, and also maintains the unitarity. The quantum-to-classical
transition occurs within 5 - 10 e-folds, faster than the curvature
perturbation.
