  The nonclassicality of primordial gravitational waves (PGWs) is characterized
in terms of sub-Poissonian graviton statistics. The sub-Poissonian statistics
are realized when quantum states are squeezed coherent states. In the presence
of matter fields, the Universe experiences the squeezed coherent state during
inflation. The condition to realize the sub-Poissonian graviton statistics is
translated into the frequency range of gravitational waves. If the initial
state is the Bunch-Davies vacuum, there is another necessary condition between
phases of squeezing and coherent parameters. Here, we extend the initial state
to entangled states. We consider $\alpha$-vacua as the initial entangled state
that are more general de Sitter invariant vacua than the Bunch-Davies vacuum.
We find that, unlike the Bunch-Davies vacuum, PGWs generated in the initial
entangled state become sub-Poissonian without requiring the condition between
the phases.
