  We propose a unified scheme to identify phase transitions out of the
$\mathbb{Z}_2$ Abelian topological order, including the transition to a
non-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y.
Lee, R. Kaneko, T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] on
the star lattice Kitaev model as an example, we compute the overlap of
minimally entangled states through transfer matrices. We demonstrate that,
similar to the anyon condensation, continuous deformation of a
$\mathbb{Z}_2$-injective projected entangled-pair state (PEPS) also allows us
to study the transition between Abelian and non-Abelian topological orders. We
show that the charge and flux anyons defined in the Abelian phase transmute
into the $\sigma$ anyon in the non-Abelian topological order. Furthermore, we
show that both the LG and SG states have infinite correlation length in the
non-Abelian regime, consistent with the recent claim that a chiral PEPS has a
gapless parent Hamiltonian.
