  By developing a cluster sampling of stochastic series expansion quantum Monte
Carlo method, we investigate a spin-$1/2$ model on a bilayer square lattice
with intra-layer ferromagnetic (FM) Ising coupling and inter-layer
antiferromagnetic Heisenberg interaction. The continuous quantum phase
transition which occurs at $g_c=3.045(2)$ between the FM Ising phase and the
dimerized phase is studied via large scale simulations. From the analyzes of
critical exponents we show that this phase transition belongs to the
(2+1)-dimensional Ising universality class. Besides, the quantum entanglement
is strong between the two layers, especially in dimerized phase. The effective
Hamiltonian of single layer seems like a transverse field Ising model. However,
we found the quantum entanglement Hamiltonian is a pure classical Ising model
without any quantum fluctuations. Furthermore, we give a more general
explanation about how a classical entanglement Hamiltonian emerges.
