  We show the boundedness of entanglement entropy for (bipartite) pure states
of quantum spin chains implies split property of subsystems. As a corollary the
infinite volume ground states for 1-dim spin chains with the spectral gap
between the ground state energy and the rest of spectrum have the split
property. We see gapless excitation exists for the spinless Fermion on $\bfZ$
if the ground state is non-trivial and translationally invariant and the U(1)
gauge symmetry is unbroken. Here we do not assume uniqueness of ground states
for all finite volume Hamiltonians.
