  Using free-fermionic techniques we study the entanglement entropy of a block
of contiguous spins in a large finite quantum Ising chain in a transverse
field, with couplings of different types: homogeneous, periodically modulated
and random. We carry out a systematic study of finite-size effects at the
quantum critical point, and evaluate subleading corrections both for open and
for periodic boundary conditions. For a block corresponding to a half of a
finite chain, the position of the maximum of the entropy as a function of the
control parameter (e.g. the transverse field) can define the effective critical
point in the finite sample. On the basis of homogeneous chains, we demonstrate
that the scaling behavior of the entropy near the quantum phase transition is
in agreement with the universality hypothesis, and calculate the shift of the
effective critical point, which has different scaling behaviors for open and
for periodic boundary conditions.
