  We study the entropy of pure shift-invariant states on a quantum spin chain.
Unlike the classical case, the local restrictions to intervals of length $N$
are typically mixed and have therefore a non-zero entropy $S_N$ which is,
moreover, monotonically increasing in $N$. We are interested in the asymptotics
of the total entropy. We investigate in detail a class of states derived from
quasi-free states on a CAR algebra. These are characterised by a measurable
subset of the unit interval. As the entropy density is known to vanishes, $S_N$
is sublinear in $N$. For states corresponding to unions of finitely many
intervals, $S_N$ is shown to grow slower than $(\log N)^2$. Numerical
calculations suggest a $\log N$ behaviour. For the case with infinitely many
intervals, we present a class of states for which the entropy $S_N$ increases
as $N^\alpha$ where $\alpha$ can take any value in $(0,1)$.
