  We study the one-dimensional transverse-field spin-1/2 Ising ferromagnet at
its critical point. We consider an $L$-sized subsystem of a $N$-sized ring, and
trace over the states of $(N-L)$ spins, with $N\to\infty$. The full $N$-system
is in a pure state, but the $L$-system is in a statistical mixture. As well
known, for $L >>1$, the Boltzmann-Gibbs-von Neumann entropy violates
thermodynamical extensivity, namely $S_{BG}(L) \propto \log L$, whereas the
nonadditive entropy $S_q$ is extensive for $q=q_c=\sqrt{37}-6 $, namely
$S_{q_c}(L) \propto L$. When this problem is expressed in terms of independent
fermions, we show that the usual thermostatistical sums emerging within
Fermi-Dirac statistics can, for $L>>1$, be indistinctively taken up to $L$
terms or up to $\log L$ terms. This is interpreted as a compact occupancy of
phase-space of the $L$-system, hence standard BG quantities with an effective
length $V \equiv \log L$ are appropriate and are explicitly calculated. In
other words, the calculations are to be done in a phase-space whose effective
dimension is $2^{\log L}$ instead of $2^L$. The whole scenario is strongly
reminiscent of a usual phase transition of a spin-1/2 $d$-dimensional system,
where the phase-space dimension is $2^{L^d}$ in the disordered phase, and
effectively $2^{L^d/2}$ in the ordered one.
