  We present details on a physical realization, in a many-body Hamiltonian
system, of the abstract probabilistic structure recently exhibited by
Gell-Mann, Sato and one of us (C.T.), that the nonadditive entropy $S_q=k [1-
Tr \hat{\rho}^q]/[q-1]$ ($\hat{\rho}\equiv$ density matrix; $S_1=-k Tr
\hat{\rho} \ln \hat{\rho}$) can conform, for an anomalous value of q (i.e., q
not equal to 1), to the classical thermodynamical requirement for the entropy
to be extensive. Moreover, we find that the entropic index q provides a tool to
characterize both universal and nonuniversal aspects in quantum phase
transitions (e.g., for a L-sized block of the Ising ferromagnetic chain at its
T=0 critical transverse field, we obtain
$\lim_{L\to\infty}S_{\sqrt{37}-6}(L)/L=3.56 \pm 0.03$). The present results
suggest a new and powerful approach to measure entanglement in quantum
many-body systems. At the light of these results, and similar ones for a d=2
Bosonic system discussed by us elsewhere, we conjecture that, for blocks of
linear size L of a large class of Fermionic and Bosonic d-dimensional many-body
Hamiltonians with short-range interaction at T=0, we have that the additive
entropy $S_1(L) \propto [L^{d-1}-1]/(d-1)$ (i.e., $ \ln L$ for $d=1$, and $
L^{d-1}$ for d>1), hence it is not extensive, whereas, for anomalous values of
the index q, we have that the nonadditive entropy $S_q(L)\propto L^d$ ($\forall
d$), i.e., it is extensive. The present discussion neatly illustrates that
entropic additivity and entropic extensivity are quite different properties,
even if they essentially coincide in the presence of short-range correlations.
