Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics

The Boltzmann-Gibbs-von Neumann entropy of a large part (of linear size L) of some (much larger) d-dimensional quantum systems follows the so-called area law (as for black holes), i.e., it is proportional to $L^{d-1}$. Here we show, for d=1,2, that the (nonadditive) entropy S_q satisfies, for a special value of $q \neq 1$, the classical thermodynamical prescription for the entropy to be extensive, i.e., $S_q \propto L^d$. Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited, by M. Gell-Mann, Y. Sato and one of us (C.T.), in mathematical models with scale-invariant correlations. Finally, we find that the system critical features are marked by a maximum of the special entropic index q.