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 Ld-1. Here we show, for d= 1 2, that the (nonadditive) entropy S-q satisfies, for a special value of q not equal 1, the classical thermodynamical prescription for the entropy to be extensive, i.e., S-q alpha L-d. Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited in mathematical models with scale-invariant. correlations [C. Tsallis, M. Gell-Mann, and Y. Sato, Proc. Natl. Acad. Sci. U.S.A. 102 15377 (2005)]. Finally, we find that the system critical features are marked by a maximum of the special entropic index q.