Geometric Entropy, Area, and Strong Subadditivity

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with non zero entropy. This geometric entropy is believed to be deeply related to the entropy of black holes. Indeed, previous calculations in the context of quantum field theory, where the result is actually ultraviolet divergent, have shown that the geometric entropy is proportional to the area for a very special type of subsets. In this work we show that the area law follows in general from simple considerations based on quantum mechanics and relativity. An essential ingredient in our approach is the strong subadditive property of the quantum mechanical entropy.