Abstract
Few facts are known about the entanglement entropy for disconnected regions
in quantum field theory. We study here the property of extensivity of the
mutual information, which holds for free massless fermions in two dimensions.
We uncover the structure of the entropy function in the extensive case, and
find an interesting connection with the renormalization group irreversibility.
The solution is a function on space-time regions which complies with all the
known requirements a relativistic entropy function has to satisfy. We show that
the holographic ansatz of Ryu and Takayanagi, the free scalar and Dirac fields
in dimensions greater than two, and the massive free fields in two dimensions
all fail to be exactly extensive, disproving recent conjectures.
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