Entanglement entropy and algebraic holography

In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cut- o) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal eld theory on the conformal boundary of AdSd+2 is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1=4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. (More precisely, RT showed this for d = 1 and adduced evidence that it also holds in many examples in d > 1.) We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography the- orem, the entanglement entropy between two complementary bulk Rehren wedges is equal to one 1=4G times the (suitably cut o) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture { that, in a theory of quantum gravity, the entanglement entropy, S between the degrees of freedom of a given region with those of its complement takes the form S = A=4G (plus lower order terms) { but only if the phrase `degrees of freedom' is replaced by `matter degrees of freedom'. It also supports related previous arguments of the author { consistent with the author's `matter-gravity entanglement hypothesis' { that the AdS/CFT correspondence is actu- ally only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.