Entanglement entropy and algebraic holography

Document Type : Letter


Department of Mathematics University of YORK YORK, YO10 5DD UK



In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS{d+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 theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to one 1/4G times the (suitably cut off) 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 actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.