Bulk Entanglement Entropy and Matrices

Abstract: Motivated by the Bekenstein Hawking formula and the area law behaviour of entanglement entropy, we propose that in any UV finite theory of quantum gravity with a smooth spacetime, the total entropy for a pure state in a co-dimension one spatial region, to leading order, is given by $S={A\over 4 G_N}$, where $A$ is the area of the co-dimension two boundary. In the context of $Dp$ brane holography we show that for some specially chosen regions bulk entanglement can be mapped to ``target space" entanglement in the boundary theory. Our conjecture then leads to a precise proposal for target space entanglement in the boundary theory at strong coupling and large $N$. In particular it leads to the conclusion that the target space entanglement would scale like $O(N^2)$ which is quite plausible in a system with $O(N^2)$ degrees of freedom. Recent numerical advances in studying the D0 brane system hold out the hope that this proposal can be tested in a precise way in the future.