A Holographic Entanglement Entropy at Spi

Defining finite entanglement entropy for a subregion in quantum field theory requires the introduction of two logically independent scales: an IR scale that controls the size of the subregion, and a UV cut-off. In AdS/CFT, the IR scale is the AdS lengthscale, the UV cut-off is the bulk radial cut-off, and the subregion is specified by dimensionless angles. This is the data that determines Ryu-Takayanagi surfaces and their areas in AdS/CFT. We argue that in asymptotically flat space there exists the notion of a "spi-subregion" that one can associate to spatial infinity (spi). Even though geometrically quite different from an AdS subregion, this angle data has the crucial feature that it allows an interpretation as a bi-partitioning of spi. Therefore, the area of the RT surface associated to the spi-subregion can be interpreted as the entanglement entropy of the reduced density matrix of the bulk state under this bi-partition, as in AdS/CFT. For symmetric spi-subregions, these RT surfaces are the waists of Asymptotic Causal Diamonds. In empty flat space they reduce to Rindler horizons, and are analogues of the AdS-Rindler horizons of Casini, Huerta \& Myers. We connect these definitions to previous work on minimal surfaces anchored to screens in empty space, but also generalize the discussion to the case where there are black holes in the bulk. The phases of black hole RT surfaces as the spi-subregion is varied, naturally connect with those of black holes (small and large) in AdS. A key observation is that the radial cut-off is associated to an IR scale in flat space -- and in fact there are no UV divergences. We argue that this is consistent with previous suggestions that in sub-AdS scales the holographic duality is an IR/IR correspondence and that the degrees of freedom are {\em not} those of a local QFT, but those of long strings. Strings are of course, famously UV finite.