We discuss two notions of entanglement of internal degrees of freedom. The first notion is target space entanglement: we show how to define entanglement of matrix degrees of freedom in a gauge invariant manner, and discuss connections to holography. The second notion relates to global symmetries in AdS/CFT settings and relates to an interpretation of Ryu-Takayanagi surfaces which are anchored on the boundary of a subregion on the internal space and smeared in the AdS spatial directions.

Most realizations of holography in String Theory involve space-times of the form of AdS X Y, where Y is an internal space which geometrizes R symmetries. The initimate connection of bulk reconstruction in AdS with the entanglement of subregions of the base space of the dual field theory suggests the possibility that a reconstruction of the internal space could be related to entanglement among internal degrees of freedom. We initiate an investigation into this issue by studying properties of RT surfaces which are smeared along the AdS directions, and anchored on the boundary of a subregion of the internal space Y. In cases where the product space appears as an IR geometry of a higher dimensional asymptotically AdS space we identify the area of the RT surface with the entanglement entropy associated with an operator subalgebra, and speculate on stand alone instances of product spaces.