Notes on the Factorisation of the Hilbert Space for Two-Sided Black Holes in Higher Dimensions

In this paper, we investigate the Hilbert space factorisation problem of two-sided black holes in high dimensions. We demonstrate that the Hilbert space of two-sided black holes can be factorized into the tensor product of two one-sided bulk Hilbert spaces when the effect of non-perturbative replica wormholes is taken into account. We further interpret the one-sided bulk Hilbert space as the Hilbert space of a one-sided black hole. Therefore, since the Hilbert space of a two-sided black hole can be obtained from the tensor product of two single-sided black hole Hilbert spaces, we consider this as an embodiment of the ER=EPR conjecture, and we show when the entanglement between the two single-sided black holes is sufficiently strong, the (Lorentzian) geometry of a two-sided black hole will emerge.