Holographic AME states in black hole interiors

TL;DR

The paper identifies a special extremal slice inside an AdS black hole as dual to an absolutely maximally entangled (AME) state and proves $S_A^{(n)}$ is independent of $n$ for all bipartitions using holographic Renyi entropy calculations. It extends the result from BTZ to higher dimensional AdS black branes, showing the interior state is AME with $S_A^{(n)}=S_A$ and that the interior Hilbert space is effectively infinite with a bond dimension set by the black hole entropy $S_{BH}$. This provides gravity-side evidence for Haar-random structures and supports non-isometric holographic codes, including a fixed-area-like perspective and a diagnostic for interior holography. The work connects interior holography to quantum information concepts and discusses limitations to pure gravity and the role of matter fields and boundary conditions in future studies.

Abstract

We argue that the special extremal slice inside an AdS black hole is dual to an absolutely maximally entangled (AME) state. We demonstrate this by confirming the $n$-independence of holographic $n$-th Renyi entropies for any bi-partite subsystems. Our result gives an AME state in an infinite-volume system, where the local bond dimension is set by the black hole entropy. In particular, our construction provides concrete support from the gravity side for the emergence of random structures and an infinite-dimensional Hilbert space in recent non-isometric holographic codes.

Figures (3)

• Figure 1: A time slice (blue solid line in shaded region) called special extremal slice in eternal AdS black hole. We view it as a holographic screen on which dual theory is defined. Orange line corresponds to the time slice dual to TFD state in CFTs. A time evolution of TFD state at the late time gives the extremal surface (orange dotted line) that has partial contact with special extremal slice.
• Figure 2: Example of potential problem in $d=3$. There are two extrema (the horizon $\kappa=0$ and the special extremal slice $\kappa=\kappa_m$), and we start from $\kappa=\kappa_m$. Note that we now require that $\kappa = \kappa_m$ at both ends of the finite time interval. Therefore, only allowed solution that comes back to $\kappa=\kappa_m$ in finite time is $\kappa(t)=\kappa_m$.
• Figure 3: On the state on special extremal slice, even if we start from a particular $U_0$ for the non-isometric code, we have natural replacement $U_0\rightarrow U^\prime_{\text{Haar}}\equiv U_{0}U_{\text{Haar}}$.