# Bulk Connectedness and Boundary Entanglement

## TL;DR

This paper proves that in holographic conformal field theories, boundary states must exhibit entanglement across any non-trivial bipartition, providing a necessary condition for states to have holographic duals.

## Contribution

It establishes a general no-go theorem showing that boundary states in holographic CFTs cannot be separable across non-trivial bipartitions, extending previous results to arbitrary partitions.

## Key findings

- Boundary states are entangled across any non-trivial bipartition.
- Separable boundary states cannot correspond to classical holographic geometries.
- Provides a necessary condition for holographic duality in boundary states.

## Abstract

We prove, for any state in a conformal field theory defined on a set of boundary manifolds with corresponding classical holographic bulk geometry, that for any bipartition of the boundary into two non-clopen sets, the density matrix cannot be a tensor product of the reduced density matrices on each region of the bipartition. In particular, there must be entanglement across the bipartition surface. We extend this no-go theorem to general, arbitrary partitions of the boundary manifolds into non-clopen parts, proving that the density matrix cannot be a tensor product. This result gives a necessary condition for states to potentially correspond to holographic duals.