  A longstanding enigma within AdS/CFT concerns the entanglement entropy of
holographic quantum fields in Rindler space. The vacuum of a quantum field in
Minkowski spacetime can be viewed as an entangled thermofield double of two
Rindler wedges at a temperature $T=1/2\pi$. We can gradually disentangle the
state by lowering this temperature, and the entanglement entropy should vanish
in the limit $T\to 0$ to the Boulware vacuum. However, holography yields a
non-zero entanglement entropy at arbitrarily low $T$, since the bridge in the
bulk between the two wedges retains a finite width. We show how this is
resolved by bulk quantum effects of the same kind that affect the entropy of
near-extremal black holes. Specifically, a Weyl transformation maps the
holographic Boulware states to near-extremal hyperbolic black holes. A
reduction to an effective two-dimensional theory captures the large quantum
fluctuations in the geometry of the bridge, which bring down to zero the
density of entangled states in the Boulware vacuum. Using another Weyl
transformation, we construct unentangled Boulware states in de Sitter space.
