A 2D conformal field theory lives on the boundary of a 3D hyperbolic disk (Poincaré model). Select an arc on the boundary — the entanglement entropy of that arc equals the length of the minimal geodesic in the bulk connecting its endpoints. Bulk geometry IS boundary entanglement.
geometry from quantum information
The Ryu-Takayanagi formula (2006):
S(A) = Area(γ_A) / (4G_N)
where A is a region on the boundary of an AdS spacetime, γ_A is the minimal-area surface in the bulk that hangs from ∂A, and G_N is Newton's gravitational constant. This is the holographic principle made quantitative: the entanglement entropy of a region in a quantum field theory equals the area of a geometric object in one higher dimension.
Drag the boundary arc. Notice: small arcs have small geodesics (low entropy — local degrees of freedom). Half-boundary arcs have maximal geodesic length (maximally entangled with the rest). Tiny arcs have logarithmically divergent entropy at the boundary — that's the ultraviolet structure of the CFT showing up as the geodesic plunging deep into the bulk.
The deep claim: spacetime geometry is built from entanglement. Maldacena and Susskind's ER = EPR conjecture extends this to wormholes (Einstein-Rosen bridges) being equivalent to entangled pairs (Einstein-Podolsky-Rosen). When two CFTs share entanglement, their dual spacetimes become geometrically connected. This is the picture of quantum gravity emerging from quantum information that the past 20 years of high-energy physics has been building toward.