  We argue that the Anti-de-Sitter (AdS) geometry in d+1 dimensions naturally
emerges from an arbitrary conformal field theory in d dimensions using the free
flow equation. We first show that an induced metric defined from the flowed
field generally corresponds to the quantum information metric, called the Bures
or Helstrom metric, if the flowed field is normalized appropriately. We next
verify that the induced metric computed explicitly with the free flow equation
always becomes the AdS metric when the theory is conformal. We finally prove
that the conformal symmetry in d dimensions converts to the AdS isometry in d+1
dimensions after d dimensional quantum averaging. This guarantees the emergence
of AdS geometry without explicit calculation.

High Energy Physics - Theory