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We argue that the Anti-de-Sitter (AdS) geometry in d+1 dimensions naturallyemerges from an arbitrary conformal field theory in d dimensions using the freeflow equation. We first show that an induced metric defined from the flowedfield generally corresponds to the quantum information metric, called the Buresor Helstrom metric, if the flowed field is normalized appropriately. We nextverify that the induced metric computed explicitly with the free flow equationalways becomes the AdS metric when the theory is conformal. We finally provethat the conformal symmetry in d dimensions converts to the AdS isometry in d+1dimensions after d dimensional quantum averaging. This guarantees the emergenceof AdS geometry without explicit calculation.

Flow equation, conformal symmetry and AdS geometry