*Seminar room changed from 160 to 433.

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 also show 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.

We next apply this method to non-relativistic systems with anisotropic scaling symmetries, such as Lifshitz field theories and Schr\”odinger invariant theories. In consequence we obtain a new hybrid geometry of Lifshitz and Schr\”odinger spacetimes as a general holographic geometry. We also show that the bulk hybrid geometry is realized by an Einstein-Maxwell-Higgs system plus a gauge fixing term for diffeomorphism, which may be interpreted as a holographic dual of a general non-relativistic system at the boundary.