In a physical system with conformal symmetry, observables depend on ??cross-ratios, measures of distance invariant under global conformal ??transformations (conformal geometry for short). We identify a quantum ??information-theoretic mechanism by which the conformal geometry ??emerges at the gapless edge of a 2+1D quantum many-body system with a ??bulk energy gap. We introduce a novel pair of information-theoretic ??quantities (?,η) that can be defined locally on the edge from the ??wavefunction of the many-body system, without prior knowledge of any ??distance measure. We posit that, for a topological ground state, the ??quantity ? is stationary under arbitrary variations of the quantum ??state, and study the logical consequences. We show that stationarity, ??modulo an entanglement-based assumption about the bulk, implies (i) ? ??is a non-negative constant that can be interpreted as the total ??central charge of the edge theory. (ii) η is a cross-ratio, obeying ??the full set of mathematical consistency rules, which further ??indicates the existence of a distance measure of the edge with global ??conformal invariance. Thus, the conformal geometry emerges from a ??simple assumption on ground state entanglement. The stationarity of ? ??is equivalent to a vector fixed-point equation involving η, making ?our ??assumption locally checkable. If time permits, we discuss a ?class of ??modular flow on a disk, which creates only edge ?excitations. We ??intuitively explain why Virasoro algebra can be ?revealed from a single ??wavefunction by analyzing such modular flows.