  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
$(\mathfrak{c}_{\mathrm{tot}}, \eta)$ 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 groundstate, the quantity
$\mathfrak{c}_{\mathrm{tot}}$ 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)
$\mathfrak{c}_{\mathrm{tot}}$ is a non-negative constant that can be
interpreted as the total central charge of the edge theory. (ii) $\eta$ 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 groundstate entanglement.
  We show that stationarity of $\mathfrak{c}_{\mathrm{tot}}$ is equivalent to a
vector fixed-point equation involving $\eta$, making our assumption locally
checkable. We also derive similar results for 1+1D systems under a suitable set
of assumptions.
