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

Polarization is an unprecedented coding technique in that it not only achieves channel capacity, but also does so at a faster speed of convergence than any other technique. This speed is measured by the "scaling exponent" and its importance is three-fold. Firstly, estimating the scaling exponent is challenging and demands a deeper understanding of the dynamics of communication channels. Secondly, scaling exponents serve as a benchmark for different variants of polar codes that helps us select the proper variant for real-life applications. Thirdly, the need to optimize for the scaling exponent sheds light on how to reinforce the design of polar code. In this paper, we generalize the binary erasure channel (BEC), the simplest communication channel and the protagonist of many polar code studies, to the "tetrahedral erasure channel" (TEC). We then invoke Mori-Tanaka’s 2 × 2 matrix over 𝔽_4 to construct polar codes over TEC. Our main contribution is showing that the dynamic of TECs converges to an almost-one-parameter family of channels, which then leads to an upper bound of 3.328 on the scaling exponent. This is the first non-binary matrix whose scaling exponent is upper-bounded. It also polarizes BEC faster than all known binary matrices up to 23 × 23 in size. Our result indicates that expanding the alphabet is a more effective and practical alternative to enlarging the matrix in order to achieve faster polarization.