I will explain that the notion of quantum entanglement is not defined for gravitationally anomalous two-dimensional theories, because they do not admit a local tensor factorization of the Hilbert space into local Hilbert spaces. Qualitatively, the modular flow cannot act consistently and unitarily in a finite region, if there are different numbers of states with a given energy traveling in the two opposite directions. I will make this precise by decomposing it into two observations: First, a two-dimensional conformal field theory admits a consistent quantization on a space with boundary only if it is not anomalous. Second, a local tensor factorization always leads to a definition of consistent, unitary, energy-preserving boundary condition. As a corollary I will establish a generalization of the Nielsen-Ninomiya theorem to all two-dimensional unitary local quantum field theories: No continuum quantum field theory in two dimensions can admit a lattice regulator unless its gravitational anomaly vanishes. I will advocate that these points be used to reinterpret the gravitational anomaly quantum-information-theoretically, as a fundamental obstruction to the localization of quantum information. [Zoom link: please ask your local triangle organizer]