Quantum Information Theory of the Gravitational Anomaly

We show that the standard notion of 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. We make this precise by decomposing it into two observations: First, a two-dimensional CFT 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 we establish a generalization of the Nielsen-Ninomiya theorem to all two-dimensional unitary local QFTs: No continuum quantum field theory in two dimensions can admit a lattice regulator unless its gravitational anomaly vanishes. We also show that the conclusion can be generalized to six dimensions by dimensional reduction on a four-manifold of nonvanishing signature. We advocate that these points be used to reinterpret the gravitational anomaly quantum-information-theoretically, as a fundamental obstruction to the localization of quantum information.