  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.
