ABSTRACT: I will show 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.