Quantum Information Theory of the Gravitational Anomaly

TL;DR

The paper proves that a nonzero gravitational anomaly in two-dimensional quantum field theories prevents a local tensor factorization of the Hilbert space, thereby obstructing a meaningful notion of entanglement. It shows that a lattice regulator necessarily induces a boundary conformal field theory with finite boundary entropy, which can only occur if the gravitational anomaly vanishes, generalizing Nielsen–Ninomiya to interacting 2D QFTs and, by dimensional reduction, to six dimensions. The authors introduce a renormalized tensor product framework and a locality criterion for factorization maps, establishing that gravitational anomalies obstruct such factorizations and the associated modular flow entanglement structure. They also discuss approximately-factorized states in theories with boundary conditions and examine holographic theories with large anomalies, highlighting implications for holography and quantum gravity. The work reframes gravitational anomalies as fundamental obstructions to localizing quantum information, with broad consequences for QFT, lattice regularization, and AdS/CFT holography.

Abstract

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.

Figures (6)

• Figure 1: The slit geometry can be conformally transformed into a strip. The crosses represent the positions at which the local operators are located. We may take whatever boundary condition as long as it has finite boundary entropy.
• Figure 2: This is (half of the) quotient geometry for our construction, where we cut the whole geometry of figure \ref{['fig:quotient-double']} in half on the right edge. The brown circle is of circumference $\epsilon$ as we regularize the trace using heat-kernels. Also, the blue circle is of circumference $\epsilon_{{\rm f}}{}$ in Ohmori:2014eia, but this scale can be shown via conformal transformation to decouple in our finite-temperature construction. The circle in yellow-green is the Euclidean-time circle whose circumference defines the inverse temperature of the thermal ensemble on the full spatial slice $\Sigma = A \cup B$, on which the boundary states live in our construction. After an open-closed modular transformation, this can be used in the dual description of Sec. \ref{['sec:factorization-implies-boundary']}hep-th/0411189.
• Figure 3: This is half of the quotient geometry in the construction of Ohmori:2014eia, where we again cut the whole geometry in half on the right edge. After conformal transformation from Fig. \ref{['quo']}, the radius of the cylinder is $2\pi$ and the length $2\pi L/\epsilon$.
• Figure 4: This is the octopus geometry, half of the geometry of the $n$-fold cover of the infinite cylinder with two $n$-order twist operators inserted; in the figure we take $n$ to be $4$. Green wavy lines indicate the positions of what used to be the branch cuts in the original geometry. The replica transformation acts as a $2\pi/n$ rotation of the large cylinder, where again $n=4$ in this figure.
• Figure 5: Geometries relevant for our discussion. In the figure above we depict the trace ${\tt tr}(\rho^ n_ I)$ of the n$^{\rm{\underline{th}}}$ power of the reduced density matrix for subregion $I$, where the state of the full system is a thermal density matrix. In the figure, we take $n=4$ and the subregion $I$ is an interval of finite length, whose boundaries are the two light blue dots. The green squiggly line between them is a branch cut whose monodromy is a cyclic permutation of the four replicas of the underlying cft. The singularity of the Riemann surface in the lower figure is resolved by the finite factorization-time $\epsilon_{{\rm f}}{}$, which cuts off the region containing the conical deficit and is represented by the finite thickness of the blue dots.In the lower figure we show the full Riemann surface realized as the covering space of the upper figure. A conformal transformation has been performed to make the blue circle the largest scale in the geometry other than the length of the interval.