Emergent Area Operators in the Boundary

TL;DR

This paper investigates area operators in boundary theories of certain bulk dimensions, showing how they emerge through quantum error correction, coarse-graining, and semiclassicality conditions, with implications for entanglement entropy.

Contribution

It introduces a framework for understanding emergent area operators via QECCs and coarse-graining, revealing their non-linear constraints and semiclassical interpretations.

Findings

Exact QECCs admit a central decomposition where the area operator vanishes.

A non-zero, coarse-grained area operator approximates entanglement entropy.

Ambiguities in the coarse-grained operator relate to fixed-area state definitions.

Abstract

In some cases in two and three bulk dimensions without bulk local degrees of freedom, I look for area operators in a fixed boundary theory. In each case, I define an exact quantum error-correcting code (QECC) and show that it admits a central decomposition. However, the area operator that arises from this central decomposition vanishes. A non-zero area operator, however, emerges after coarse-graining. The expectation value of this operator approximates the actual entanglement entropy for a class of states that do not form a linear subspace. These non-linear constraints can be interpreted as semiclassicality conditions. The coarse-grained area operator is ambiguous, and this ambiguity can be matched with that in defining fixed-area states.