Towards a Topological Proof of the Strong Subadditivity

TL;DR

This paper establishes a topological proof that the strong subadditivity property holds for topological entanglement entropy within the framework of topological quantum field theory, supported by verification across known categories.

Contribution

It provides a topological analysis linking SSA of TEE to a specific inequality in TQFT, and verifies this for all known UMTCs up to rank 11.

Findings

SSA for TEE is equivalent to a TQFT inequality

The inequality holds for all known UMTCs up to rank 11

SSA may serve as a consistency criterion for UMTCs

Abstract

Topological entanglement entropy (TEE) represents an intrinsic contribution to the entanglement entropy (EE) in topologically ordered systems. In quantum information theory, strong subadditivity (SSA) is a fundamental property of EE, reflecting the non-negativity of conditional mutual information. TEE was originally believed to be a universal correction to the area law of EE, suggesting that its SSA would directly follow from the SSA of EE. However, due to spurious contributions, the correction term is not universal; consequently, the value predicted by topological quantum field theory (TQFT) provides only a lower bound. In this work, we present a topological analysis showing that the SSA for TEE is equivalent to a specific inequality within the TQFT framework. We further verify that this inequality holds for all known unitary modular tensor categories (UMTCs) up to rank 11, supporting the conjecture that SSA holds universally in the TQFT framework. Conversely, assuming the validity of the SSA condition, the inequality can be interpreted as a consistency criterion for candidate UMTCs.