Physical proof of the topological entanglement entropy inequality

TL;DR

This paper provides a direct physical proof of the universal inequality relating topological entanglement entropy and quantum dimension, using entropy properties and applicable to various complex systems.

Contribution

It offers a new, more straightforward proof of the TEE inequality that extends to inhomogeneous, higher-dimensional, and mixed quantum states.

Findings

Proof relies solely on strong subadditivity of von Neumann entropy.

The method generalizes to systems with defects, boundaries, and higher dimensions.

Validates the universal TEE inequality across diverse quantum systems.

Abstract

Recently it was shown that the topological entanglement entropy (TEE) of a two-dimensional gapped ground state obeys the universal inequality $\gamma \geq \log \mathcal{D}$, where $\gamma$ is the TEE and $\mathcal{D}$ is the total quantum dimension of all anyon excitations, $\mathcal{D} = \sqrt{\sum_a d_a^2}$. Here we present an alternative, more direct proof of this inequality. Our proof uses only the strong subadditivity property of the von Neumann entropy together with a few physical assumptions about the ground state density operator. Our derivation naturally generalizes to a variety of systems, including spatially inhomogeneous systems with defects and boundaries, higher dimensional systems, and mixed states.