Topological entanglement entropy meets holographic entropy inequalities

Topological entanglement entropy (TEE) is an efficient way to detect topological order in the ground state of gapped Hamiltonians. The seminal work of Kitaev and Preskill~\cite{preskill-kitaev-tee} and simultaneously by Levin and Wen~\cite{levin-wen-tee} proposed information quantities that can probe the TEE. In the present work, we explain why the subtraction schemes in the proposed information quantities~\cite{levin-wen-tee,preskill-kitaev-tee} work for the computation of TEE and generalize them for arbitrary number of subregions by explicitly noting the necessary conditions for an information quantity to capture TEE. Our conditions differentiate the probes defined by Kitaev-Preskill and Levin-Wen into separate classes. While there are infinitely many possible probes of TEE, we focus particularly on the cyclic quantities $Q_{2n+1}$ and multi-information $I_n$. We also show that the holographic entropy inequalities are satisfied by the quantum entanglement entropy of the non-degenerate ground state of a topologically ordered two-dimensional medium with a mass gap.