A statistical approach to topological entanglement: Boltzmann machine representation of high-order irreducible correlation

Strongly interacting systems can be described in terms of correlation functions at various orders. A quantum analog of high-order correlations is the topological entanglement in topologically ordered states of matter at zero temperature, usually quantified by topological entanglement entropy (TEE). In this work, we propose a statistical interpretation that unifies the two under the same information-theoretic framework. We demonstrate that the existence of a non-zero TEE can be understood in the statistical view as the emergent $n$th order mutual information $I_n$ (for arbitrary integer $n\ge 3$) reflected in projectively measured samples, which also makes explicit the equivalence between the two existing methods for its extraction -- the Kitaev-Preskill and the Levin-Wen construction. To exploit the statistical nature of $I_n$, we construct a restricted Boltzmann machine (RBM) which captures the high-order correlations and correspondingly the topological entanglement that are encoded in the distribution of projected samples by representing the entanglement Hamiltonian of a local region under the proper basis. Furthermore, we derive a closed form which presents a method to interrogate the trained RBM, making explicit the analytical form of arbitrary order of correlations relevant for $I_n$. We remark that the interrogation method for extracting high-order correlation can also be applied to the construction of auxiliary fields that disentangle many-body interactions relevant for diverse interacting models.