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Strongly interacting systems can be described in terms of correlationfunctions at various orders. A quantum analog of high-order correlations is thetopological entanglement in topologically ordered states of matter at zerotemperature, usually quantified by topological entanglement entropy (TEE). Inthis work, we propose a statistical interpretation that unifies the two underthe same information-theoretic framework. We demonstrate that the existence ofa non-zero TEE can be understood in the statistical view as the emergent $n$thorder mutual information $I_n$ (for arbitrary integer $n\ge 3$) reflected inprojectively measured samples, which also makes explicit the equivalencebetween the two existing methods for its extraction -- the Kitaev-Preskill andthe Levin-Wen construction. To exploit the statistical nature of $I_n$, weconstruct a restricted Boltzmann machine (RBM) which captures the high-ordercorrelations and correspondingly the topological entanglement that are encodedin the distribution of projected samples by representing the entanglementHamiltonian of a local region under the proper basis. Furthermore, we derive aclosed form which presents a method to interrogate the trained RBM, makingexplicit the analytical form of arbitrary order of correlations relevant for$I_n$. We remark that the interrogation method for extracting high-ordercorrelation can also be applied to the construction of auxiliary fields thatdisentangle many-body interactions relevant for diverse interacting models.

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