  Recent investigations have unveiled exotic quantum phases that elude
characterization by simple bipartite correlation functions. In these phases,
long-range entanglement arising from tripartite correlations plays a central
role. Consequently, the study of multipartite correlations has become a focal
point in modern physics. In these, Conditional Mutual Information (CMI) is one
of the most well-established information-theoretic measures, adept at
encapsulating the essence of various exotic phases, including topologically
ordered ones. Within the realm of quantum many-body physics, it has been a
long-sought goal to establish a quantum analog to the Hammersley-Clifford
theorem that bridges the two concepts of the Gibbs state and the Markov
network. This theorem posits that the correlation length of CMI remains
short-range across all thermal equilibrium quantum phases. In this work, we
demonstrate that CMI exhibits exponential decay concerning distance, with its
correlation length increasing polynomially with the inverse temperature. While
this clustering theorem has previously been established for high temperatures
devoid of thermal phase transitions, it has remained elusive at low
temperatures, where genuine long-range entanglement is corroborated to exist by
the quantum topological order. Our findings unveil that, even at low
temperatures, a broad class of tripartite entanglement cannot manifest in the
long-range regime. To achieve the proof, we establish a comprehensive formalism
for analyzing the locality of effective Hamiltonians on subsystems, commonly
known as the `entanglement Hamiltonian' or `Hamiltonian of mean force.' As one
outcome of our analyses, we enhance the prior clustering theorem concerning
bipartite entanglement. In essence, this means that we investigate genuine
bipartite entanglement that extends beyond the limitations of the Positive
Partial Transpose (PPT) class.
