Quantum Hammersley-Clifford Theorem 

The classical Hammersley-Clifford theorem allows us to write the joint probability distribution of random variables defined at the vertices of an undirected graph as thermal states of positive functions defined over cliques of the graph. This is interesting because we obtain a mathematical form that is, in essence, a Gibbs' state, a concept widely used in physics, from something that is completely abstract and not necessarily motivated by physics. The quantum analog of this theorem would describe the joint density operator of an undirected graph, where each vertex corresponds to a quantum system, in terms of positive operators acting on its cliques. One direction of the proof was established in [Ann. Phys. 323, 1899 (2008)], showing that the joint density operator can be decomposed as a product of such clique operators. [arXiv:1206.0755, 2012] Extended the result by showing the existence of a family of commuting Hamiltonians for which both directions of the quantum Hammersley-Clifford theorem hold for a graph with no triangles. In this work, we aim to present a proof of this theorem for general undirected graphs, including those containing triangles.