Investigation of commuting Hamiltonian in quantum Markov network

Noname manuscript No. (will be inserted by the editor) Abstract Graphical Models have various applications in science and engineering which include physics, bioinformatics, telecommunication and etc. Usage of graphical models needs complex computations in order to evaluation of marginal functions, so there are some powerful methods including mean field approximation, belief propagation algorithm and etc. Quantum graphical models have been recently developed in context of quantum information and computation, and quantum statistical physics, which is possible by generalization of classical probability theory to quantum theory. The main goal of this paper is preparing a primary generalization of Markov network, as a type of graphical models, to quantum case and applying in quantum statistical physics. We have investigated the Markov network and the role of commuting Hamiltonian terms in conditional independence with simple examples of quantum statistical physics. Keywords Graphical models · Quantum graphical models · Conditional independence · Quantum conditional independence · Commuting Hamiltonian · Quantum Markov network 1 Introduction In 1988, Pearl devised belief propagation algorithm to solve marginalization and other inference problems.