Quantum Markov Networks and Commuting Hamiltonians

Abstract

Quantum Markov networks are a generalization of quantum Markov chains to arbitrary graphs. They provide a powerful classification of correlations in quantum many-body systems---complementing the area law at finite temperature---and are therefore useful to understand the powers and limitations of certain classes of simulation algorithms. Here, we extend the characterization of quantum Markov networks and in particular prove the equivalence of positive quantum Markov networks and Gibbs states of Hamiltonians that are the sum of local commuting terms on graphs containing no triangles. For more general graphs we demonstrate the equivalence between quantum Markov networks and Gibbs states of a class of Hamiltonians of intermediate complexity between commuting and general local Hamiltonians.

References in corpus (7)

• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Quantum Nonlocality without Entanglement
• Area laws in quantum systems: mutual information and correlations
• Quantum Belief Propagation
• Cavity method for quantum spin glasses on the Bethe lattice
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Cited by in corpus (6)

• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
• Renormalization group constructions of topological quantum liquids and beyond
• The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions
• Exponential clustering of bipartite quantum entanglement at arbitrary temperatures
• Matrix Product Density Operators: when do they have a local parent Hamiltonian?
• Classical restrictions of generic matrix product states are quasi-locally Gibbsian