The quantum entanglement $E$ of a bipartite quantum Ising chain is compared with the mutual information $I$ between the two parts after a local measurement of the classical spin configuration. As the model is conformally invariant, the entanglement m
easured in its ground state at the critical point is known to obey a certain scaling form. Surprisingly, the mutual information of classical spin configurations is found to obey the same scaling form, although with a different prefactor. Moreover, we find that mutual information and the entanglement obey the inequality $Ileq E$ in the ground state as well as in a dynamically evolving situation. This inequality holds for general bipartite systems in a pure state and can be proven using similar techniques as for Holevos bound.
We construct equilibrium networks by introducing an energy function depending on the degree of each node as well as the product of neighboring degrees. With this topological energy function, networks constitute a canonical ensemble, which follows the
Boltzmann distribution for given temperature. It is observed that the system undergoes a topological phase transition from a random network to a star or a fully-connected network as the temperature is lowered. Both mean-field analysis and numerical simulations reveal strong first-order phase transitions at temperatures which decrease logarithmically with the system size. Quantitative discrepancies of the simulation results from the mean-field prediction are discussed in view of the strong first-order nature.
