Separability for lattice systems at high temperature

Equilibrium states of infinite extended lattice systems at high temperature are studied with respect to their entanglement. Two notions of separability are offered. They coincide for finite systems but differ for infinitely extended ones. It is shown that for lattice systems with localized interaction for high enough temperature there exists no entanglement localized in a finite region of given size. Even more quasifree states at high temperature are also not distillably entangled for all local regions of arbitrary size. For continuous systems entanglement survives at all temperatures. In mean-field theories it is possible that local regions are not entangled but we have still entanglement hidden in the non local fluctuation algebra.