  Using the thermal Green's function approach we propose a general method to
investigate entanglement of the vacuum state or thermal ground states in an
arbitrary dimensional space-time. As an application we show quantum
separability of the massive thermal scalar field vacuum in the 1+1 dimensional
cylindrical space-time. Separability is demonstrated using the positive partial
transpose criterion for effective two-mode Gaussian states of collective
operators. In this case, for all mass and temperature values entanglement is
absent between the collective operators.
