  We study different aspects of quantum von Neumann and R\'enyi entanglement
entropy of one dimensional long-range harmonic oscillators that can be
described by well-defined non-local field theories. We show that the
entanglement entropy of one interval with respect to the rest changes
logarithmically with the number of oscillators inside the subsystem. This is
true also in the presence of different boundary conditions. We show that the
coefficients of the logarithms coming from different boundary conditions can be
reduced to just two different universal coefficients. We also study the effect
of the mass and temperature on the entanglement entropy of the system in
different situations. The universality of our results is also confirmed by
changing different parameters in the coupled harmonic oscillators. We also show
that more general interactions coming from general singular Toeplitz matrices
can be decomposed to our long-range harmonic oscillators. Despite the
long-range nature of the couplings we show that the area law is valid in two
dimensions and the universal logarithmic terms appear if we consider subregions
with sharp corners. Finally we study analytically different aspects of the
mutual information such as its logarithmic dependence to the subsystem, effect
of mass and influence of the boundary. We also generalize our results in this
case to general singular Toeplitz matrices and higher dimensions.
