  We initiate an investigation into separable, but physically reasonable,
states in relativistic quantum field theory. In particular we will consider the
minimum amount of energy density needed to ensure the existence of separable
states between given spacelike separated regions. This is a first step towards
improving our understanding of the balance between entanglement entropy and
energy (density), which is of great physical interest in its own right and also
in the context of black hole thermodynamics. We will focus concretely on a
linear scalar quantum field in a topologically trivial, four-dimensional
globally hyperbolic spacetime. For rather general spacelike separated regions
$A$ and $B$ we prove the existence of a separable quasi-free Hadamard state. In
Minkowski spacetime we provide a tighter construction for massive free scalar
fields: given any $R>0$ we construct a quasi-free Hadamard state which is
stationary, homogeneous, spatially isotropic and separable between any two
regions in an inertial time slice $t=\mathrm{const.}$ all of whose points have
a distance $>R$. We also show that the normal ordered energy density of these
states can be made $\le 10^{31}\frac{m^4}{(mR)^8}e^{-\frac14mR}$ (in Planck
units). To achieve these results we use a rather explicit construction of
test-functions $f$ of positive type for which we can get sufficient control on
lower bounds on $\hat{f}$.
