On separable states in relativistic quantum field theory

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 $\hat{f}$.