Structure of entanglement in regulated Lorentz invariant field theories

Regulated Lorentz invariant quantum field theories satisfy an area law for the entanglement entropy $S$ of a spatial subregion in the ground state in $d>1$ spatial dimensions; nevertheless, the full density matrix contains many more than $e^{S}$ non-zero eigenvalues. We ask how well the state of a subregion $R$ in the ground state of such a theory can be approximated when keeping only the $e^{S}$ largest eigenvalues of the reduced density matrix of $R$. We argue that by taking the region $R$ big enough, we can always ensure that keeping roughly $e^{S}$ states leads to bounded error in trace norm even for subregions in gapless ground states. We support these general arguments with an explicit computation of the error in a half-space geometry for a free scalar field in any dimension. Along the way we show that the Renyi entropy of a ball in the ground state of any conformal field theory at small Renyi parameter is controlled by the conventional \textit{thermal entropy density} at low temperatures. We also reobtain and generalize some old results relevant to DMRG on the decay of Schmidt coefficients of intervals in one dimensional ground states. Finally, we discuss the role of the regulator, the insensitivity of our arguments to the precise ultraviolet physics, and the role of adiabatic continuity in our results.