  We consider a model for a Planck scale ultraviolet cutoff which is based on
Shannon sampling. Shannon sampling originated in information theory, where it
expresses the equivalence of continuous and discrete representations of
information. When applied to quantum field theory, Shannon sampling expresses a
hard ultraviolet cutoff in the form of a bandlimitation. This introduces
nonlocality at the cutoff scale in a way that is more subtle than a simple
discretization of space: quantum fields can then be represented as either
living on continuous space or, entirely equivalently, as living on any one
lattice whose average spacing is sufficiently small. We explicitly calculate
vacuum entanglement entropies in 1+1 dimension and we find a transition between
logarithmic and linear scaling of the entropy, which is the expected 1+1
dimensional analog of the transition from an area to a volume law. We also use
entanglement entropy and mutual information as measures to probe in detail the
localizability of the field degrees of freedom. We find that, even though
neither translation nor rotation invariance are broken, each field degree of
freedom occupies an incompressible volume of space, indicating a finite
information density.
