  The Sorkin-Johnston (SJ) state is a candidate physical vacuum state for a
scalar field in a generic curved spacetime. It has the attractive feature that
it is covariantly and uniquely defined in any globally hyperbolic spacetime,
often reflecting the underlying symmetries if there are any. A potential
drawback of the SJ state is that it does not always satisfy the Hadamard
condition. In this work, we study the extent to which the SJ state in a $1+1$D
causal diamond is Hadamard, finding that it is not Hadamard at the boundary. We
then study the softened SJ state, which is a slight modification of the
original state to make it Hadamard. We use the softened SJ state to investigate
whether some peculiar features of entanglement entropy in causal set theory may
be linked to its non-Hadamard nature.
