Any state analytic in energy (which includes most physical states since they have bounded energy) contains non-local correlations described by the Reeh-Schlieder theorem in AQFT. It is further shown that decreasing the distance between wedges will increase the entanglement as measured by a Bell-type inequality, until it reaches a maximum for tangent wedges. In this situation all analytic states are maximally entangled.
Does maximal entanglement translate into infinite entropy of entanglement?
My intuition is based on the fact that the vacuum state (which is analytic hence subject to RS correlations) follows a UV-divergent area law or a UV-divergent log law for entanglement entropy, depending on the dimension, criticality or the bosonic/fermionic character of the field, but anyway, entropy diverges. Vacuum being maximally entangled, it means that all maximally entangled states have the same entropy of entanglement i.e. infinite entropy. Another point that backs my intuition is the result that the reduced state has norm dense components i.e. is highly mixed. But I can't find or formulate a rigorous statement on the entropy of the RS correlations.