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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.

Correct me if I am wrong.