  Dimensional regularization is a common method used to regulate the UV
divergence of field theoretic quantities. When it is used in the context of
Renyi entropy, however, it is important to consider whether such a procedure
eliminates the statistical interpretation thereof as a measure of entanglement
of states living on a Hilbert space. We therefore examine the dimensionally
regularized Renyi entropy of a 4d unitary CFT and show that it admits no
underlying Hilbert space in the state-counting sense. This gives a concrete
proof that dimensionally regularized Renyi entropy cannot always be obtained as
a limit of the Renyi entropy of some finite-dimensional quantum system.
