  Entanglement entropy has proven invaluable to our understanding of quantum
criticality. It is natural to try to extend the concept to non-unitary quantum
mechanics, which has seen growing interest from areas as diverse as open
quantum systems, non-interacting electronic disordered systems, or non-unitary
conformal field theory (CFT). We propose and investigate such an extension
here, by focussing on the case of one-dimensional quantum group symmetric or
supergroup symmetric spin chains. We show that the consideration of left and
right eigenstates combined with appropriate definitions of the trace leads to a
natural definition of R\'enyi entropies in a large variety of models. We
interpret this definition geometrically in terms of related loop models and
calculate the corresponding scaling in the conformal case. This allows us to
distinguish the role of the central charge and effective central charge in
rational minimal models of CFT, and to define an effective central charge in
other, less well understood cases. The example of the $sl(2|1)$ alternating
spin chain for percolation is discussed in detail.
