Entanglement entropy of multipartite pure states

Consider a system consisting of n d-dimensional quantum particles and an arbitrary pure state |Ψ〉 of the whole system. Suppose we simultaneously perform complete von Neumann measurements on each particle. The Shannon entropy of the outcomes' joint probability distribution is a functional of the state |Ψ〉 and of n measurements chosen for each particle. Denote S[Ψ] the minimum of this entropy over all choices of the measurements. We show that S[Ψ] coincides with the entropy of entanglement for bipartite states. We compute S[Ψ] for some special multipartite states: the hexacode state $|H〉$ $(n=6,$ $d=2)$ and the determinant states |Detₙ〉 $(d=n).$ The computation yields S[H]=4log2 and S[Detₙ]=log(n!). Counterparts of the determinant state defined for $d<n$ are also considered.