Entanglement entropy of multipartite pure state

Consider a system consisting of n d-dimensional quantum particles and arbitrary pure state $|\Psi\ra$ of the whole system. Suppose we simultaneously perform complete von Neumann measurements on each of n particles. One can ask: what is the minimal possible value $S[\Psi]$ of the entropy of outcomes joint probability distribution? We show that $S[\Psi]$ coincides with entanglement entropy $E[\Psi]$ for n=2. We compute $S[\Psi]$ for two sample multipartite states~: the hexacode state $|{\rm H}\ra$, n=6, d=2, $S[H]=4\log 2$ and determinant state $|{\rm Det}_n\ra$, d=n, $S[{\rm Det}_n]=\log(n!)$. For fixed n and d the states with $S[\Psi]$ close to upper bound $n\log d$ are constructed