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 |Det_n> (d=n). The computation yields S[H]=4 log 2 and S[Det_n]=log(n!). Counterparts of the determinant state defined for d<n are also considered.