  Consider a system consisting of $n$ $d$-dimensional quantum particles and
arbitrary pure state $\Psi$ of the whole system. Suppose we simultaneously
perform complete von Neumann measurements on each particle. 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 for bipartite states. We compute $S[\Psi]$ for two sample multipartite
states: the hexacode state ($n=6, d=2$) and determinant states ($n=d$). The
generalization of determinant states to the case $d<n$ is considered.
