Entanglement entropy of multipartite pure states

Abstract

Consider a system consisting of n d-dimensional quantum particles and an arbitrary pure state $|\ensuremath{\Psi}〉$ 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 $|\ensuremath{\Psi}〉$ and of n measurements chosen for each particle. Denote $S[\ensuremath{\Psi}]$ the minimum of this entropy over all choices of the measurements. We show that $S[\ensuremath{\Psi}]$ coincides with the entropy of entanglement for bipartite states. We compute $S[\ensuremath{\Psi}]$ for some special multipartite states: the hexacode state $|H〉$ $(n=6,$ $d=2)$ and the determinant states $|{\mathrm{Det}}_{n}〉$ $(d=n).$ The computation yields $S[H]=4\mathrm{log}2$ and $S[{\mathrm{Det}}_{n}]=\mathrm{log}(n!).$ Counterparts of the determinant state defined for $d<n$ are also considered.