  The so-called permutation separability criteria are simple operational
conditions that are necessary for separability of mixed states of multipartite
systems: (1) permute the indices of the density matrix and (2) check if the
trace norm of at least one of the resulting operators is greater than one. If
it is greater than one then the state is necessarily entangled. A shortcoming
of the permutation separability criteria is that many permutations give rise to
dependent separability criteria. Therefore, we introduce a necessary condition
for two permutations to yield independent criteria called combinatorical
independence. This condition basically means that the map corresponding to one
permutation cannot be obtained by concatenating the map corresponding to the
second permutation with a norm-preserving map. We characterize completely
combinatorically independent criteria, and determine simple permutations that
represent all independent criteria. The representatives can be visualized by
means of a simple graphical notation. They are composed of three basic
operations: partial transpose, and two types of so-called reshufflings. In
particular, for a four-partite system all criteria except one are composed of
partial transpose and only one type of reshuffling; the exceptional one
requires the second type of reshuffling. Furthermore, we show how to obtain
efficiently for every permutation a simple representative. This method allows
to check easily if two permutations are combinatorically equivalent or not.

Quantum Physics