Concurrence classes for general pure multipartite states

We propose concurrence classes for general pure multipartite states based on an orthogonal complement of a positive operator-valued measure on quantum phase. In particular, we construct W m class, GHZ m , and GHZ m −1 class concurrences for general pure m -partite states. We give explicit expressions for W 3 and GHZ 3 class concurrences for general pure three-partite states and for W 4 , GHZ 4 and GHZ 3 class concurrences for general pure four-partite states.