for all unitary operators U acting on d-dimensional Hilbert space.
are the projectors and
is the permutation operator that exchanges the two subsystems.
Werner states are separable for psym ≥ 1⁄2 and entangled for psym < 1⁄2. All entangled Werner states violate the PPT separability criterion, but for d ≥ 3 no Werner states violate the weaker reduction criterion. Werner states can be parametrized in different ways. One way of writing them is
where the new parameter α varies between −1 and 1 and relates to psym as
Multipartite Werner states
Werner states can be generalized to the multipartite case. An N-party Werner state is a state that is invariant under for any unitary U on a single subsystem. The Werner state is no longer described by a single parameter, but by N! − 1 parameters, and is a linear combination of the N! different permutations on N systems.