# Werner state

A Werner stateWerner:1989 is a d × d dimensional bipartite quantum state that is invariant under the unitary U ⊗ U for any unitary U. That is, a state ρ that satisfies

ρ = (U ⊗ U)ρ(U †  ⊗ U † ) for all U on the d-dimensional subsystems.

The Werner states are mixtures of projectors onto the symmetric- and anti-symmetric subspaces, with the relative weight psymbeing the only parameter that defines the state.

$$\rho = p_{sym} \frac{2}{d^2 + d} P_{sym} + (1-p_{sym}) \frac{2}{d^2 - d} P_{as},$$ where

$$P_{sym} = \frac{1}{2}(1+P),$$ $P_{as} = \frac{1}{2}(1-P),$ are the projectors and

P = ∑iji⟩⟨j∣ ⊗ ∣j⟩⟨i∣ is the permutation operator that exchanges the two subsystems.

Werner states are separable for psym ≥ 1/2 and entangled for pas < 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

$$\rho = \frac{1}{d^2-d \alpha}(1 - \alpha P),$$ where the new parameter α varies between -1 and 1 and relates to the psym above as α = ((1 − 2psym)d + 1)/(1 − 2psym + d).

### Multipartite Werner states

Werner states can be generalized to the multipartite case Eggeling.etal:2008. An N-party Werner state is a state that is invariant under U⊗U⊗...⊗U 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.