# Werner state

A **Werner state**Werner: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 *p**s**y**m*being 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* = ∑*i**j*∣*i*⟩⟨*j*∣ ⊗ ∣*j*⟩⟨*i*∣ is the permutation operator that exchanges the two subsystems.

Werner states are separable for *p**s**y**m* ≥ 1/2 and entangled for *p**a**s* < 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 *p**s**y**m* above as *α* = ((1 − 2*p**s**y**m*)*d* + 1)/(1 − 2*p**s**y**m* + *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.