# Werner state

A '''Werner state'''Werner:1989 is a $d \times d$ dimensional bipartite quantum state that is invariant under the unitary $U \otimes U$ for any unitary $U$. That is, a state $\rho$ that satisfies :$\rho = \left(U \otimes U\right) \rho \left(U^\dagger \otimes U^\dagger\right)$ 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 $p_\left\{sym\right\}$being the only parameter that defines the state. :$\rho = p_\left\{sym\right\} \frac\left\{2\right\}\left\{d^2 + d\right\} P_\left\{sym\right\} + \left(1-p_\left\{sym\right\}\right) \frac\left\{2\right\}\left\{d^2 - d\right\} P_\left\{as\right\},$ where :$P_\left\{sym\right\} = \frac\left\{1\right\}\left\{2\right\}\left(1+P\right),$ $P_\left\{as\right\} = \frac\left\{1\right\}\left\{2\right\}\left(1-P\right),$ are the projectors and :$P = \sum_\left\{ij\right\} |i\rangle \langle j| \otimes |j\rangle \langle i|$ is the permutation operator that exchanges the two subsystems. Werner states are separable for $p_\left\{sym\right\} \geq 1/2$ and entangled for $p_\left\{as\right\} < 1/2$. All entangled Werner states violate the PPT separability criterion, but for $d \geq 3$ no Werner states violate the weaker reduction criterion. Werner states can be parametrized in different ways. One way of writing them is :$\rho = \frac\left\{1\right\}\left\{d^2-d \alpha\right\}\left(1 - \alpha P\right),$ where the new parameter $\alpha$ varies between -1 and 1 and relates to the $p_\left\{sym\right\}$ above as $\alpha = \left(\left(1-2p_\left\{sym\right\}\right)d+1\right)/\left(1-2p_\left\{sym\right\}+d\right)$. == 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. == See also == * Isotropic state == References == Category:Quantum States