Werner state

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