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 projectors onto the symmetric- and anti-symmetric subspaces, 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 $p\_\{as\}\; <\; 1/2$. All entangled Werner states violate the 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

## Last modified:

Monday, October 26, 2015 - 17:56