Werner state

A Werner state[1] 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 ρ that satisfies

\rho = (U \otimes U) \rho (U^\dagger \otimes U^\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 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 = \sum_{ij} |i\rangle \langle j| \otimes |j\rangle \langle i|

is the permutation operator that exchanges the two subsystems.

Werner states are separable for p_{sym} \geq 1/2 and entangled for pas < 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{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 [2]. 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

References

[1] Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, R. F. Werner, Publication Phys. Rev. A 40, 4277 - 4281 (1989)
[2] Separability properties of tripartite states with U⊗U⊗U symmetry, T. Eggeling and R. F. Werner, Publication Phys. Rev. A 63, 042111 (2001) pre-print

Retrieved from "http://www.quantiki.org/wiki/index.php/Werner_state"