# Werner state

A Werner state[1] is a d × d-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form $U \otimes U$. That is, it is a quantum state ρ that satisfies

$\rho = (U \otimes U) \rho (U^\dagger \otimes U^\dagger)$

for all unitary operators U acting on d-dimensional Hilbert space.

Every Werner state is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight psym being the only parameter that defines the state.

$\rho = p_\text{sym} \frac{2}{d^2 + d} P_\text{sym} + (1-p_\text{sym}) \frac{2}{d^2 - d} P_\text{as},$

where

$P_\text{sym} = \frac{1}{2}(1+P),$ $P_\text{as} = \frac{1}{2}(1-P),$

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 psym12 and entangled for psym < 12. 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 psym as

$\alpha = ((1-2p_\text{sym})d+1)/(1-2p_\text{sym}+d) .$

## 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 $U \otimes U \otimes ... \otimes U$ 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.

## References

1. ^ Reinhard F. Werner (1989). "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model". Physical Review A 40 (8): 4277–4281. Bibcode:1989PhRvA..40.4277W. doi:10.1103/PhysRevA.40.4277. PMID 9902666.
2. ^ Eggeling et al. (2008)