# 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 ${\displaystyle U\otimes U}$. That is, it is a quantum state ρ that satisfies

${\displaystyle \rho =(U\otimes U)\rho (U^{\dagger }\otimes U^{\dagger })}$

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.

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

where

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

are the projectors and

${\displaystyle P=\sum _{ij}|i\rangle \langle j|\otimes |j\rangle \langle i|}$

is the permutation operator that exchanges the two subsystems.

Werner states are separable for psym ≥ ​12 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

${\displaystyle \rho ={\frac {1}{d^{2}-d\alpha }}(1-\alpha P),}$

where the new parameter α varies between −1 and 1 and relates to psym as

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

## 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 ${\displaystyle U\otimes U\otimes ...\otimes U}$ 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)