Werner state
A Werner state is a -dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form. That is, it is a bipartite quantum state that satisfies
for all unitary operators U acting on d-dimensional Hilbert space. These states were first developed by Reinhard F. Werner in 1989.
General definition
Every Werner state is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight being the main parameter that defines the state, in addition to the dimension :where
are the projectors and
is the permutation or flip operator that exchanges the two subsystems A and B.
Werner states are separable for p ≥ and entangled for p <. All entangled Werner states violate the PPT separability criterion, but for d ≥ 3 no Werner state violates the weaker reduction criterion. Werner states can be parametrized in different ways. One way of writing them is
where the new parameter α varies between −1 and 1 and relates to p as
Two-qubit example
Two-qubit Werner states, corresponding to above, can be written explicitly in matrix form asEquivalently, these can be written as a convex combination of the totally mixed state with a Bell state: where is related to by. Then, two-qubit Werner states are separable for and entangled for.Werner-Holevo channels
A Werner-Holevo quantum channel with parameters and integeris defined as
where the quantum channels and
are defined as
and denotes the partial transpose map on system A. Note that the
Choi state of the Werner-Holevo channel
is a Werner state:
where.