A Werner state[1] 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.
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 ≥ 1⁄2 and entangled for p < 1⁄2.
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 Werner states, corresponding to
above, can be written explicitly in matrix form as
{\displaystyle W_{AB}^{(p,2)}={\frac {p}{6}}{\begin{pmatrix}2&0&0&0\\0&1&1&0\\0&1&1&0\\0&0&0&2\end{pmatrix}}+{\frac {(1-p)}{2}}{\begin{pmatrix}0&0&0&0\\0&1&-1&0\\0&-1&1&0\\0&0&0&0\end{pmatrix}}={\begin{pmatrix}{\frac {p}{3}}&0&0&0\\0&{\frac {3-2p}{6}}&{\frac {-3+4p}{6}}&0\\0&{\frac {-3+4p}{6}}&{\frac {3-2p}{6}}&0\\0&0&0&{\frac {p}{3}}\end{pmatrix}}.}
Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state:
(or, confining oneself to positive values,
Then, two-qubit Werner states are separable for
A Werner-Holevo quantum channel
is defined as [2] [3] [4] where the quantum channels
denotes the partial transpose map on system A.
Note that the Choi state of the Werner-Holevo channel
is a Werner state: where
Werner states can be generalized to the multipartite case.
[5] An N-party Werner state is a state that is invariant under
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.
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