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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 ≥ 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 example
Two-qubit Werner states, corresponding to above, can be written explicitly in matrix form as
Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state:
where (or, confining oneself to positive values, ) is related to by . Then, two-qubit Werner states are separable for and entangled for .
and denotes the partial transpose map on system A. Note that the
Choi state of the Werner-Holevo channel
is a Werner state:
where .
Multipartite Werner states
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.
^
Fannes, Mark; Haegeman, B.; Mosonyi, Milan; Vanpeteghem, D. (2004). "Additivity of minimal entropy out- put for a class of covariant channels". unpublished. arXiv:quant-ph/0410195. Bibcode:2004quant.ph.10195F.