Werner state
A Werner state is a -dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form formula_1. That is, it is a bipartite quantum state formula_2 that satisfies for all unitary operators "U" acting on "d"-dimensional Hilbert space. First developed by Reinhard F. Werner in 1989. General definition. Every Werner state formula_4 is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight formula_5 being the main parameter that defines the state, in addition to the dimension formula_6: 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 formula_13 above, can be written explicitly in matrix form asformula_14Equivalently, these can be written as a convex combination of the  totally mixed state with (the projection onto) a Bell state:formula_15where formula_16 is related to formula_17 by formula_18. Werner-Holevo channels. A Werner-Holevo quantum channel formula_19 with parameters formula_20 and integer formula_21 is defined as where the quantum channels formula_23 and formula_24 are defined as and formula_27 denotes the partial transpose map on system "A". Note that the Choi state of the Werner-Holevo channel formula_28 is a Werner state: where formula_30. Multipartite Werner states. Werner states can be generalized to the multipartite case. An "N"-party Werner state is a state that is invariant under formula_31 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.