A Werner state is a d^2 × d^2-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form U \otimes U. That is, it is a bipartite quantum state \rho_{AB} 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 p \in [0,1] being the main parameter that defines the state, in addition to the dimension d \geq 2: 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 d=2 above, can be written explicitly in matrix form asEquivalently, 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 p by. Then, two-qubit Werner states are separable for and entangled for.

Werner-Holevo channels

A Werner-Holevo quantum channel with parameters and integer d\geq2 is defined as where the quantum channels and are defined as and T_{A} 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. 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.