Werner state

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A Werner state[1] is a d × d-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form . That is, it is a quantum state ρ that satisfies

for all unitary operators U acting on d-dimensional Hilbert space.

Every Werner state is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight psym being the only parameter that defines the state.


are the projectors and

is the permutation operator that exchanges the two subsystems.

Werner states are separable for psym ≥ ​12 and entangled for psym < ​12. All entangled Werner states violate the PPT separability criterion, but for d ≥ 3 no Werner states violate 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 psym as

Multipartite Werner states[edit]

Werner states can be generalized to the multipartite case.[2] 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.


  1. ^ Reinhard F. Werner (1989). "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model". Physical Review A. 40 (8): 4277–4281. Bibcode:1989PhRvA..40.4277W. doi:10.1103/PhysRevA.40.4277. PMID 9902666. 
  2. ^ Eggeling et al. (2008)