  Werner states are multipartite quantum states that are invariant under the
diagonal conjugate action of the unitary group. This paper gives a complete
characterization of their entanglement that is independent of the underlying
local Hilbert space: for every entangled Werner state there exists a
dimension-free entanglement witness. The construction of such a witness is
formulated as an optimization problem. To solve it, two semidefinite
programming hierarchies are introduced. The first one is derived using real
algebraic geometry applied to positive polynomials in the entries of a Gram
matrix, and is complete in the sense that for every entangled Werner state it
converges to a witness. The second one is based on a sum-of-squares certificate
for the positivity of trace polynomials in noncommuting variables, and is a
relaxation that involves smaller semidefinite constraints.
