  We express the positive partial transpose (PPT) separability criterion for
symmetric states of multi-qubit systems in terms of matrix inequalities based
on the recently introduced tensor representation for spin states. We construct
a matrix from the tensor representation of the state and show that it is
similar to the partial transpose of the density matrix written in the
computational basis. Furthermore, the positivity of this matrix is equivalent
to the positivity of a correlation matrix constructed from tensor products of
Pauli operators. This allows for a more transparent experimental interpretation
of the PPT criteria for an arbitrary spin-j state. The unitary matrices
connecting our matrix to the partial transpose of the state generalize the
so-called magic basis that plays a central role in Wootters' explicit formula
for the concurrence of a 2-qubit system and the Bell bases used for the
teleportation of a one or two-qubit state.
