  Using the finite Fourier transform, we introduce a generalization of
Pauli-spin matrices for $d$-dimensional spaces, and the resulting set of
unitary matrices $S(d) $ is a basis for $d\times d$ matrices. If $N=d_{1}\times
d_{2}\times...\times d_{b}$ and $H^{[ N]}=\bigotimes H^{% [ d_{k}]}$, we give a
sufficient condition for separability of a density matrix $\rho $ relative to
the $H^{[ d_{k}]}$ in terms of the $L_{1}$ norm of the spin coefficients of
$\rho >.$ Since the spin representation depends on the form of the tensor
product, the theory applies to both full and partial separability on a given
space $H^{[ N]}$% . It follows from this result that for a prescribed form of
separability, there is always a neighborhood of the normalized identity in
which every density matrix is separable. We also show that for every prime $p$
and $n>1$ the generalized Werner density matrix $W^{[ p^{n}]}(s) $ is fully
separable if and only if $s\leq (1+p^{n-1}) ^{-1}$.
