Separability and Fourier representations of density matrices

Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for d-dimensional spaces, and the resulting set of unitary matrices $S\left(d\right)\mathrm{}$ is a basis for $d\times d$ matrices. If ${N=d}_1\times d_2\times \cdot \cdot \cdot \times d_b$ and $H^{\left[N\right]}=\otimes H^{\left[d_k\right]},$ we give a sufficient condition for separability of a density matrix $\rho$ relative to the $H^{\left[d_k\right]}$ 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 $H^{\left[N\right]}.$ 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 $n>\mathrm{}1,$ the generalized Werner density matrix $W^{\left[p^n\right]}\mathrm{}\left(s\right)\mathrm{}$ is fully separable if and only if $s<~{(1+p}^{n-1}{)}^{-1}.$