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 $S(d)$ is a basis for d×d matrices. If N=d₁×d₂×···×db and H[N]=H^[dₖ], we give a sufficient condition for separability of a density matrix ρ relative to the H^[dₖ] in terms of the L₁ norm of the spin coefficients of ρ. 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^[pⁿ](s) is fully separable if and only if s<~(1+p^n-1)^-1.