Separability and Fourier representations of density matrices

Abstract

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 $d\ifmmode\times\else\texttimes\fi{}d$ matrices. If ${N=d}_{1}\ifmmode\times\else\texttimes\fi{}{d}_{2}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}\ifmmode\times\else\texttimes\fi{}{d}_{b}$ and ${H}^{[N]}=\ensuremath{\bigotimes}{H}^{[{d}_{k}]},$ we give a sufficient condition for separability of a density matrix $\ensuremath{\rho}$ relative to the ${H}^{[{d}_{k}]}$ in terms of the ${L}_{1}$ norm of the spin coefficients of $\ensuremath{\rho}.$ 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 $s<~{(1+p}^{n\ensuremath{-}1}{)}^{\ensuremath{-}1}.$