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 $d\times d$ 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 $s\le (1+p^{n-1}{)}^{-1}$.

References in corpus (4)

• Classification of multi-qubit mixed states: separability and distillability properties
• Evidence for Bound Entangled States with Negative Partial Transpose
• Qutrit Entanglement
• Complete Separability and Fourier representations of n-qubit states

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