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A new proof is presented of a theorem of L.~Gurvits, which states that thecone of positive block-Toeplitz matrices with matrix entries has no entangledelements. The proof of the Gurvits separation theorem is achieved by making useof the structure of the operator system dual of the operator system$C(S^1)^{(n)}$ of $n\times n$ Toeplitz matrices over the complex field, and bydetermining precisely the structure of the generators of the extremal rays ofthe positive cones of the operator systems $C(S^1)^{(n)}\omin\B(\H)$ and$C(S^1)_{(n)}\omin\B(\H)$, where $\H$ is an arbitrary Hilbert space and$C(S^1)_{(n)}$ is the operator system dual of $C(S^1)^{(n)}$. Our approach alsohas the advantage of providing some new information concerning positiveToeplitz matrices whose entries are from $\B(\H)$ when $\H$ has infinitedimension. In particular, we prove that normal positive linear maps $\psi$ on$\B(\H)$ are partially completely positive in the sense that $\psi^{(n)}(x)$ ispositive whenever $x$ is a positive $n\times n$ Toeplitz matrix with entriesfrom $\B(\H)$. We also establish a certain factorisation theorem for positiveToeplitz matrices (of operators), showing an equivalence between the Gurvitsapproach to separation and an earlier approach of T.~Ando to universality.

Toeplitz separability, entanglement, and complete positivity using operator system duality