• Corpus ID: 251371350

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

@inproceedings{Farenick2022ToeplitzSE,
  title={Toeplitz separability, entanglement, and complete positivity using operator system duality},
  author={Douglas R. Farenick and Michelle McBurney},
  year={2022}
}

• D. Farenick, Michelle McBurney
• Published 5 August 2022
• Mathematics

. A new proof is presented of a theorem of L. Gurvits, which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. We also show that in the cone of positive Toeplitz matrices with Toeplitz entries, entangled elements exist in all dimensions. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C ( S 1 ) ( n ) of n × n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C ( S 1 ) ( n ) ⊗ min B ( H ) and C ( S 1 ) ( n ) ⊗ min 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 also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from the type I factor B ( H ). In particular, we prove that normal positive linear maps ψ on B ( H ) are partially completely positive in the sense that ψ ( n ) ( x ) is positive whenever x is a positive n × n Toeplitz matrix with entries from B ( H ). We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), and explain how the factorisation theorem can be used to show an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando to universality.