Quantum Teleportation and Super-Dense Coding in Operator Algebras

We show for any d,m ≥ 2 with (d,m) ≠ (2,2), the matrix-valued generalization of the (tensor product) quantum correlation set of d inputs and m outputs is not closed. Our argument uses a reformulation of super-dense coding and teleportation in terms of C∗-algebra isomorphisms. Namely, we prove that for certain actions of cyclic group {Zd, 'Equation Presented' where Bd is the universal unital C∗-algebra generated by the elements ujk, 0 ≤ i, j ≤ d-1, satisfying the relations that [uj,k] is a unitary operator, and C∗(Fd2) is the universal C∗-algebra of d2 unitaries. These isomorphisms provide a nice connection between the embezzlement of entanglement and the non-closedness of quantum correlation sets.