Quantum Teleportation and Super-dense Coding in Operator Algebras

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Let Bd be the unital C*-algebra generated by the elements ujk, 0 ≤ i, j ≤ d-1, satisfying the relations that [uⱼ,ₖ] is a unitary operator, and let C*(Fd₂) be the full group C*-algebra of free group of d² generators. Based on the idea of teleportation and super-dense coding in quantum information theory, we exhibit the two *-isomorphisms Md(C*(Fd₂)) Bd Zd Zd and Md(Bd) C*(Fd₂) Zd Zd, for certain actions of Zd. As an application, we show that 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.