Quantum Teleportation and Super-dense Coding in Operator Algebras

Let $\mathcal{B}_d$ be the unital $C^*$-algebra generated by the elements $u_{jk}, \, 0 \le i, j \le d-1$, satisfying the relations that $[u_{j,k}]$ is a unitary operator, and let $C^*(\mathbb{F}_{d^2})$ be the full group $C^*$-algebra of free group of $d^2$ generators. Based on the idea of teleportation and super-dense coding in quantum information theory, we exhibit the two $*$-isomorphisms $M_d(C^*(\mathbb{F}_{d^2}))\cong \mathcal{B}_d\rtimes \mathbb{Z}_d\rtimes \mathbb{Z}_d$ and $M_d(\mathcal{B}_d)\cong C^*(\mathbb{F}_{d^2})\rtimes \mathbb{Z}_d\rtimes \mathbb{Z}_d$, for certain actions of $\mathbb{Z}_d$. As an application, we show that for any $d,m\ge 2$ with $(d,m)\neq (2,2)$, the matrix-valued generalization of the (tensor product) quantum correlation set of $d$ inputs and $m$ outputs is not closed.