Quantum Teleportation and Super-dense Coding in Operator Algebras

Abstract

Let ${\mathcal{B}}_d$ be the unital $C^{\ast }$-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 $C^{\ast }({\mathbb{F}}_{d^2})$ be the full group $C^{\ast }$-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 $\ast$-isomorphisms $M_d(C^{\ast }({\mathbb{F}}_{d^2}))\cong {\mathcal{B}}_d⋊{\mathbb{Z}}_d⋊{\mathbb{Z}}_d$ and $M_d({\mathcal{B}}_d)\cong C^{\ast }({\mathbb{F}}_{d^2})⋊{\mathbb{Z}}_d⋊{\mathbb{Z}}_d$, for certain actions of ${\mathbb{Z}}_d$. As an application, we show that for any $d,m\ge 2$ with $(d,m)\ne (2,2)$, the matrix-valued generalization of the (tensor product) quantum correlation set of $d$ inputs and $m$ outputs is not closed.

28pages. Comments are welcome!

References in corpus (2)

• The mother of all protocols: Restructuring quantum information's family tree
• Connes' embedding problem and Tsirelson's problem