Quantum Teleportation and Super-Dense Coding in Operator Algebras
Quantum Teleportation and Super-Dense Coding in Operator Algebras
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DOI：
10.1093/imrn/rnz095
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发表时间：
2017-09
影响因子：
1
通讯作者：
Li Gao;Samuel J. Harris;M. Junge
中科院分区：
文献类型：
--
作者：
Li Gao;Samuel J. Harris;M. Junge
We show 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. 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 ${{\mathbb{Z}}}_d$, $$\begin{equation*}M_d(C^*({{\mathbb{F}}}_{d^2}))\cong{{\mathcal{B}}}_d\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d , M_d({{\mathcal{B}}}_d)\cong C^*({{\mathbb{F}}}_{d^2})\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d,\end{equation*}$$where ${{\mathcal{B}}}_d$ is the universal 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 $C^*({{\mathbb{F}}}_{d^2})$ is the universal $C^*$-algebra of $d^2$ unitaries. These isomorphisms provide a nice connection between the embezzlement of entanglement and the non-closedness of quantum correlation sets.
We show 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. 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 ${{\mathbb{Z}}}_d$, $$\begin{equation*}M_d(C^*({{\mathbb{F}}}_{d^2}))\cong{{\mathcal{B}}}_d\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d , M_d({{\mathcal{B}}}_d)\cong C^*({{\mathbb{F}}}_{d^2})\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d,\end{equation*}$$where ${{\mathcal{B}}}_d$ is the universal 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 $C^*({{\mathbb{F}}}_{d^2})$ is the universal $C^*$-algebra of $d^2$ unitaries. These isomorphisms provide a nice connection between the embezzlement of entanglement and the non-closedness of quantum correlation sets.