几乎幂等量子信道与近似$C^*$-代数
Almost-idempotent quantum channels and approximate $C^*$-algebras
AI总结：
针对$η$-幂等的单位完全正（UCP）映射，构造了关联的$\u03b5=O(η)$-$C^*$代数，证明其与真正$C^*$代数的$O(\u03b5)$同构性，且同构可由UCP映射实现，给出量子信道的近似分解。
摘要：
设$Φ$为某希尔伯特空间上算子空间的单位完全正（UCP）映射。我们假设$Φ$是$η$-幂等的，即$\|Φ^2-Φ\|_{\rm cb} ≤η$，并为$\u03b5=O(η)$构造了一个关联的$\u03b5$-$C^*$代数（几乎不变可观测量的代数）。这类结构具有单位$C^*$代数的公理，但结合律及其他涉及乘法和单位元的公理仅在$\u03b5$范围内成立。我们证明，任何有限维$\u03b5$-$C^*$代数$A$都与某个真正的$C^*$代数$B$是$O(\u03b5)$同构的。这些界是普适的，即不依赖于维数或其他参数。当$A$来自有限维$η$-幂等UCP映射$Φ$时，$O(η)$同构及其逆可由UCP映射实现，这给出了量子信道$Φ^*$分解为解码信道、生成$B$上态和编码信道的近似分解。
英文摘要：
Let $Φ$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $Φ$ is $η$-idempotent, namely, $\|Φ^2-Φ\|_{\mathrm{cb}} \leη$, and construct an associated $\varepsilon$-$C^*$ algebra (of almost-invariant observables) for $\varepsilon=O(η)$. This type of structure has the axioms of a unital $C^*$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\varepsilon$. We prove that any finite-dimensional $\varepsilon$-$C^*$ algebra $A$ is $O(\varepsilon)$-isomorphic to some genuine $C^*$ algebra $B$. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When $A$ comes from a finite-dimensional $η$-idempotent UCP map $Φ$, the $O(η)$-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel $Φ^*$ into a decoding channel, producing a state on $B$, and an encoding channel.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
