渐近量子信道近似的范畴论方法和Birkhoff定理
A category theoretic approach to asymptotic quantum channel approximation and Birkhoff's Theorem
摘要：
Birkhoff定理指出双随机矩阵是置换矩阵的凸组合。量子力学上，这些矩阵是双随机通道，也就是说，它们是保迹和保恒等的完全正映射。我们期望这些通道是酉通道的凸组合，但已知某些通道不能以这种方式写入。最近的研究表明，单个通道的$n$拷贝可能近似于酉的混合（凸组合）。在本文中，我们表明，$n(n+1)/2$副本的对称单位量子信道可以任意很好地近似的混合物（凸组合）的单位实现的信道。此外，我们证明了任何极端属性的通道上$n$（因此$n(n+1)/2$）副本保存。结果有可能被完全推广到包括非对称通道。
英文摘要：
Birkhoff's Theorem states that doubly stochastic matrices are convex combinations of permutation matrices. Quantum mechanically these matrices are doubly stochastic channels, i.e. they are completely positive maps preserving both the trace and the identity. We expect these channels to be convex combinations of unitary channels and yet it is known that some channels cannot be written that way. Recent work has suggested that $n$ copies of a single channel might approximate a mixture (convex combination) of unitaries. In this paper we show that $n(n+1)/2$ copies of a symmetric unital quantum channel may be arbitrarily-well approximated by a mixture (convex combination) of unitarily implemented channels. In addition, we prove that any extremal properties of a channel are preserved over $n$ (and thus $n(n+1)/2$) copies. The result has the potential to be completely generalized to include non-symmetric channels.
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。
