论量子信道：极点、拓扑与范畴性质
On quantum channels: extreme points, topology, and categorical properties
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中文总结 AI 辅助
该论文从数学角度研究量子信道，构造了不同秩的极点CPTP与UCPTP映射，证明了CPTP映射张量积的极点性质，补充了其范畴分类并探究了相关拓扑。
中文摘要 AI 辅助
本论文从数学视角研究量子信道。我们研究了CPTP（完全正定保迹）信道与UCPTP信道（后者为单位信道），这两类信道的集合均为紧凸集，因此是其极点的闭包。我们针对所有可能的秩构造了极点CPTP映射；针对每个可能的维度构造了秩为2的极点UCPTP映射，针对秩至少为3的情况，为多个维度与秩构造了极点UCPTP映射。我们还证明，极点CPTP映射的张量积仍为极点CPTP映射；而对于UCPTP映射，我们证明该结论并非总是成立，并给出了多个反例。已知CPTP映射的范畴是半笛卡尔的，我们在此基础上补充了二元积的分类，证明仅在平凡情形下存在积；对于二元余积，我们得到了其存在时应具备的维度。上述研究均使用了拓扑技术，因此我们还探究了CPTP映射极点集合的闭包的拓扑。拓扑不变量可通过空间分解（如CW结构）计算，CW结构是将空间表示为球的粘合的一类分解。在寻找极点CPTP映射集合的闭包的CW结构的过程中，我们得到了一种类似复射影空间CW结构的部分分解。
英文摘要
In this thesis, quantum channels are studied from the point of view of Mathematics. We studied the CPTP (completely positive trace preserving) and UCPTP channels, the latter being the unital channels. In both cases, the sets are compact convex, so they are the closure of their extreme points. We constructed extreme CPTP maps for every possible rank. We also constructed extreme UCPTP maps with rank 2 for each possible dimension. For ranks of at least 3, we constructed extreme UCPTP maps for many dimensions and ranks. We also proved that the tensor product of extreme CPTP maps is an extreme CPTP map. For UCPTP maps, we proved that the same is not always true, with many counterexamples. It is known that the category of CPTP maps is semicartesian. We add to this a classification of binary products. We proved that products only exist in trivial cases. For binary coproducts, we obtained the dimensions they should have, if they exist. In both cases, topological techniques were used. For this reason, we also investigated the topology of the closure of the set of extreme points of the CPTP maps. Topological invariants can be computed from decompositions of a space such as CW structures. These are a type of decomposition in which the space is expressed as a gluing of balls. In the search for a CW structure for the closure of the set of extreme CPTP maps, we obtained a partial decomposition that resembles a CW structure of the complex projective space.
