Dilations of Higher Rank Operator Tuples

在过去的三十年里，算子和算子代数理论有了巨大的发展。这两个领域都与量子理论有着密切的联系，并构成了快速发展的量子信息理论的数学支柱。在这个理论中，信息通过量子通道传递，量子通道可以用称为算子元组的数学对象来描述。这些都是算子和算子代数理论的研究对象。伸缩理论是分析算子元组的一个非常有力的工具。到目前为止，只研究了单个通道，即单个元组。在这个项目中，我们要考虑一种新类型的操作元组（高阶元组），在量子信息理论中对应于几个独立的量子通道，并为这样的元组发展一个膨胀理论。为此，我们可以利用最近发展起来的强有力的高阶图代数理论。除了这些应用，我们的结果也将在抽象算子和算子代数理论的兴趣。

There has been a tremendous development of operator and operator algebra theory in the past three decades. Both areas have strong connections to quantum theory and form the mathematical backbone of the rapidly developing theory of quantum information. In this theory information is passed on through quantum channels which can be described by mathematical objects called operator tuples. These are objects studied in operator and operator algebra theory. Dilation theory is a very powerful tool to analyze operator tuples. Up to now only single channels have been investigated i.e. single tuples. In this project we want to consider a new type of operator tuple (higher rank tuples) which in quantum information theory corresponds to several independent quantum channels and develop a dilation theory for such tuples. To this end we can make use of the powerful theory of higher rank graph algebras recently developed. Besides these applications our results would also be of interest in abstract operator and operator algebra theory.