论文标题
分类量子协议的图表结石
A diagrammatic calculus for categorical quantum protocols
论文作者
论文摘要
如Abramsky和Coecke所示，可以根据具有双脂肪的匕首紧凑的类别来研究量子力学。在这种结构中，可以描述许多众所周知的量子协议，并且可以通过在该类别中建立某些图的交换性来显示其有效性。在本文中，我们提出了具有足够结构的类别的明确实现，以检查某些类别的量子协议的有效性。为了做到这一点，我们根据有限集自由生成的一组元素的一维元素构建类别。我们将此类别用作图形语言，并表明它是用双脂肪封闭的匕首紧凑。然后，依靠凯利（Kelly）和拉普拉扎（Laplaza）证明的紧凑型封闭类别的相干结果，我们显示了连贯性结果，这使我们能够仅通过绘制图表来检查量子协议的有效性。特别是，我们显示了量子传送，纠缠交换（如在艾布拉姆斯基和科克的工作中提出的）和超密集编码方案的有效性。
As shown by Abramsky and Coecke, quantum mechanics can be studied in terms of dagger compact closed categories with biproducts. Within this structure, many well-known quantum protocols can be described and their validity can be shown by establishing the commutativity of certain diagrams in that category. In this paper, we propose an explicit realisation of a category with enough structure to check the validity of a certain class of quantum protocols. In order to do this, we construct a category based on 1-dimensional cobordisms with attached elements of a certain group freely generated by a finite set. We use this category as a graphical language, and we show that it is dagger compact closed with biproducts. Then, relying on the coherence result for compact closed categories, proved by Kelly and Laplaza, we show the coherence result, which enables us to check the validity of quantum protocols just by drawing diagrams. In particular, we show the validity of quantum teleportation, entanglement swapping (as formulated in the work of Abramsky and Coecke) and superdense coding protocol.
