论文标题
Kirchhoff关系类别
Categories of Kirchhoff relations
论文作者
论文摘要
众所周知，仿射拉格朗关系（Affine Lagrangian关系）类别，afflagrel_f，在一个字段，f，整数a modulo a prime p（带有p> 2）是p-dits稳定器量子回路类别的同构。此外，众所周知，电路（概括为field F）是afflagrel_f的自然子类别。本文的目的是在错误检测中使用的奇偶校验检查和生成器矩阵中提供此子类别中关系的表征。 特别是，我们介绍了包括基希霍夫关系组成的子类别（亲切地）那些保留了总动量或同等地满足基希霍夫现行法律的拉格朗日关系。该子类别中的地图可以由电气组件（概括为f）生成：电阻器，当前分隔线以及电流和电压源。 “源”电源组件提供了地图的仿射性，而当前的分隔线则增加了一个有趣的准故事方面。 我们从奇偶校验和检查矩阵方面表征了这些Kirchhoff的关系，此外，还表征了两个重要的子类别：确定性的Kirchhoff关系和无损关系。确定性的基尔chhoff关系的类别是电路，由电阻电路产生。确定性的基尔霍夫（Kirchhoff）的无损关系完全提供了这些设置的基本超分类结构。
It is known that the category of affine Lagrangian relations, AffLagRel_F, over a field, F, of integers modulo a prime p (with p > 2) is isomorphic to the category of stabilizer quantum circuits for p-dits. Furthermore, it is known that electrical circuits (generalized for the field F) occur as a natural subcategory of AffLagRel_F. The purpose of this paper is to provide a characterization of the relations in this subcategory -- and in important subcategories thereof -- in terms of parity-check and generator matrices as used in error detection. In particular, we introduce the subcategory consisting of Kirchhoff relations to be (affinely) those Lagrangian relations that conserve total momentum or equivalently satisfy Kirchhoff's current law. Maps in this subcategory can be generated by electrical components (generalized for the field F): namely resistors, current dividers, and current and voltage sources. The "source" electrical components deliver the affine nature of the maps while current dividers add an interesting quasi-stochastic aspect. We characterize these Kirchhoff relations in terms of parity-check matrices and in addition, characterizes two important subcategories: the deterministic Kirchhoff relations and the lossless relations. The category of deterministic Kirchhoff relations as electrical circuits are generated by resistors circuits. Lossless relations, which are deterministic Kirchhoff, provide exactly the basic hyper-categorical structure of these settings.
