Symplectic vector spaces are the phase space of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays of the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.
翻译:中位矢量空间是线性机械系统的阶段空间。 共位形式描述了位置和动力以及当前和电压之间的关系。 相位矢量空间之间的线性拉格朗格关系类别是一个对称的单向子关系类别, 它为各种物理系统的演化提供了一种语义, 也为各种物理系统的演化提供了更一般的线性限制。 我们用一个“ 双向” 的线性关系类别, 将拉格朗格人关系类别作为任意的域。 更确切地说, 我们显示, 由Selinger 的 CPM 构造的变异性适用于线性关系, 在线性关系中, 共位性或共位调的配方关系是共振作用的一个对称。 此外, 在正位关系上的线性关系中, 这与CPM 的构造完全对应, 以适当选择匕首端。 我们还可以通过一个单一的松式转换操作者来扩展这一构造, 以获得“ 双向” 线性关系类别。 。 更确切地说, 我们用这一新的演示来证明, 离心性平极的平极理论的平流( 包括平流的平流的平流的平流的平极) 。