We give generators and relations for the hypergraph props of Gaussian relations and positive affine Lagrangian relations. The former extends Gaussian probabilistic processes by completely-uninformative priors, and the latter extends Gaussian quantum mechanics with infinitely-squeezed states. These presentations are given by adding a generator to the presentation of real affine relations and of real affine Lagrangian relations which freely codiscards effects, as well as certain rotations. The presentation of positive affine Lagrangian relations provides a rigorous justification for many common yet informal calculations in the quantum physics literature involving infinite-squeezing. Our presentation naturally extends Menicucci et al.'s graph-theoretic representation of Gaussian quantum states with a representation for Gaussian transformations. We interpret the LOv-calculus, a diagrammatic calculus for reasoning about passive linear-optical quantum circuits in our presentation of positive affine Lagrangian relations. Moreover, we show how our presentation allows for additional optical operations such as active squeezing.

翻译：我们给出了高斯关系与正仿射拉格朗日关系的超图性质的发生元和关系式。前者通过完全无信息先验扩展了高斯概率过程，后者通过无限压缩态扩展了高斯量子力学。这些表示通过在实仿射关系与实仿射拉格朗日关系的表示中增加一个自由协弃效应的发生器以及某些旋转操作而得到。正仿射拉格朗日关系的表示对量子物理文献中大量涉及无限压缩的非正式计算提供了严格的理论基础。我们的表示自然地扩展了Menicucci等人关于高斯量子态的图论表示，将其推广至高斯变换的表示。我们在正仿射拉格朗日关系的表示框架下解释了LOv-演算（一种用于分析被动线性光学量子电路的图解演算）。此外，我们展示了该表示如何容纳主动压缩等附加光学操作。