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. Using this graphical calculus, we also give a graphical proof of Braunstein and Kimble's continuous-variable quantum teleportation protocol. We also interpret the LOv-calculus, a diagrammatic calculus for reasoning about passive linear-optical quantum circuits in our graphical calculus. 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操作（hypergraph props）的生成元与关系式。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。这些表示通过向实仿射关系与实仿射拉格朗日关系的表示中添加自由余丢弃效应（freely codiscards effects）的生成元及特定旋转得到。正仿射拉格朗日关系的表示为量子物理文献中涉及无限压缩的常见但非形式化的计算提供了严格数学基础。我们的表示自然地将Menicucci等人的高斯量子态图论表示扩展至高斯变换的表示。利用这一图式演算，我们还给出了Braunstein与Kimble连续变量量子隐形传态协议的图式证明。此外，我们阐释了LOv-演算（一种用于推理无源线性光学量子电路的图式演算）如何嵌入我们的图式演算中。最后，我们展示了该表示如何支持主动压缩等额外光学操作。