Continuous-variable multipartite entanglement is a key resource for quantum technologies. This paper considers the multipartite entanglement generated in separated spatial modes of the same light beam by three different parametric sources: a standard ${\chi }^{\left(2\right)}$ medium pumped by two pumps, a single-pump nonlinear photonic crystal, and a doubly pumped nonlinear photonic crystal. These sources have in common the coexistence of several concurrent nonlinear processes in the same medium, which allows the generation of nonstandard three- and four-mode couplings. We test the genuine nature of the multipartite entangled states thereby generated in a common framework, using criteria based both on proper bounds for the variances of nonlocal observables and on the positive partial transpose criterion. The relative simplicity of these states allows a (hopefully) useful comparison of the different inseparability tests.

中文翻译：

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由并发非线性过程产生的多方空间纠缠

连续变量多部分纠缠是量子技术的关键资源。本文考虑了三个不同参量源在同一光束的分离空间模式中产生的多分纠缠：一个标准${\chi }^{\left(2\right)}$由两个泵泵浦的介质，一个单泵非线性光子晶体，一个双泵浦非线性光子晶体。这些源的共同点是在同一介质中同时存在多个并发非线性过程，这允许生成非标准的三模和四模耦合。我们使用基于非局部可观察量方差的适当界限和正部分转置标准的标准来测试由此在公共框架中生成的多部分纠缠状态的真实性质。这些状态的相对简单性允许（希望）对不同的不可分离性测试进行有用的比较。