Complementarity between one-particle visibility and two-particle visibility in discrete systems can be extended to bipartite quantum-entangled Gaussian states implemented with continuous-variable quantum optics. The meaning of the two-particle visibility originally defined by Jaeger, Horne, Shimony, and Vaidman with the use of an indirect method that first corrects the two-particle probability distribution by adding and subtracting other distributions with varying degree of entanglement, however, deserves further analysis. Furthermore, the origin of complementarity between one-particle visibility and two-particle visibility is somewhat elusive and it is not entirely clear what is the best way to associate particular two-particle quantum observables with the two-particle visibility. Here, we develop a direct method for quantifying the two-particle visibility based on measurement of a pair of two-particle observables that are compatible with the measured pair of single-particle observables. For each of the two-particle observables from the pair is computed corresponding visibility, after which the absolute difference of the latter pair of visibilities is considered as a redefinition of the two-particle visibility. Our approach reveals an underlying mathematical symmetry as it treats the two pairs of one-particle or two-particle observables on equal footing by formally identifying all four observable distributions as rotated marginal distributions of the original two-particle probability distribution. The complementarity relation between one-particle visibility and two-particle visibility obtained with the direct method is exact in the limit of infinite Gaussian precision where the entangled Gaussian state approaches an ideal Einstein-Podolsky-Rosen state. The presented results demonstrate the theoretical utility of rotated marginal distributions for elucidating the nature of two-particle visibility and provide tools for the development of quantum applications employing continuous variables.

中文翻译：

---

二分纠缠高斯态中的单粒子和双粒子可见性

离散系统中单粒子可见性和两粒子可见性之间的互补性可以扩展到用连续可变量子光学实现的二分量子纠缠高斯态。最初由 Jaeger、Horne、Shimony 和 Vaidman 使用间接方法定义的双粒子可见性的含义，该方法首先通过添加和减去具有不同纠缠度的其他分布来校正双粒子概率分布，但是，值得进一步的分析。此外，单粒子可见性和双粒子可见性之间互补性的起源有些难以捉摸，并且不完全清楚将特定的双粒子量子可观测性与双粒子可见性相关联的最佳方式是什么。这里，我们开发了一种直接方法来量化双粒子能见度，该方法基于对与所测量的单粒子可观察值对兼容的一对双粒子可观察值的测量。对于来自这对双粒子可观测量中的每一个，计算相应的可见性，然后将后一对可见性的绝对差视为双粒子可见性的重新定义。我们的方法揭示了潜在的数学对称性，因为它通过将所有四个可观测分布正式识别为原始双粒子概率分布的旋转边缘分布来平等对待两对单粒子或双粒子可观测值。直接法得到的单粒子能见度和双粒子能见度之间的互补关系在无限高斯精度极限内是精确的，其中纠缠的高斯态接近理想的爱因斯坦-波多尔斯基-罗森态。所呈现的结果证明了旋转边缘分布在阐明两粒子可见性的性质方面的理论效用，并为使用连续变量的量子应用的开发提供了工具。