We discuss the Gaussian and the mixture of Gaussians in the limit of open quantum random walks. The central limit theorems for the open quantum random walks under certain conditions were proven by Attal et al (Ann Henri Poincaré 16(1):15–43, 2015) on the integer lattices and by Ko et al (Quantum Inf Process 17(7):167, 2018) on the crystal lattices. The purpose of this paper is to investigate the general situation. We see that the Gaussian and the mixture of Gaussians in the limit depend on the structure of the invariant states of the intrinsic quantum Markov semigroup whose generator is given by the Kraus operators which generate the open quantum random walks. Some concrete models are considered for the open quantum random walks on the crystal lattices. Due to the intrinsic structure of the crystal lattices, we can conveniently construct the dynamics as we like. Here, we consider the crystal lattices of \(\mathbb {Z}^2\) with intrinsic two points, hexagonal, triangular, and Kagome lattices. We also discuss Fourier analysis on the crystal lattices which gives another method to get the limit theorems.

中文翻译：

---

开放量子随机游动中的高斯混合

我们讨论在开放量子随机游动极限中的高斯和高斯混合。Attal等人（Ann HenriPoincaré16（1）：15–43，2015）在整数格上以及Ko等人（Quantf Inf Process 17（7）证明了在某些条件下开放量子随机游走的中心极限定理。 ）：167，2018）在晶格上。本文的目的是调查一般情况。我们看到高斯和极限中的高斯混合取决于本征量子马尔可夫半群的不变态的结构，其固有生成器由产生开放量子随机步态的克劳斯算子给出。对于晶格上的开放量子随机游动，考虑了一些具体模型。由于晶格的固有结构，我们可以根据需要方便地构建动力学。在这里，我们考虑\（\ mathbb {Z} ^ 2 \）具有固有的两个点，即六角形，三角形和Kagome晶格。我们还讨论了对晶格的傅立叶分析，这提供了另一种获得极限定理的方法。