The [Formula: see text]-bit is the [Formula: see text]-deformation of the [Formula: see text]-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the [Formula: see text]-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of [Formula: see text]-deformation (generalizing the Hudson–Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right [Formula: see text]-JW-embeddings of [Formula: see text] ([Formula: see text] complex matrices) into the infinite tensor product [Formula: see text], proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of [Formula: see text]-Levy processes on ∗-bi-algebras. For [Formula: see text], left [Formula: see text]-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For [Formula: see text], they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all [Formula: see text]. The moments of the limit random variable coincide with those found by Parthasarathy in the case [Formula: see text]. We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case [Formula: see text] and a non-trivial deformation of it for [Formula: see text]. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paper1 deals with the functional extension of this CLT, leading to a process. In paper2 the left [Formula: see text]-JW–embeddings are replaced by symmetric [Formula: see text]-embeddings. The radical differences between the results of the present paper and those of2 raise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paper3 for CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.

中文翻译：

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qq-bit (I)：左 q-Jordan-Wigner 嵌入的中心极限，单调交互 Fock 空间，Azema 随机变量，q 的概率含义

[公式：见文]-位是[公式：见文]-位的[公式：见文]-变形。它典型地源于伯努利随机变量的量子分解，并且 [公式：见正文] 参数具有作为给定随机变量的不对称指数的自然概率和物理解释。Parthasarathy 建立了一种新型的[公式：见正文]-变形（概括了 Hudson-Parthasarathy 玻色子化技术，与通常的不同）与 Azema 鞅之间的联系。受这个结果的启发，Schürmann首先将[Formula: see text]（[Formula: see text]复矩阵）的left和right [Formula: see text]-JW-embeddings引入到无穷张量积[Formula: see text]中，在 *-bi-algebras 的上下文中证明了基于这些嵌入的中心极限定理 (CLT)，并构建了 [Formula: see text]-Levy processes on *-bi-algebras 的一般理论。对于 [Formula: see text]，left [Formula: see text]-JW-embeddings 定义了 Jordan-Wigner 变换，用于构建费米反对易关系（玻色子化）的张量表示。对于 [公式：见正文]，它们简化为通常的张量嵌入，这是由于 von Waldenfels 而成为第一个量子 CLT 的基础。本文是我们在张量积背景下研究这些定理的四篇系列论文中的第一篇。我们证明了所有 [公式：见正文] 的 CLT 的收敛性。极限随机变量的矩与 Parthasarathy 在案例 [公式：见正文] 中发现的矩一致。我们证明了表示极限随机变量的空间不是像 Parthasarathy 中那样的 Boson Fock 空间，而是在 [Formula: see text] 的情况下的单调 Fock 空间和对于 [Formula: see文本]。证明中的主要分析工具是最近证明的多维高阶 Cesaro 型定理的重要扩展。本论文处理标准CLT，即极限是单个随机变量。纸 本论文处理标准CLT，即极限是单个随机变量。纸 本论文处理标准CLT，即极限是单个随机变量。纸1处理这个 CLT 的功能扩展，导致一个过程。在纸上2左边的[公式：见正文]-JW-嵌入被对称的[公式：见正文]-嵌入替换。本文的结果与本文的结果之间的根本差异2提出问题来表征那些限制空间提供所有基础经典随机变量的规范分解的 CLT（有关此问题的起源，请参见本论文的引言 4.5 和第 5 节）。这个问题在论文中得到解决3对于与满足广义 Fock 属性的状态相关联的 CLT。本系列中考虑的状态具有此属性。