The Haar measure plays a vital role in quantum information, but its study often requires a deep understanding of representation theory, posing a challenge for beginners. This tutorial aims to provide a basic introduction to Haar measure tools in quantum information, utilizing only basic knowledge of linear algebra and thus aiming to make this topic more accessible. The tutorial begins by introducing the Haar measure with a specific emphasis on characterizing the moment operator, an essential element for computing integrals over the Haar measure. It also covers properties of the symmetric subspace and introduces helpful tools like tensor network diagrammatic notation, which aid in visualizing and simplifying calculations. Next, the tutorial explores the concept of unitary designs, providing equivalent definitions, and subsequently explores approximate notions of unitary designs, shedding light on the relationships between these different notions. Practical examples of Haar measure calculations are illustrated, including the derivation of well-known formulas such as the twirling of a quantum channel. Lastly, the tutorial showcases the applications of Haar measure calculations in quantum machine learning and classical shadow tomography.

中文翻译：

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量子信息中的 Haar 测量工具简介：初学者教程

Haar测度在量子信息中起着至关重要的作用，但其研究往往需要对表示论有深入的理解，这对初学者提出了挑战。本教程旨在仅利用线性代数的基础知识，对量子信息中的 Haar 测量工具进行基本介绍，从而使该主题更容易理解。本教程首先介绍 Haar 测度，特别强调矩算子的特征，这是计算 Haar 测度积分的基本元素。它还涵盖了对称子空间的属性，并引入了有用的工具，例如张量网络图表符号，有助于可视化和简化计算。接下来，本教程探讨单一设计的概念，提供等效定义，随后探索单一设计的近似概念，阐明这些不同概念之间的关系。阐述了哈尔测度计算的实际例子，包括众所周知的公式的推导，例如量子通道的旋转。最后，本教程展示了 Haar 测度计算在量子机器学习和经典阴影断层扫描中的应用。