We explore the interplay between symmetry and randomness in quantum information. Adopting a geometric approach, we consider states as $H$-equivalent if related by a symmetry transformation characterized by the group $H$. We then introduce the Haar measure on the homogeneous space $\mathbb{U}/H$, characterizing true randomness for $H$-equivalent systems. While this mathematical machinery is well-studied by mathematicians, it has seen limited application in quantum information: we believe our work to be the first instance of utilizing homogeneous spaces to characterize symmetry in quantum information. This is followed by a discussion of approximations of true randomness, commencing with $t$-wise independent approximations and defining $t$-designs on $\mathbb{U}/H$ and $H$-equivalent states. Transitioning further, we explore pseudorandomness, defining pseudorandom unitaries and states within homogeneous spaces. Finally, as a practical demonstration of our findings, we study the expressibility of quantum machine learning ansatze in homogeneous spaces. Our work provides a fresh perspective on the relationship between randomness and symmetry in the quantum world.

翻译：我们探索量子信息中对称性与随机性之间的相互作用。采用几何方法，我们将通过群$H$表征的对称变换相关联的状态视为$H$等价。随后，我们在齐性空间$\mathbb{U}/H$上引入哈尔测度，用以表征$H$等价系统的真正随机性。尽管这一数学工具已被数学家深入研究，但其在量子信息中的应用有限：我们相信，本文是首次利用齐性空间来刻画量子信息中的对称性。接着，我们讨论了真正随机性的近似，从$t$阶独立近似开始，定义了$\mathbb{U}/H$上的$t$设计以及$H$等价状态。进一步，我们探索了伪随机性，在齐性空间内定义了伪随机酉算子与状态。最后，作为我们研究成果的实际演示，我们研究了量子机器学习拟设在齐性空间中的表达能力。我们的工作为量子世界中随机性与对称性之间的关系提供了全新视角。