A quantum tug of war between randomness and symmetries on homogeneous spaces

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.