Quantum tug of war between randomness and symmetries on homogeneous spaces - Rahul Arvind; Kishor Bharti; Jun Yong Khoo; Dax Enshan Koh; Jian Feng Kong (Physical Review Research)
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 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. Our work addresses this gap, providing an example 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 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 Ansätze in homogeneous spaces. Our work provides a fresh perspective on the relationship between randomness and symmetry in the quantum world.