On construction of finite averaging sets for $SL(2, \mathbb{C})$ via its Cartan decomposition

Abstract: Averaging physical quantities over Lie groups appears in many contexts across the rapidly developing branches of physics like quantum information science or quantum optics. Such an averaging process can be always represented as averaging with respect to a finite number of elements of the group, called a finite averaging set. In the previous research such sets, known as $t$-designs, were constructed only for the case of averaging over unitary groups (hence the name unitary $t$-designs). In this work we investigate the problem of constructing finite averaging sets for averaging over general non-compact matrix Lie groups, which is much more subtle task due to the fact that the the uniform invariant measure on the group manifold (the Haar measure) is infinite. We provide a general construction of such sets based on the Cartan decomposition of the group, which splits the group into its compact and non-compact components. The averaging over the compact part can be done in a uniform way, whereas the averaging over the non-compact one has to be endowed with a suppresing weight function, and can be approached using generalised Gauss quadratures. This leads us to the general form of finite averaging sets for semisimple matrix Lie groups in the product form of finite averaging sets with respect to the compact and non-compact parts. We provide an explicit calculation of such sets for the group $SL(2, \mathbb{C})$, although our construction can be applied to other cases. Possible applications of our results cover finding finite ensambles of random operations in quantum information science and quantum optics, which can be used in constructions of randomised quantum algorithms, including optical interferometric implementations.