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 suppressing weight function, and can be approached using generalized 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 , although our construction can be applied to other cases. Possible applications of our results cover finding finite ensembles of random operations in quantum information science and quantum optics, which can be used in constructions of randomised quantum algorithms, including optical interferometric implementations.

在快速发展的物理学分支（如量子信息科学或量子光学）的许多背景下，李群上的物理量取平均值出现。这样的平均过程总是可以表示为对群中有限数量的元素求平均，称为有限平均集。在之前的研究中，这种称为t- designs 的集合仅在对酉组求平均的情况下构建（因此名称为unitary t- designs）。在这项工作中，我们研究了构造有限平均集以对一般非紧矩阵 Lie 群求平均的问题，由于群流形上的一致不变测度（Haar 测度）是无限的，因此这是一项更加微妙的任务. 我们基于群的 Cartan 分解提供了此类集合的一般构造，它将群分为紧致和非紧致组件。对紧凑部分的平均可以以统一的方式完成，而对非紧凑部分的平均必须具有抑制权重函数，并且可以使用广义高斯求积来实现。这使我们得到关于紧和非紧部分的有限平均集的乘积形式的半单矩阵李群的有限平均集的一般形式。，虽然我们的构造可以应用于其他情况。我们的结果的可能应用包括在量子信息科学和量子光学中寻找随机操作的有限集合，这些集合可用于构建随机量子算法，包括光学干涉实现。