A parametrized family of normal states on a von Neumann algebra is called a statistical experiment, which generalizes the corresponding concepts in classical statistics and finite-dimensional quantum systems. We introduce randomization preorder and equivalence relations for statistical experiments with a fixed parameter set and for normal channels with a fixed input space by post-processing completely positive channels. In this paper, we prove that the set of equivalence classes of statistical experiments or those of normal channels is an upper and lower directed-complete partially ordered set with respect to the randomization order, i.e. any increasing or decreasing net of statistical experiments or channels has its supremum or infimum in the randomization order. We also show that if the outcome space of each statistical experiment or channel of a randomization-monotone net is commutative, the outcome space of the supremum or infimum can also be taken to be commutative. We consider two examples of homogeneous Markov processes of channels on infinite-dimensional separable Hilbert spaces, namely block-diagonalization with irrational translation and ideal quantum linear amplifier channels, and explicitly derive their infima. Throughout the paper, the concept of channel conjugation is used to obtain results for decreasing channels from those for increasing channels.

中文翻译：

---

随机顺序中量子统计实验的有向完备性

冯诺依曼代数上的参数化正常状态族称为统计实验，它概括了经典统计和有限维量子系统中的相应概念。我们通过对完全正通道进行后处理，为具有固定参数集的统计实验和具有固定输入空间的正常通道引入随机化前序和等价关系。在本文中，我们证明了统计实验或正常通道的等价类集合是关于随机化顺序的上下有向完全偏序集，即任何增加或减少的统计实验或通道网络都有它在随机化顺序中的上确界或下确界。我们还表明，如果随机化单调网络的每个统计实验或通道的结果空间是可交换的，那么上确界或下确界的结果空间也可以被认为是可交换的。我们考虑了无限维可分希尔伯特空间上通道的齐次马尔可夫过程的两个例子，即具有非理性平移的块对角化和理想的量子线性放大器通道，并明确地推导出它们的infima。在整篇论文中，通道共轭的概念用于从增加通道的结果中获得减少通道的结果。即具有非理性平移和理想量子线性放大器通道的块对角化，并显式推导出它们的infima。在整篇论文中，通道共轭的概念用于从增加通道的结果中获得减少通道的结果。即具有非理性平移和理想量子线性放大器通道的块对角化，并显式推导出它们的infima。在整篇论文中，通道共轭的概念用于从增加通道的结果中获得减少通道的结果。