A prepare-and-measure scenario is naturally described by a communication matrix that collects all conditional outcome probabilities of the scenario into a row-stochastic matrix. The set of all possible communication matrices is partially ordered via the possibility to transform one matrix to another by pre- and post-processings. By considering maximal elements in this preorder for a subset of matrices implementable in a given theory, it becomes possible to identify communication matrices of maximum utility, i.e., matrices that are not majorized by any other matrices in the theory. The identity matrix of an appropriate size is the greatest element in classical theories, while the maximal elements in quantum theory have remained unknown. We completely characterize the maximal elements in quantum theory, thereby revealing the essential structure of the set of quantum communication matrices. In particular, we show that the identity matrix is the only maximal element in quantum theory but, as opposed to a classical theory, it is not the greatest element. Quantum theory can hence be seen to be distinct from classical theory by the existence of incompatible communication matrices.

中文翻译：

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量子通信的最大元素


准备和测量场景自然由通信矩阵描述，该矩阵将场景的所有条件结果概率收集到行随机矩阵中。所有可能的通信矩阵的集合通过前处理和后处理将一个矩阵转换为另一个矩阵的可能性，部分排序。通过考虑给定理论中可实现的矩阵子集的此前序中的最大元素，可以识别具有最大效用的通信矩阵，即理论中未被任何其他矩阵主修的矩阵。适当大小的单位矩阵是经典理论中的最大元素，而量子理论中的最大元素仍然是未知的。我们完全描述了量子理论中的最大元素，从而揭示了量子通信矩阵集的基本结构。特别是，我们表明单位矩阵是量子理论中唯一的最大元素，但与经典理论相反，它不是最大的元素。因此，量子理论与经典理论的不同之处在于存在不兼容的通信矩阵。