AI 中文总结
本文证明了与量子理论相关的两个随机矩阵新定理，即玻恩表示定理和单随机矩阵定理，揭示了单随机矩阵与量子酉演化的联系，并提及了其在离散时间过程中的潜在应用。
AI 中文摘要
本文针对随机矩阵给出了两个新定理的自包含构造性证明，这两个定理与量子理论直接相关。第一个定理被称为玻恩表示定理，表明任意随机矩阵的每个元素都可表示为矩阵对乘积的迹，其中对乘积的第一个因子属于正算子值测度（POVM），第二个因子属于投影值测度（PVM）。顾名思义，该定理意味着任意随机矩阵的元素可通过量子理论玻恩规则的广义形式来表示。由此可推导出一个推论：若第一个定理中的POVM是PVM，则该随机矩阵是单随机矩阵，即其每个元素都是同规模酉矩阵对应元素的模平方。本文证明的第二个定理被称为单随机矩阵定理，表明在必要时通过用有限个额外维度扩张底层向量空间，任意随机矩阵的每个元素都可表示为对乘积的迹，其中两个因子均属于PVM，因此可通过从更大的单随机矩阵中进行边缘化得到。该定理确立了单随机矩阵相对于随机矩阵的某种首要性，并暗示其与量子理论中的酉时间演化存在密切联系。本文最后简要讨论了其在离散时间确定性过程和马尔可夫链中的潜在应用。
英文摘要
This paper presents self-contained, constructive proofs of two new theorems about stochastic matrices, with direct relevance to quantum theory. The first theorem, herein called the Born Representation Theorem, shows that each entry of any stochastic matrix can be expressed as the trace of a pairwise product of matrices, where the first factor in the pairwise product belongs to a positive-operator-valued measure (POVM) and the second factor belongs to a projection-valued measure (PVM). As its name suggests, this theorem entails that the entries of any stochastic matrix can be expressed in terms of a generalized version of the quantum-theoretic Born rule. It follows as a corollary that if the POVM in this first theorem is a PVM, then the stochastic matrix is unistochastic, meaning that its entries are each the modulus square of the corresponding entry of a unitary matrix of the same size. The second theorem proved in this paper, called the Unistochastic Theorem, then shows that by dilating the underlying vector space by a bounded number of additional dimensions if necessary, each entry of any stochastic matrix can be expressed in terms of the trace of a pairwise product for which both factors belong to PVMs, and can thus be derived via marginalization from a larger unistochastic matrix. This second theorem therefore establishes a kind of primacy of unistochastic matrices over stochastic matrices, and hints at a close connection with unitary time evolution in quantum theory. The paper concludes with a brief discussion of potential applications to discrete-time deterministic processes and Markov chains.
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