联合可测性与Birkhoff-von Neumann定理
Joint measurability meets Birkhoff-von Neumann's theorem
AI总结：
本文通过将量子测量类比为概率向量，定义双重归一化张量，证明联合可测性是其数学特征，并建立类似Birkhoff-von Neumann定理的刻画，同时揭示DNTs在算子理论中的自然涌现。
摘要：
量子测量可以被解释为概率向量的一种推广，其中非负实数被正半定算子所取代。我们外推这一类比，定义了一类双随机矩阵的推广，称之为双重归一化张量（DNTs），并提出了相应的Birkhoff-von Neumann定理版本，该定理指出置换是双随机矩阵集合的极值点。我们证明，在此背景下，联合可测性作为DNTs的一个数学特征出现，这对于建立类似于Birkhoff-von Neumann定理的刻画是必要的。反之，我们也表明DNTs自然地从联合可测性问题的特定实例中产生，并指出其在一般算子理论中的相关性。
英文摘要：
Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
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