AI 中文总结
本文通过最小修改自由薛定谔流使其对所有正动量支撑态非负，导出Kijowski到达时间POVM，并证明在成对最小构造下结果唯一。
AI 中文摘要
量子回流在此指对于动量支撑完全为正的态，出现负的薛定谔流。我们寻求对自由薛定谔流的最小修改，使得它对每个这样的态都非负，同时保留每个单独动量分量的流。我们证明所需的最小修改将自由粒子动量核按照 \\[ K_{\rm Sch}(p,p')=\frac{p+p'}{2m} \\;\longrightarrow\\; K_{\min}(p,p')=\frac{\sqrt{pp'}}{m} \\] 改变。所得流是正的且归一化的，因此定义了一个到达时间POVM。将方向性无回流要求扩展到包含两种动量符号的态，迫使跨扇区核消失，\\[ K_{\min}^{+-}=K_{\min}^{-+}=0, \\] 从而完整流是两个独立方向贡献之和。所得POVM正是Kijowski到达时间POVM，为其方向性核及其分离提供了基于流的物理动机。在本文考虑的对角保持成对最小构造中，结果是唯一的。该构造本身不施加首次到达条件。
英文摘要
Quantum backflow refers here to the appearance of a negative Schrödinger current for a state whose momentum support is entirely positive. We ask for the smallest modification of the free Schrödinger current that makes it nonnegative for every such state, while preserving the current of each individual momentum component. We show that the required minimal modification changes the free-particle momentum kernel according to \[ K_{\rm Sch}(p,p')=\frac{p+p'}{2m} \;\longrightarrow\; K_{\min}(p,p')=\frac{\sqrt{pp'}}{m}. \] The resulting current is positive and normalized and therefore defines an arrival-time POVM. Extending the directional no-backflow requirement to states containing both momentum signs forces the cross-sector kernel to vanish, \[ K_{\min}^{+-}=K_{\min}^{-+}=0, \] so that the full current is the sum of two independent directional contributions. The resulting POVM is exactly the Kijowski time-of-arrival POVM, providing a current-based physical motivation for both its directional kernels and their separation. Within the diagonal-preserving pairwise-minimal construction considered here, the result is unique. The construction itself does not impose a first-arrival condition.
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