发表机构
Baruch College（巴鲁克学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文为量子记录历史构造了超度量几何，通过Bures缺陷的收敛指数定义一致性速率，并证明其等价于扇区分离度量和区分错误概率指数。
AI 中文摘要
量子力学的一致历史表述在相应干涉项消失时，为粗粒化事件序列赋予经典概率。在有限模型中，这些干涉项通常不必精确为零，而其微小性本身并不为历史空间提供渐近几何。我们为累积量子记录的历史构造了这样一种几何。当两个历史之间的归一化Dowker-Halliwell干涉因子化为其记录的累积保真度时，其衰减由累积平方Bures距离控制。在无限登记极限下，Kakutani-von Neumann二分法给出了一个尖锐的扇区选择：可和的Bures缺陷产生弱等价的记录序列，而不可和的缺陷则产生正交的超选择扇区和消失的归一化干涉。Bures缺陷的收敛指数在历史上定义了一个伪超度量。在对其零距离简并取商后，它成为一个完备的超度量，既不需要结果空间的紧致性，也不需要记录族的连续性，并且与\cite{LesTP}中引入的规范不变扇区度量$\tilde d$一致。其值，即“一致性速率”，具有三种等价解释：它是扇区分离度量、归一化干涉负对数的多项式增长指数，并且根据Helstrom公式，它是区分累积记录的最小错误概率的负对数的相应指数。经典分离轮廓决定速率：正的长时平均缺陷给出最大值$\delta=1$，而边界轮廓$d_m^2\asymp m^{-1}$在零多项式速率$\delta=0$下产生不同的超选择扇区。
英文摘要
The consistent histories formulation of quantum mechanics assigns classical probabilities to sequences of coarse-grained events when the corresponding interference terms vanish. In finite models these terms generically need not vanish exactly, and their smallness alone supplies no asymptotic geometry on the space of histories. We construct such a geometry for histories that accumulate quantum records. When the normalized Dowker - Halliwell interference between two histories factorizes as the cumulative fidelity of their records, its decay is controlled by the accumulated squared Bures distance. In the infinite registration limit, the Kakutani - von Neumann dichotomy gives a sharp sector alternative: summable Bures defect yields weakly equivalent record sequences, whereas nonsummable defect produces orthogonal superselection sectors and vanishing normalized interference. The convergence exponent of the Bures defect defines a pseudo-ultrametric on histories. After quotienting its zero distance degeneracy, it becomes a complete ultrametric, requiring neither compactness of the outcome space nor continuity of the record family, and coincides with the gauge invariant sector metric $\tilde d$ introduced in \cite{LesTP}. Its value, the \emph{consistency rate}, has three equivalent interpretations: it is the sector separation metric, the polynomial growth exponent of the negative logarithm of normalized interference, and, by the Helstrom formula, the corresponding exponent of the negative logarithm of the minimum error probability for discriminating the cumulative records. Classical separation profiles determine the rate: positive long time mean defect gives the maximal value $δ=1$, while the borderline profile $d_m^2\asymp m^{-1}$ produces distinct superselection sectors at zero polynomial rate, $δ=0$.
Comments45 pages
