AI 中文总结
本文对含固定结果投影的投影测量的酉轨道进行分类，证明精细化一致性等价于上下文无关性，结合Gleason定理得到概率的Born形式，给出轨道图的结构刻画。
AI 中文摘要
设$\u2702{H}$为有限维复希尔伯特空间，$w(P,\u2702{M})$为分配给投影测量$\u2702{M}$中出现的结果投影$P$的归一化概率权重。对于固定的$P$，令$G_P\frown\u2702{U}(P^\u2702)$为在$\text{ran}P$上作用相同、在$P^\u2702$上任意作用的酉群。我们对包含$P$的投影测量的$G_P$轨道进行分类：两个测量处于同一轨道当且仅当它们的互补结果的秩多重集一致。轮廓为$\u2702=(r_1,\u2026,r_k)$的轨道是实维数为$(d-\text{rank}P)^2-\u2211_j r_j^2$的紧齐性空间。在最大秩一测量上只有一个轨道，因此上下文无关性等价于$G_P$不变性，且对应的均匀缺陷相等；仅在两能级互补酉下的不变性就已足够。对于任意投影测量，每个包含结果$P\neq I$的上下文都可粗化为唯一的二元上下文$\u2702P,I-P\u2703$。因此，仅精细化一致性就等价于上下文无关性。有限轨道空间具有自然的秩轮廓精细化图，当$P\neq I$时其直径为$d-\text{rank}P-1$。我们证明了一个稳定性定理，将二元粗化路径与该图中最短路径后接一个互补酉的情况进行比较。在维数至少为3时，Gleason定理将各自定义域上的最大上下文不变性条件和全上下文精细化条件转化为Born形式$\text{Tr}(\u2702P)$。这些结果是结构性刻画，而非上下文无关性的独立物理推导。
英文摘要
Let $\mathcal H$ be a finite-dimensional complex Hilbert space, and let $w(P,\mathsf M)$ be a normalized probability weight assigned to an outcome projection $P$ as it occurs in a projective measurement $\mathsf M$. For fixed $P$, let $G_P\cong\mathcal U(P^\perp)$ be the group of unitaries acting identically on $\operatorname{ran}P$ and arbitrarily on $P^\perp$. We classify the $G_P$-orbits of projective measurements containing $P$: two measurements lie in the same orbit exactly when the multisets of ranks of their complementary outcomes agree. The orbit with profile $λ=(r_1,\ldots,r_k)$ is a compact homogeneous space of real dimension $(d-\operatorname{rank}P)^2-\sum_j r_j^2$. On maximal rank-one measurements there is one orbit, so context independence is equivalent to $G_P$-invariance, with equality of the corresponding uniform defects; invariance under two-level complementary unitaries already suffices. For arbitrary projective measurements, every context containing an outcome $P\neq I$ coarsens to the unique binary context $\{P,I-P\}$. Consequently, refinement consistency alone is equivalent to context independence. The finite orbit space carries a natural rank-profile refinement graph, whose diameter is $d-\operatorname{rank}P-1$ when $P\neq I$. We prove a stability theorem that compares the binary-coarsening path with a shortest path in this graph followed by one complementary unitary. In dimension at least three, Gleason's theorem converts the maximal-context invariance condition and the all-context refinement condition, on their respective domains, into the Born form $\operatorname{Tr}(ρP)$. The results are structural characterizations, not independent physical derivations of context independence.
Comments14 pages, no figures. Finite-dimensional projective measurements; unitary-orbit classification, compact homogeneous-space geometry, a rank-profile refinement graph, quantitative stability bounds, and consequences of Gleason's theorem
