Unitary-orbit classification and a refinement theorem for context-independent projective probabilities

Abstract

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 $\mathrm{ran}P$ 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 $\lambda =(r_1,\dots ,r_k)$ is a compact homogeneous space of real dimension $(d-\mathrm{rank}P{)}^2-{\sum }_j{r}_j^2$. 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 $P\ne I$ 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 $d-\mathrm{rank}P-1$ when $P\ne I$. 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 $\mathrm{Tr}(\rho P)$. The results are structural characterizations, not independent physical derivations of context independence.

14 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