Michael P. Rubin | Searcharxiv

arXiv2026

A Uniqueness Theorem for Exact Separable Log-Partition Decompositions

Let $\mathcal{X}$ be a finite set, let $p$ and $Q$ be strictly positive probability distributions on $\mathcal{X}$, and let $u\in\mathbb{R}^{\mathcal{X}}$ be a log-weight. Define $Ψ_p(u)=\log\sum_x p(x)e^{u(x)}$ and $π_{p,u}(x)=p(x)e^{u(x)-Ψ_p(u)}$. The variational free energy $F(Q;p,u)=D(Q\|p)-\mathbb{E}_Q[u]$ satisfies the exact identity $Ψ_p(u)=-F(Q;p,u)+D(Q\|π_{p,u})$, whose residual is the canonical divergence of the dually flat simplex. We prove a converse. Suppose $G(Q;p,u)=A(Q,p)+B(Q,u)$ is additively separable and its gap $Ψ_p(u)+G(Q;p,u)$ is nonnegative and vanishes whenever $Q=π_{p,u}$. Then $G=F$ and the gap equals $D(Q\|π_{p,u})$; $A$ and $B$ are unique up to a $Q$-dependent additive gauge. The gap is not assumed to be a divergence or to depend only on the target, and no continuity, measurability, differentiability, convexity, or membership in a prescribed divergence family is assumed. The proof bounds increments of a corrected accuracy term by centered log-moment-generating costs of order $k^{-2}$ per step. Telescoping $k$ equal exponential tilts gives an $O(k^{-1})$ bound and forces every increment to vanish. Thus target factorization and strictness are conclusions. Counterexamples show that separability, nonnegativity, and tightness are each necessary. Within this class, no $α$-divergence distinct from Kullback--Leibler and no Rényi divergence of order other than one can occur as the residual.

cs.IT↗

arXiv2026

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

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.

quant-ph↗