bibkey: gibiliscoisola2005scalarcurvature authors: Paolo Gibilisco; Tommaso Isola year: 2005 title: “On the monotonicity of scalar curvature in classical and quantum information geometry” doi: 10.1063/1.1834693 url: https://arxiv.org/abs/math-ph/0407007v2 claim: “Conjecture 4.1 asks whether the scalar curvature of the logarithmic alpha-geometry is Schur-increasing on the positive simplex; this note records the p = infinity endpoint.” strata_touched:

• D5/S3/Geometry/AlphaGeometry/MajorizationTransfer
• D5/S3/Geometry/AlphaGeometry/PairCurvatureInequality
• D5/S3/Geometry/AlphaGeometry/PositiveSimplexGeometryAndMixing
• D5/S3/Geometry/AlphaGeometry/LogScalarCurvatureMonotone license: citation-only triage: anchor

Gibilisco–Isola scalar curvature

The paper uses the positive simplex (\mathcal P^1_n), whose coordinates are positive and sum to one. Majorization orders vectors by being more mixed; a Schur-increasing function increases along that order, and a strictly Schur-increasing function increases unless the vectors differ only by a permutation.

Definition 2.1 (printed p. 3) defines

$${\mathcal{P}}_n^1:=\{\rho \in {\mathbb{R}}^n\mid \sum _i{\rho }_i=1,{\rho }_i>0\}.$$

Definition 2.3 (printed p. 3): “$x$ is more mixed (more chaotic,…) than $y$ (denoted by $x\succ y$) if and only if”

$$x_1^{\downarrow }+\cdots +x_k^{\downarrow }\le y_1^{\downarrow }+\cdots +y_k^{\downarrow }(k<n),x_1^{\downarrow }+\cdots +x_n^{\downarrow }=y_1^{\downarrow }+\cdots +y_n^{\downarrow }.$$

Definition 2.4 (printed p. 4): “A real-valued function $f$ defined on a set $\mathcal{A}\subset {\mathbb{R}}^n$ is said to be Schur-increasing on $\mathcal{A}$ if” $x\succ y\Rightarrow f(x)\ge f(y)$. “If in addition $f(x)>f(y)$ whenever $x\succ y$ but $x$ is not a permutation of $y$ then $f$ is said to be strictly Schur-increasing.”

Definition 4.3 (printed p. 10): “The $\alpha$-geometry on ${\mathcal{P}}_n^1$ is the pull-back geometry induced by the map $A_p(\rho ):{\mathcal{P}}_n^1\to {\mathbb{R}}^n$ defined by”

$$A_p(\rho ):=\{\begin{array}{ll}p{\rho }^{1/p}& p\in \mathbb{R}\setminus \{0\},\\ \log (\rho )& p=\infty ,\end{array}p=\frac{2}{1-\alpha }.$$

Definition 4.4 (printed p. 10): “We denote by ${\mathrm{Scal}}_p(\rho )$ the scalar curvature of the $\alpha$-geometry (with $p=\frac{2}{1-\alpha }$) at the point $\rho \in {\mathcal{P}}_n^1$.”

Conjecture 4.1 (printed p. 11): “Suppose $n>2$. For the function ${\mathrm{Scal}}_p(\cdot ):{\mathcal{P}}_n^1\to R$ one has the following properties:

• if $p\in (1,2)$ then ${\mathrm{Scal}}_p(\cdot )$ is a strictly Schur-decreasing function;
• if $p\in (2,+\infty ]$ then ${\mathrm{Scal}}_p(\cdot )$ is a strictly Schur-increasing function.”

The repository encoding in research line #15145 uses strictly positive normalized Fin n real vectors, decreasing merge-sort initial sums for majorization, the logarithmic pull-back metric defined with Fréchet derivatives, and the usual coordinate Christoffel–Ricci contraction. The settling statement concerns only $p=\infty$ and $n\ge 3$. Its proof reduces this literal curvature to a rational power-sum formula and strictly increases it under a nontrivial equalizing T-transform. Finite-$p$ clauses and quantum Conjecture 4.2 remain open; the quantum $p=3$ slice has separate research line #15179.

Verified locator

DOI: 10.1063/1.1834693 (https://doi.org/10.1063/1.1834693)

Source: https://arxiv.org/abs/math-ph/0407007v2