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math.FA — Functional Analysis

Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric

Contributed by Suvrit Sra

We study positive definiteness of Hua-Bellman matrices (i.e., matrices of the form $[\det(I-A_i^*A_j)^{-α}]_{ij}$ , where $A_i$ and $A_j$ are strict contractions and $α\in\mathbb{R}$ ). We start by revisiting a 1959 work of Bellman (Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies positive definiteness of Hua-Bellman matrices and claims a strengthening of Hua's representation-theoretic results (Inequalities involving determinants; Acta Mathematica Sinica, 5 (1955)). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date, and we show by an explicit example that his claim itself is false. We then provide conditions under which $\det(I-A^*B)^{-α}$ is a positive definite function. For complex contractions our condition is the known sharp one, with the integer exponents below Hua's range reached by an elementary argument. For real symmetric contractions we recover the part of Bellman's claims that survives. Building on our result, we introduce a new Poincaré-like distance metric on noncommuting strict contractions, and yet another `log-det' based distance in the appendix.

Primarily human-written textHuman understanding: all partsHua-Bellman matricesdistance metricnoncommuting contractionsoperator inequalitiespositive definite functions

Oct 29, 2026PDF ↗View work →

hexagon:2610.00602v1

math.FA — Functional Analysis

Tsallis Metrics on Density Matrices via the S-Divergence

Contributed by Suvrit Sra

We study symmetrized divergences induced by centroids of $\alpha$ -Tsallis relative entropies between density matrices $\sigma$ and $\rho$ ; up to scaling, the induced divergence equals $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|$ . Earlier work (S. Sra, "Metrics induced by Jensen–Shannon and related divergences on positive definite matrices," Linear Algebra Appl., 2021) studied such divergences and metrics induced by them. That paper allowed centroids to range over the positive definite cone, while we require that the centroid itself be a density matrix. This (more natural) choice makes the metricity in this paper considerably stronger than that of the aforementioned paper. Specifically, we prove that $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|^{1/2}$ is a metric on density matrices for $1/2\le\alpha\le2$ , $\alpha\ne1$ . For $\alpha<1$ , the proof uses a new four-point theorem for the S-divergence that might be of independent interest. For $\alpha>1$ , we first prove the endpoint $\alpha=2$ , and then transfer it to the other orders by a tensoring trick.

A mix of human-written and AI-generated textHuman understanding: all partsS-divergenceSchatten normSibson information radiusTsallis relative entropylog-determinant inequalitynegative typequantum Jensen-Shannon divergencequantum Renyi entropy

Oct 29, 2026PDF ↗View work →

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Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric

Tsallis Metrics on Density Matrices via the S-Divergence

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