Universal Super-Tensorization of Jensen–Shannon Divergence Contraction Coefficients

Abstract

We prove that Jensen–Shannon divergence (JSD) contraction coefficients exhibit universal strict super-tensorization: for every finite channel W with nontrivial contraction 0 < η_JSD(W) < 1, one has η_JSD(W⊗2) > η_JSD(W). The sequence η_n(W) := η_JSD(W⊗n) is nondecreasing, strictly increases along doubling, and satisfies lim η_n(W) = 1, while for η_JSD(W) ∈ {0, 1} it is identically 0 or 1. This contrasts sharply with the multiplicative tensorization η_f(W⊗n) = η_f(W)^n enjoyed by operator-convex f-divergences (KL, χ², squared Hellinger), for which contraction decays exponentially to zero. To our knowledge, this is the first f-divergence for which a universal strict super-tensorization law is established. The proof uses the Ordentlich–Polyanskiy binary edge reduction, expresses the binary JSD SDPI constant as a normalized posterior-variance functional, and shows strict amplification via the law of total variance. Convergence rate is controlled by the Bhattacharyya coefficient: 1 − η_n(W) ≤ 2A^n. Numerical verification over 4729 random channels across 26 configurations confirms zero violations. **Update v1.1:** Includes addendum with three targeted clarifications: (1) precise assumptions for binary edge reduction lemma replacing informal "mild regularity conditions," (2) explicit two-case split in the key strictness argument (Lemma 5.2, Step 2), (3) refined table caption for operator-convex divergences.