Full majorization of depolarizing-channel outputs

Prove that, for every blocklength n, every input state \(\rho^n\) on \((\mathbb C^d)^{\otimes n}\), and the quantum depolarizing channel \({\mathcal D}_{p,d}^{\otimes n}\), the output spectrum for an arbitrary input is majorized by the output spectrum for a fixed pure product input, namely \(\lambda^\downarrow({\mathcal D}_{p,d}^{\otimes n}(\rho^n))\prec\lambda^\downarrow({\mathcal D}_{p,d}^{\otimes n}(|0^n\rangle\langle0^n|))\).

Background

The paper conjectures a spectral ordering stronger than King's multiplicativity result for Schatten norms. For the d-dimensional quantum depolarizing channel, the conjecture asserts that the output generated by a pure product input majorizes the output generated by any possibly entangled input across n channel uses. Exact majorization would control all convex unitarily invariant spectral functionals and could yield sharper finite-blocklength communication bounds. The paper proves only an approximate majorization relation with a defect of order $O_{d,p}(\log\log n/\sqrt n)$, so the exact relation remains unresolved.