Constant-order remainder and fourth-order capacity behavior

Improve the \(O(\log\log n)\) remainder in the third-order expansion of the classical capacity of quantum depolarizing channels to \(O(1)\), and characterize the corresponding fourth-order behavior.

Background

The paper establishes a third-order expansion for the logarithm of the maximum message size with an $O_{d,p,\varepsilon}(\log\log n)$ remainder. This remainder arises from the approximate-majorization defect and the resulting perturbation of the hypothesis-testing error parameter. The authors explicitly identify reducing this remainder to constant order and determining the next, fourth-order asymptotic term as unresolved directions for future work.