Exact series formula for the quantum capacity of the amplitude damping channel

In its degradable regime, the quantum capacity of the qubit amplitude damping channel admits a single-letter expression that retains a maximization over the excited-state population of the input. We evaluate this optimization analytically and obtain an explicit series that is absolutely convergent in the interior of the degradable regime. The series contains neither an input-state optimization nor an implicitly defined root. We characterize the unique maximizing population in the interior of this regime and determine the endpoint limits and the leading behavior at the degradable-antidegradable threshold. The same solution evaluates a squashed-entanglement upper bound on two-way assisted quantum communication, entanglement distribution, and secret-key generation. For a balanced amplitude damping squashing channel, the remaining input optimization coincides exactly with the ordinary quantum-capacity problem for an amplitude damping channel with half the physical damping probability. Consequently, the corresponding two-way assisted capacity bound also admits an explicit, absolutely convergent series representation throughout the interior of the physical damping range, with the endpoint values obtained by continuity.