Dimension-Independent Approximate Additivity and Single-Letter Quantum Capacity Bounds Under Diamond-Norm Perturbations

Evaluating the quantum capacity $Q(N)$ of a noisy quantum channel is generally intractable due to the superadditivity of coherent information, which necessitates an infinite-letter regularization over tensor powers N⊗ n. Building on recent discoveries of non-degradable channels with strictly additive coherent information and the conditional mutual information (CMI) continuity framework of Winter (2016) and Shirokov (2017), we establish an explicit, dimension-independent approximate additivity and single-letter capacity bounding framework for quantum channels in the diamond-norm neighborhood of any strongly additive channel D. While the general Alicki-Fannes-Winter-Shirokov (AFWS) CMI method yields a symmetric single-step continuity modulus fWS(ε) = 2εlog₂ dB + 2g(ε) for arbitrary perturbations 1/2\|N - D\|_ ≤ ε—already avoiding the O(√εlog(1/ε)) Stinespring square-root penalty of Leung and Smith (2009)—its orthogonal state decomposition incurs a 2g(ε) entropic overhead. Here we prove that for physical convex-mixture noise perturbations N_ε = (1-ε)D +, direct state-level mixture concavity eliminates the orthogonal decomposition, cutting the entropic overhead in half to h₂(ε) (fₘᵢₓ(ε) = 2εlog₂ dB + h₂(ε)). Furthermore, when M is a replacer noise process (such as depolarizing, erasure, or thermal-reset noise), its CMI vanishes identically and the quantum data-processing inequality restricts the linear slope to dₘᵢₙ := min(dA, dB), yielding a strongly asymmetric two-sided bound -2εlog₂ dₘᵢₙ - h₂(ε) ≤ Q⁽¹⁾(N_ε ⊗ M) - Q⁽¹⁾(D ⊗ M) ≤ h₂(ε). Telescoping over N_ε⊗ k ⊗ D⊗ (n-k) tightly sandwiches the regularized capacity Q(N_ε) via Q⁽¹⁾(N_ε) ≤ Q(N_ε) ≤ Q⁽¹⁾(N_ε) + 2εlog₂ dₘᵢₙ + 2h₂(ε). Numerical benchmarks on depolarizing-perturbed erasure channels confirm that our sharpened bound comes within 1.45× of the exact numerical shift at ε = 10⁻³, improving upon the general Winter-Shirokov CMI bound by nearly 2× and the Leung-Smith Stinespring bound by over 45×.