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

Abstract

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 $N^{\otimes n}$. 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 $f_{\mathrm{WS}}(\epsilon) = 2\epsilon\log_2 d_B + 2g(\epsilon)$ for arbitrary perturbations $\frac{1}{2}\|N - D\|_\diamond \le \epsilon$—already avoiding the $\mathcal{O}(\sqrt{\epsilon}\log(1/\epsilon))$ Stinespring square-root penalty of Leung and Smith (2009)—its orthogonal state decomposition incurs a $2g(\epsilon)$ entropic overhead. Here we prove that for physical convex-mixture noise perturbations $N_\epsilon = (1-\epsilon)D + \epsilon\mathcal{M}$, direct state-level mixture concavity eliminates the orthogonal decomposition, cutting the entropic overhead in half to $h_2(\epsilon)$ ($f_{\mathrm{mix}}(\epsilon) = 2\epsilon\log_2 d_B + h_2(\epsilon)$). Furthermore, when $\mathcal{M}$ 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 $d_{\min} := \min(d_A, d_B)$, yielding a strongly asymmetric two-sided bound $-2\epsilon\log_2 d_{\min} - h_2(\epsilon) \le Q^{(1)}(N_\epsilon \otimes M) - Q^{(1)}(D \otimes M) \le h_2(\epsilon)$. Telescoping over $N_\epsilon^{\otimes k} \otimes D^{\otimes (n-k)}$ tightly sandwiches the regularized capacity $Q(N_\epsilon)$ via $Q^{(1)}(N_\epsilon) \le Q(N_\epsilon) \le Q^{(1)}(N_\epsilon) + 2\epsilon\log_2 d_{\min} + 2h_2(\epsilon)$. Numerical benchmarks on depolarizing-perturbed erasure channels confirm that our sharpened bound comes within $1.45\times$ of the exact numerical shift at $\epsilon = 10^{-3}$, improving upon the general Winter-Shirokov CMI bound by nearly $2\times$ and the Leung-Smith Stinespring bound by over $45\times$.