有限维度中的无界 Holevo 可加性间隙
Unbounded Holevo additivity gaps in finite dimensions
AI总结：
本研究在有限维度下构造通道，证明两次使用Holevo可加性间隙无界且线性于输出量子比特，并展示单次容量趋于零而两次容量发散的通道，方法基于互补混合酉通道的张量积与强收敛估计。
AI中文摘要：
我们在有限维度中建立了无界的两次使用 Holevo 可加性间隙。对于每个足够大的固定整数 $K$ 和所有足够大的 $n$，我们构造了输出维度为 $K^n$、输入维度为 $\exp(\Theta_K(n^2))$ 的通道 $T_n$，并且满足 \\[ \chi(T_n^{\otimes2})-2\chi(T_n) \ge n\left[\frac{\log_2K}{K}-2\log_2(1+9/K)\right]-O(1/n). \\] 该间隙在输出量子比特上是线性的，具有明确的二次输入量子比特代价和明确的 $n$ 阈值。我们还获得了单次使用 Holevo 量趋于零的通道，而它们的两次使用 Holevo 信息（因此经典容量）发散。这些通道源于 Collins 的自由概率证明中使用的互补混合酉通道的结构化张量积。我们通过将 Collins-Youn 的乘积群 Haagerup 不等式与 Bordenave-Collins 的定量强收敛估计相结合，建立了最小输出熵间隙。
英文摘要：
We establish unbounded two-use Holevo additivity gaps in finite dimensions. For each sufficiently large fixed integer $K$ and all sufficiently large $n$, we construct channels $T_n$ with output dimension $K^n$, input dimension $\exp(Θ_K(n^2))$, and \[ χ(T_n^{\otimes2})-2χ(T_n) \ge n\left[\frac{\log_2K}{K}-2\log_2(1+9/K)\right]-O(1/n). \] The gap is linear in output qubits, with an explicit quadratic input-qubit cost and an explicit threshold on $n$. We also obtain channels whose single-use Holevo quantity tends to zero while their two-use Holevo information, and hence classical capacity, diverges. The channels arise from structured tensor products of the complementary mixed-unitary channels used in Collins's free-probabilistic proof. We establish minimum-output-entropy gaps by combining Collins--Youn's product-group Haagerup inequality with Bordenave--Collins's quantitative strong-convergence estimates. We also provide a Lean certificate for our proofs.
