基于Haagerup不等式与近自由置换表示的确定性最小输出熵非加性
Deterministic Minimum-Output-Entropy Nonadditivity via Haagerup's Inequality and Near-Free Permutation Representations
- The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
- QudeLeap Research（量子跃迁研究）
- Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究结合Haagerup不等式与O'Donnell和Wu的谱逼近，构造确定性多项式时间算法实现最小输出熵非加性，证明常数3渐近尖锐，将熵间隙转化为Holevo量超加性，为输出体几何研究提供方向。
AI中文摘要：
我们给出了Collins混合酉证明最小输出熵非加性所基于的有限维二次证书的确定性实现。对于每个固定整数K≥2和满足log K>2(3+η)²的有理数η>0，存在确定性多项式时间算法，对每个足够大的目标规模N，输出N'=N+o_{K,η}(N)个点上的K个置换。将其置换矩阵限制在非平凡标准表示上，得到实正交Stinespring块和信道Φ_{N'}:M_{N'-1}(ℂ)→M_K(ℂ)，满足2H_min(Φ_{N'})−H_min(Φ_{N'}^{⊗2})≥(log K)/K −2log(1+(3+η)²/K)>0。该构造结合了Haagerup的长度二不等式与O'Donnell和Wu的同时确定性谱逼近。我们进一步证明，常数3在相关Hermitian零对角系数类上是渐近尖锐的，且有限谱转移几乎达到饱和，从而将完整输出体的更精细几何确定为超越标量半径方法的自然下一个细化层级。最后，通过标准协变扩展，相同的确定性熵间隙精确转化为单次Holevo量的自张量超加性。
英文摘要：
We give a deterministic realization of the finite-dimensional quadratic certificate underlying Collins's mixed-unitary proof of minimum-output-entropy nonadditivity. For every fixed integer $K\ge 2$ and rational $η>0$ satisfying $\log K>2(3+η)^2$, a deterministic polynomial-time algorithm, for every sufficiently large target size $N$, outputs $K$ permutations on $N'=N+o_{K,η}(N)$ points. Restricting their permutation matrices to the nontrivial standard representation yields real orthogonal Stinespring blocks and a channel $Φ_{N'}:M_{N'-1}(\mathbb{C})\to M_K(\mathbb{C})$ such that \[ 2H_{\min}(Φ_{N'}) -H_{\min}(Φ_{N'}^{\otimes 2}) \ge \frac{\log K}{K} -2\log\left(1+\frac{(3+η)^2}{K}\right) >0. \] The construction combines Haagerup's length-two inequality with the simultaneous deterministic spectral approximation of O'Donnell and Wu. We further show that the constant $3$ is asymptotically sharp on the relevant Hermitian zero-diagonal coefficient class and that the finite spectral transfer is nearly saturated, thereby isolating the finer geometry of the full output body as the natural next level of refinement beyond the scalar-radius method. Finally, a standard covariant extension converts the same deterministic entropy gap exactly into self-tensor superadditivity of the one-shot Holevo quantity.
