AI 中文总结
该研究针对奇数局域维度下的差集魏尔信道，刻画了其输出纯度界等式条件，推导了精确容量、有限块长强逆等，发现完全维格纳正性可与持续信道纠缠及可扩展纠缠辅助优势共存。
AI 中文摘要
在奇数局域维度D中，完全维格纳正性会在维格纳非负态上产生随机相空间动力学，但无法控制带符号输入、多信道使用间的纠缠或联合解码。利用子系统分解的魏尔分解，我们刻画了张量稳定输出纯度界的等式条件。循环差集恰好是饱和最坏非平凡碰撞模式通用帕塞瓦尔下界的均匀平移支撑。对于平移支撑为R、|R|=r且碰撞半径不超过给定值的相位分布h的因子化平移-相位噪声，我们对所有n≥1和0≤α≤2得到S_{α,min}(Φ_{R,h}^{⊗n})=n log₂ r，无约束容量C=log₂(D/r)，以及有限块长强逆。对于平衡双差集插值h_ε=(1-ε)q_H+εu_D，0≤ε≤1时无辅助容量恒定，而Choi态对所有ε<1为非正部分转置（NPT），在ε=1时变为纠缠破缺态。在ε=0时，所有张量幂最小输出态是两个相互无偏魏尔基之一的局域因子的乘积；ε>0时仅计算基保留。对于辛格参数D=q²+q+1和r=q+1，C_E/C→2-ε。最后，对于身份-退相轮廓，我们确定了精确的张量幂碰撞熵相图，推导了严格容量界，并分离出独特的冯·诺依曼交叉，数值支持张量幂雷尼猜想。因此，完全维格纳正性可与持续信道纠缠及可扩展纠缠辅助优势共存。
英文摘要
In odd local dimension $D$, complete Wigner positivity yields stochastic phase-space dynamics on Wigner-nonnegative states but does not control signed inputs, entanglement across channel uses, or collective decoding. Using a subsystem-resolved Weyl decomposition, we characterize the equality conditions of the tensor-stable output-purity bound. Cyclic difference sets are precisely the uniform shift supports saturating the universal Parseval lower bound on the worst nontrivial collision mode. For factorized shift--phase noise with shift support $R$, $|R|=r$, and phase distribution $h$ of no larger collision radius, we obtain $S_{α,\min}(Φ_{R,h}^{\otimes n})=n\log_2 r$ for all $n\ge1$ and $0\leα\le2$, the unrestricted capacity $C=\log_2(D/r)$, and a finite-blocklength strong converse. For a balanced bi-difference-set interpolation $h_\varepsilon=(1-\varepsilon)q_H+\varepsilon u_D$, the unassisted capacity is constant for $0\le\varepsilon\le1$, while the Choi state is NPT for every $\varepsilon<1$ and becomes entanglement breaking at $\varepsilon=1$. At $\varepsilon=0$, all tensor-power minimum-output states are products with local factors in one of two mutually unbiased Weyl bases; for $\varepsilon>0$, only the computational basis remains. For Singer parameters $D=q^2+q+1$ and $r=q+1$, $C_{\mathrm E}/C\to2-\varepsilon$. Finally, for an identity--dephasing profile we determine the exact tensor-power collision-entropy phase diagram, derive rigorous capacity bounds, and isolate a distinct von Neumann crossover, with a tensor-power R'enyi conjecture supported by numerics. Thus complete Wigner positivity can coexist with persistent channel entanglement and a scalable entanglement-assisted advantage.
