Difference-Set Weyl Channels: Exact Capacity, Optimizer Bifurcation, and Scalable Entanglement Separation

Abstract

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 $S_{\alpha ,\min }({\Phi }_{R,h}^{\otimes n})=n{\log }_{2}r$ for all $n\ge 1$ and $0\le \alpha \le 2$, the unrestricted capacity $C={\log }_2(D/r)$, and a finite-blocklength strong converse. For a balanced bi-difference-set interpolation $h_{\epsilon }=(1-\epsilon )q_H+\epsilon u_D$, the unassisted capacity is constant for $0\le \epsilon \le 1$, while the Choi state is NPT for every $\epsilon <1$ and becomes entanglement breaking at $\epsilon =1$. At $\epsilon =0$, all tensor-power minimum-output states are products with local factors in one of two mutually unbiased Weyl bases; for $\epsilon >0$, only the computational basis remains. For Singer parameters $D=q^2+q+1$ and $r=q+1$, $C_{\mathrm{E}}/C\to 2-\epsilon$. 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.