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 $n\ge1$ and $0\le\alpha\le2$, 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.