Flat-input entanglement-assisted achievability by Weyl coding

Fix a $D$-dimensional subspace of the input, with embedding $J$ and flat state $\tau =J{J}^{\ast }\mathrm{/}D$. Share $\mathrm{\mid }{\mathrm{\Phi }}_D⟩$ between the sender's subspace and a receiver reference $R$. On the sender half apply the $D^2$ Weyl unitaries, each uniformly likely, before the input embedding. The Weyl matrices in the fixed ordered basis are $U_{ab}\mathrm{\mid }j⟩=e^{2\pi ibj\mathrm{/}D}\mathrm{\mid }j+a(\mathrm{m}\mathrm{o}\mathrm{d}D)⟩$. Reorder the shared state as $R\otimes {\mathbb{C}}^D$ using [F5]. Applying [F9] with retained factor $R$ and twirled factor the sender's $D$-dimensional register gives $I_R\mathrm{/}D\otimes I_D\mathrm{/}D$, since the retained marginal is $I_R\mathrm{/}D$. Reordering back gives $I_D\mathrm{/}D\otimes I_R\mathrm{/}D$. The supplied finite root-of-unity calculation establishes the phase cancellation for every $D\ge 1$.

Let ${\sigma }_{ab}$ be the joint output on $BR$ of this finite cq alphabet through $\mathcal{N}$. Because $(U_{ab}\otimes I)\mathrm{\mid }{\mathrm{\Phi }}_D⟩=(I\otimes U_{ab}^T)\mathrm{\mid }{\mathrm{\Phi }}_D⟩$ in coefficient coordinates, all ${\sigma }_{ab}$ are related by a unitary on $R$ and have identical entropy $S({\mathcal{N}}^c(\tau ))$ by purity of the full Stinespring output and [F4]. Their average is $\mathcal{N}(\tau )\otimes I_R\mathrm{/}D$. The spectral theorem [F8] and logarithm law [F7] give $S(\mathcal{N}(\tau )\otimes I_R\mathrm{/}D)=S(\mathcal{N}(\tau ))+\ln D$. Thus [F6] gives the ensemble Holevo information exactly as $\chi =\ln D+S(\mathcal{N}(\tau ))-S({\mathcal{N}}^c(\tau ))=f_{\mathcal{N}}(\tau )\mathrm{.}$ The HSW theorem applies to this fixed finite cq channel $ab\mapsto {\sigma }_{ab}$, with output $BR$. Each cq codeword is physically implemented by its sequence of Weyl operations on independent shared maximally entangled pairs. Its collective POVM is a permitted receiver measurement and its rate/error are exactly those of HSW. Therefore every nonnegative rate strictly below $f_{\mathcal{N}}(\tau )$ is entanglement-assisted achievable. Dimension $D=1$ is included: the one-letter ensemble then has Holevo information zero.