Gaussian-Packet Mixtures Attain Asymptotic Capacity in a Resource-Constrained Gaussian Channel

When does Gaussian encoding miss the communication limit? This paper identifies an encoding class that attains the asymptotic capacity of arbitrary quantum block codes, and an exact resource threshold below which individual Gaussian signal states cease to be optimal. The result gives communication theorists a source-selection rule: classical mixtures of Gaussian packets suffice, while the shape of their displacement distribution can be essential. Channel and operating regime. One use is a two-mode additive Gaussian-noise channel with covariance \(Y_{n,c}=\begin{pmatrix}nI_2&cZ\\cZ&nI_2\end{pmatrix}\), where \(Z=\operatorname{diag}(1,-1)\) and vacuum covariance is \(I_2/2\). For fixed \(y>0\) and resource budget \(r>0\), \[c_n=\sqrt{n^2-y^2},\qquad E_n\ge c_n+2\sqrt n+\tfrac12,\qquad n\to\infty.\] The mean photon budget is \(2E_n\) per pair. The resource is relative entropy to normalized PPT states across the two input streams, averaged over transmitted messages at at most \(r\) nats per pair. Arbitrary mixed, non-Gaussian and cross-use-entangled messages and joint decoding are allowed. The coding limit is taken at each fixed channel before the strong-correlation limit. 1. Exact asymptotic capacity reduction. With \(g(x)=(x+1)\ln(x+1)-x\ln x\) and \(A_n=2g(E_n+n)\), \[C=C_{\mathrm{hull}}=A_n-\ln n-1-\mathcal H(r)+o(1).\] Here \(C_{\mathrm{hull}}\) restricts every transmitted message to the trace-closed Gaussian convex hull. The regularized classical variational function \(\mathcal H\) is the lower boundary, at cost at most \(r\), of the closed convex upper hull of \[\left(\frac{D(f\Vert w)}m,\frac{h(f)}m-\ln\pi\right),\qquad w=W_\sigma\ge0,\quad m\ge1.\] The displacement law \(f\) is a classical probability density; its reference \(w\) is the nonnegative Wigner function of a physical state. The exact dual is \[\mathcal H(r)=\sup_{\alpha>1}\frac{-\mathcal P(\alpha)-\alpha r}{\alpha-1},\] \[\mathcal P(\alpha)=\sup_{m\ge1,\,W_\sigma\ge0}\frac1m\ln\left[\pi^{m(\alpha-1)}\int W_\sigma^\alpha\right].\] This converts optimization over quantum block codes into an entropy variational problem with an explicit quantum admissibility condition. 2. A sharp decision boundary for Gaussian signals. Define \(k-1-\ln k=r\), \(0 0,\] where \(\mathfrak h_{\mathrm{W+}}=\inf_{m,\sigma}[h(W_\sigma)/m-\ln\pi]\), over finite-energy physical states with \(W_\sigma\ge0\), satisfies \(\ln2\le\mathfrak h_{\mathrm{W+}}<1\). The comparator \(C_G^{(0)}\) allows Gaussian messages with vanishing resource per use. The achieving messages are simultaneously PPT, Wigner-positive and mixtures of Gaussian states. All rates and logarithms use nats.