Classical capacity of squeezed-noise channels: optimal photon allocation without entangled encoding

Directional optical noise makes photons pay for both squeezing and signal modulation. This paper determines the optimal transmitter and the ultimate classical communication rate, even when arbitrary entangled and non-Gaussian encodings across many channel uses are allowed. For a finite collection of independent squeezed thermal attenuators with different losses and noise covariances, the complete quantum optimization reduces exactly to a scalar allocation of the total mean photon budget. The result covers every input energy, including the low-energy regime in which only one quadrature should be modulated, channels receiving zero photons, and pure squeezed environments. 1. Exact capacity and all-block additivity. Let \(\Phi=\bigotimes_{j=1}^{d}\Phi_j\), let \(E\ge0\) be the total average input photon number per block, and let \(C_j(E_j)\) be the explicitly attained one-channel rate specified in the paper. Then \[ C(\Phi,E)=\chi_\Phi(E)=\max_{E_j\ge0,\,\sum_jE_j\le E}\sum_{j=1}^{d} C_j(E_j), \] \[ \chi_{\Phi^{\otimes n}}(nE)=nC(\Phi,E)\qquad(n\ge1). \] The optimization includes arbitrary joint input ensembles, with an average physical photon constraint and unrestricted decoding. Independent Gaussian modulation of pure squeezed states attains the answer. Thus neither non-Gaussian codewords nor entanglement across the specified independent channels or repeated blocks improves the capacity. Information quantities use natural logarithms; division by \(\ln2\) gives bits. 2. From measured channel noise to a transmitter design. In quadratures with vacuum covariance \(I/2\), each channel acts as \(V\mapsto\eta V+\operatorname{diag}(y_q,y_p)\), where \(0<\eta<1\), \(y_q\ge y_p>0\), and \(y_qy_p\ge(1-\eta)^2/4\). Define \[ u_e=\frac12\sqrt{\frac{y_q}{y_p}},\qquad E_{\mathrm{th}}=u_e-\frac12+\frac{y_q-y_p}{2\eta}. \] Below this threshold, a bounded one-dimensional maximization determines the optimal seed squeezing and the modulation of the quieter quadrature. Above it, both quadratures can be modulated and the average output is isotropic. Writing \(g(x)=(x+1)\ln(x+1)-x\ln x\), the high-energy rate is \[ C_j(E)=g\!\left(\eta E+\frac{\eta+y_q+y_p-1}{2}\right)-g\!\left(\frac{\eta}{2}+\sqrt{y_qy_p}-\frac12\right),\quad E\ge E_{\mathrm{th}}. \] Both seed squeezing and signal modulation are charged to the same physical input budget. The manuscript gives the low-energy variational formula and the attaining covariances explicitly. 3. When should another channel receive photons? The one-channel capacity curves are increasing and concave. For strictly mixed environments, the optimal allocation equalizes marginal information gains: \[ C_j'(E_j^*)=\lambda\quad(E_j^*>0),\qquad C_j'(0+)\le\lambda\quad(E_j^*=0). \] A previously unused channel turns on when its initial information gain becomes competitive with the declining gain of the channels already in use. Pure-environment endpoints follow by continuity of the attained capacity. In the two-channel example in Figure 1, the second channel activates at total budget approximately \(E=0.436\) photons. The figures evaluate the analytic allocation law and display how photons divide between squeezing, modulation and different links. 4. A converse for the complete correlated input. A joint quadratic output-entropy support excludes all competing encodings. Its proof develops a sharp thermal-reference contraction using both Bogoliubov orientations and independently adjusted thermal references. The appendix supplies the regularized minimizer, Gibbs identity, full-state Hessian comparison, interpolation domains, residual propagation and limiting arguments. This connects a computable Gaussian transmitter design to a bound on every allowed quantum code. Use and scope. Communication theorists and optical-source designers can use measured loss and quadrature noise to allocate a common photon budget, choose the seed squeezing and modulation, identify inactive channels, and calculate an attained capacity benchmark for these memoryless channels. The all-energy result addresses the low-energy Holevo additivity question for a squeezed thermal attenuator, and extends it to heterogeneous finite products under one total budget. Version 1.0, 13 October 2026. The 16-page PDF contains the article, two vector figures, complete supplementary proofs and references.