Classical communication through phase-sensitive thermal qubit noise

When a thermal environment damps the two transverse components of a qubit differently, can entangling signals across many channel uses improve the classical communication rate? This article gives a complete coding answer for an explicit phase-sensitive class and determines the optimal exponential cost of transmitting above capacity. Physical model and communication permissions. The channel acts as \[\mathcal N\!\begin{pmatrix}\rho_0&X\\X^*&\rho_1\end{pmatrix}=\begin{pmatrix}a\rho_0+b\rho_1&kX+\ell X^*\\kX^*+\ell X&c\rho_0+d\rho_1\end{pmatrix}.\] All coefficients are nonnegative, \(a+c=b+d=1\), and the main class satisfies \[k\ge\ell\ge0,\qquad ad-bc=k^2-\ell^2,\qquad bc\ge\ell(k+\ell).\] The class contains actual squeezed thermal reservoirs, including nonunital channels that are not entanglement breaking. The coding model is unassisted and memoryless, without feedback; arbitrary entangled block codewords and collective decoding are allowed. The coefficient condition is a sufficient region, not a classification of every physically allowed squeezed bath. An exact capacity from one-use data. Write \(h_2\) for binary entropy in bits, \(q\) for the excited-state input population, and define \[t(q)=c+(d-c)q,\qquad s_x(q)=h_2\!\left(\frac{1+\sqrt{[1-2t(q)]^2+4(k+\ell)^2q(1-q)}}{2}\right).\] With \(\operatorname{co}s_x\) the lower convex envelope on \([0,1]\), the unrestricted capacity is \[C(\mathcal N)=\chi(\mathcal N)=\max_{0\le q\le1}\{h_2[t(q)]-(\operatorname{co}s_x)(q)\}.\] At most two opposite-real-phase signal pairs, hence at most four pure signals, attain the one-use optimum. The convex envelope retains optimization over all ensembles. Holevo information is additive with every finite-dimensional channel partner, so arbitrarily long entangled encodings cannot exceed this rate. The best success exponent above capacity. Let \(P_n^\star(R)\) be the optimal average success probability over all \(n\)-use codes with at least \(\lceil2^{nR}\rceil\) messages. For \(R>C(\mathcal N)\), \[E_{\rm sc}(R,\mathcal N)=\lim_{n\to\infty}-\frac1n\log_2P_n^\star(R)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\,[R-\widetilde\chi_\alpha(\mathcal N)].\] Here \(\widetilde\chi_\alpha\) is the sandwiched Rényi information radius, reduced in the paper to a scalar reference/population optimization. It is strongly additive at every real \(\alpha>1\). Product codewords with collective decoding attain the asymptotic exponent, while the converse covers arbitrary entangled codewords. The finite-block inequality \(P_n^\star(R)\le2^{-nE_{\rm sc}}\) is an upper bound, not a finite-block equality. Why the reduction works. Normal and reversed positive block factors retain the same quantum source and admit a common scalar Gram bound. This yields maximum-output norm multiplicativity with arbitrary completely positive partners and unnormalized diagonal filters. Binary phase symmetry then suffices to make the information-radius reference diagonal, even though continuous phase covariance is absent. The same input-filter comparison also determines the formation cost of the specified diagonal-input Stinespring states across the output and the full purified environment. Relation to existing results. We independently obtained the additivity and strong-converse results for the axially symmetric Davies family that are also established by L. Eriksson, Capacities, strong-converse exponents and complementary simulation for qubit Davies channels, version 3, 26 October 2026. These results and their proofs are retained as a reference case. The phase-sensitive class and the full axial Davies class share the generalized-amplitude-damping boundary; neither is presented as a strict enlargement of the other. Fang's triangular norm-compression theorem and the general information-radius and coding theorems retain their attribution. The 16-page integrated article contains the full proofs, two figure groups and references. The figures separate the physical parameter region, the globally evaluated capacity formula and explicit finite-block success upper bounds.