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.
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.
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 Φ=⨂j=1dΦj, let E≥0 be the total average input photon number per block, and let Cj(Ej) be the explicitly attained one-channel rate specified in the paper. Then C(Φ,E)=χΦ(E)=maxEj≥0,∑jEj≤E∑j=1dCj(Ej),χΦ⊗n(nE)=nC(Φ,E)(n≥1). 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↦ηV+diag(yq,yp), where 0<η<1, yq≥yp>0, and yqyp≥(1−η)2/4. Define ue=21ypyq,Eth=ue−21+2ηyq−yp. 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)−xlnx, the high-energy rate is Cj(E)=g(ηE+2η+yq+yp−1)−g(2η+yqyp−21),E≥Eth. 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: Cj′(Ej∗)=λ(Ej∗>0),Cj′(0+)≤λ(Ej∗=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.
Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with those of any finite-dimensional partner channel. This class includes all qubit-to-qubit channels of Kraus rank at most two. For the amplitude damping channel with damping probability p, the unassisted classical capacity equals the known single-use Holevo information, attained by a binary pure-state ensemble with collective decoding, and the entanglement cost is h2((1+p)/2) ebits per use. The common mechanism is a support criterion for strong superadditivity of entanglement of formation: one marginal has no support on the sector in which both local systems are orthogonal to fixed distinguished vectors. We prove this criterion in arbitrary finite dimensions using a triangular block-matrix entropy inequality and decompositions preserving two expectations. For amplitude damping, we also derive an exact finite-block Holevo deficit, identify the unique optimal average input for p<1, and construct a binary Kraus representation attaining the uniform formation bound.
How much classical information can a noisy optical channel carry when both signal energy and preshared entanglement are limited? This work gives an exact capacity frontier for this joint resource problem. It allows one to determine the best asymptotic communication rate, price the benefit of additional entanglement, and distinguish the effect of tolerating errors from that of changing how entanglement is supplied. Exact capacity under two resource constraints The setting is any fixed bosonic Gaussian channel Φ with finitely many input and output modes. Signal energy is measured by a positive quadratic Hamiltonian HG=21RTGR, with G>0 and ground energy E0; R collects the canonical quadratures. The budgets satisfy E≥E0 and P≥0. For an n-use code, the mean total signal energy is at most nE, while the same initially shared maximally entangled state has Schmidt rank Kn satisfying logKn≤nP. Thus P is a strictly supplied entanglement rate. Encoders may be arbitrary joint quantum operations, and decoding may be collective. The classical capacity is C(E,P)=maxU≥Vphysical21Tr(GU)≤E,SG(V)≤P[SG(ΦU)+SG(V)−SG(ΦcV)]. Here V is a Gaussian seed covariance, U is the average-input covariance, and U−V is the displacement-modulation covariance. A covariance is physical when V+iΩ/2≥0, where Ω is the canonical commutation matrix and the vacuum covariance is I/2. The notation SG denotes Gaussian von Neumann entropy, and Φc is the full complementary channel, including any environment purification. All logarithms are natural, so rates are in nats per channel use. This formula replaces optimization over arbitrary quantum block codes by a finite-dimensional covariance optimization. It supplies both an achievable rate and a converse against non-Gaussian, correlated encodings. At P=0, it also determines the unassisted capacity of general fixed finite-mode Gaussian channels, including channels without a passive decomposition into independent modes. The reduction specifies the exact capacity value; it does not require a closed-form optimizer for every device. Energy and strict entanglement supply respond differently to errors For a fixed allowed average error 0≤ε<1, the exact capacity under the mean-energy constraint is Cε(E,P)=C(E0+1−εE−E0,P). Allowing failure rescales the mean energy above the ground state, but leaves the strict initial entanglement budget unchanged. The proof uses an exponential strong converse for codes whose inputs obey a hard total-energy cutoff. In a different model, where a readable flag permits random entanglement supply constrained only on average, the work establishes an achievable bound and a strict small-supply advantage for thermal attenuation at positive energy and fixed nonzero error. That comparison identifies the operational importance of the supply rule; it is not a complete capacity characterization of the flagged model. The communication value of entanglement For a thermal attenuator, let 0<η<1 be the transmissivity, b>0 the output mean photon number for vacuum input, and N the input mean photon number. Write its capacity as CN(P), and define g(N)=(N+1)log(N+1)−NlogN. The gain from entanglement obeys the sharp bound 0≤CN(P)−CN(0)≤1+bηP. For every fixed N>0, the coefficient η/(1+b) is attained as the initial right derivative at P=0. In the low-photon limit, the stronger resource law is CN(P)−CN(0)=1+bηmin{P,g(N)}+O(N),N↓0. The remainder is uniform over P≥0 at fixed η and positive b. Noise therefore determines the conversion efficiency, while g(N)∼Nlog(1/N) sets the entanglement scale sufficient for saturation. This uniform limit concerns the entanglement budget, not a simultaneous limit of vanishing thermal noise. Multimode resource allocation and scope A coupled two-mode example demonstrates how the frontier can guide resource allocation. A global resource-price certificate proves that a legal covariance pair exceeds the entire common-passive-frame Gaussian profile hull by at least 10−5 nat per two-mode use at specified budgets. The comparison includes all common passive frames, local squeezing and average-resource mixtures. The mechanism is the freedom to choose different mode allocations for the seed and average input; the benchmark is this precisely defined Gaussian class, rather than all unentangled encodings. The universal scope is the stated family of fixed finite-mode Gaussian channels, positive quadratic signal costs and strict initial maximally entangled Schmidt-rank supply, with unrestricted joint encoding and collective decoding. The model has no feedback or credit for returned entanglement. The capacity frontier and its resource consequences provide a common way to assess optical communication designs under these operational constraints.
Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in (0,1), we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an m-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm ∥ρ∥(m), the sum of the m largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever m is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves log2(e)−1 bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to 2/e.
How does the shape of detector noise affect optical communication? Two receivers can have the same noise covariance yet support different communication rates. This work determines both the quantum noise floor and the photon scale for approaching the capacity ceiling of specified non-Gaussian optical measurements. The central distinction is between the size of a high-energy communication advantage and the finite signal energy needed to realize it. Exact quantum noise floor. The receiver is a displacement-covariant measurement with a fixed, unflagged ancillary state. For every vacuum–single-photon seed σt=(1−t)∣0⟩⟨0∣+t∣1⟩⟨1∣, with 0≤t≤1/2, the output Shannon entropy is minimized uniquely by coherent states over all normal quantum inputs, without an input energy or photon-number cutoff. A separate entropy theorem covers three-level seeds σ=p0∣0⟩⟨0∣+p1∣1⟩⟨1∣+p2∣2⟩⟨2∣ with p1≤0.1, p2≤0.001, and p0=1−p1−p2. A capacity decomposition with a matched thermal benchmark. Let E be the mean signal photon budget per use, μ the fixed mean ancillary photon number, k the coherent-input response, and G1+μ the circular Gaussian density with the same second moment. With natural logarithms, the unassisted classical capacity satisfies Cσ(E)=ln(1+1+μE)+D(k∥G1+μ)−Δσ(E). The first term is the exact capacity of the matched thermal receiver, allowing arbitrary quantum encoders. The relative entropy is the limiting high-energy capacity gain. The deficit Δσ(E)=Uσ(E)−Cσ(E), where Uσ(E)=1+ln(E+1+μ)−h(k), is the finite-energy shortfall from that gain. The converse permits mixed, non-Gaussian and entangled block codewords under an average total signal-photon constraint. Response zeros determine the sharp convergence rate. For each fixed nonvacuum seed in the stated families, write the coherent-response transform as k(ξ)=e−uP(u), where u=∣ξ∣2, P(u)=1−μu+vu2, and v=p2/2. Then limE→∞E−lnΔσ(E)=2minP(ζ)=0∣ζ∣. Positive classical modulation of coherent states attains this exponent with exact signal-energy accounting. In the nonvacuum two-level family, ordinary Gaussian modulation instead leaves an algebraic deficit. Uniform estimates cover compact three-level parameter families through the merger of two real zeros into a complex pair. Thus the response's relative entropy determines the high-energy gain, while its zeros determine the leading photon cost of approaching that gain. Finite-energy guarantees. Explicit bounds compare ordinary Gaussian modulation with a positive variance-mixture preparation and an upper bound valid for every quantum code. For t=1/2 and E=1, the capacity gain over the thermal receiver with the same ancillary mean photon number obeys 0.009742<Cσ1/2(1)−Cτ1/2(1)<0.010469nat/use. The better achievable bound depends on signal energy; no exact finite-energy optimizer is assumed. The matched-covariance noise comparison builds on classical additive-noise results of Ihara and Diggavi–Cover. The contribution here connects an exact all-input quantum noise floor to a sharp capacity remainder and positive, energy-accounted preparations. The results concern the stated receiver families and ideal measurement model; ancillary preparation and implementation overhead are separate resources. The integrated article includes complete proofs and five figures.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number N and allowed average decoding error ε equals the vanishing-error capacity at mean photon number N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.