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arXiv:2608.17239

Gaussian Optimality of Energy-Constrained One-Shot Communication through Single-Mode Bosonic Gaussian Channels

Se-Wan Ji

quant-phSubmitted Aug 18, 2026• Announced Aug 19, 2026• 0 citations• Revised Sep 8, 2026 (v2)
v1 · Aug 18, 2026v2 · Sep 8, 2026
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Abstract

We prove Gaussian optimality for the energy-constrained one-shot Holevo capacity of single-mode bosonic Gaussian channels, including phase-sensitive channels with arbitrarily squeezed thermal noise. The unresolved regime is the low energy branch, where the Gaussian optimizer modulates only one quadrature and minimum-output-entropy arguments cannot decouple the average state from the letters. For pure one-mode dilations we retain a stronger pointwise result: moment-matched Gaussianization improves the fixed average Holevo function for every input. For mixed environments, whose purification produces a 1:2 entanglement-of-formation problem, we avoid any generic 1:2 Gaussian extremality conjecture. The optimal Gaussian letter selects an effective environmental Schmidt mode and an affine two-mode EPR witness. Its null direction is exactly the modulated quadrature, while its slope equals the negative derivative of the Gaussian letter-output entropy. This produces a supporting lower bound on the dilated entanglement of formation; Gaussian maximum entropy and concavity then give a global upper bound tangent at the Gaussian optimizer. Consequently the known Gaussian formulas for attenuating, amplifying, phase-conjugating, and additive-noise fiducial channels are exact over unrestricted ensembles at every input energy. Via the passive-input fiducial decomposition and energy-constrained continuity, the result extends to every single-mode Gaussian channel, including lower-rank canonical limits. No additivity across channel uses is assumed.

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We prove Gaussian optimality for the energy-constrained one-shot Holevo capacity of single-mode bosonic Gaussian channels, including phase-sensitive channels with arbitrarily squeezed thermal noise. The unresolved regime is the low energy branch, where the Gaussian optimizer modulates only one quadrature and minimum-output-entropy arguments cannot decouple the average state from the letters. For pure one-mode dilations we retain a stronger pointwise result: moment-matched Gaussianization improves the fixed average Holevo function for every input. For mixed environments, whose purification produces a 1:2 entanglement-of-formation problem, we avoid any generic 1:2 Gaussian extremality conjecture. The optimal Gaussian letter selects an effective environmental Schmidt mode and an affine two-mode EPR witness. Its null direction is exactly the modulated quadrature, while its slope equals the negative derivative of the Gaussian letter-output entropy. This produces a supporting lower bound on the dilated entanglement of formation; Gaussian maximum entropy and concavity then give a global upper bound tangent at the Gaussian optimizer. Consequently the known Gaussian formulas for attenuating, amplifying, phase-conjugating, and additive-noise fiducial channels are exact over unrestricted ensembles at every input energy. Via the passive-input fiducial decomposition and energy-constrained continuity, the result extends to every single-mode Gaussian channel, including lower-rank canonical limits. No additivity across channel uses is assumed.
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 NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
The quantum capacity of the bosonic thermal attenuator, which is given by the regularization of its coherent information, is unknown. The seminal work of Holevo and Werner established in 1999 the standard one-use lower bound obtained from input thermal states. We first prove that this long-standing lower bound is the exact supremum over all single-mode Gaussian states and then show that, crucially, a non-Gaussian state can do better. As a consequence, we prove positivity of the quantum capacity in a parameter region where the channel is not antidegradable, yet its coherent information optimized over single-mode Gaussian states vanishes. For example, with one thermal photon in the environment and at transmissivity η=0.8η=0.8η=0.8, the coherent information is non-positive for every single-mode Gaussian input. We give an explicit rank-two non-Gaussian state, supported on only six Fock levels, whose coherent information is certified to be at least 4.7×10−44.7\times10^{-4}4.7×10−4 qubits per channel use. This short witness is far from numerically optimal: a numerical optimization over fixed non-Gaussian families reaches at least 8.4×10−38.4\times 10^{-3}8.4×10−3 qubits per channel use at the same point. More generally, at ν=1ν=1ν=1, using non-Gaussian inputs we certify positivity of the coherent information, and therefore of the quantum capacity, down to η=0.7841η=0.7841η=0.7841; by contrast, the channel is antidegradable, and hence has zero quantum capacity, for η≤0.75η\leq0.75η≤0.75. Overall, our work identifies new high-noise regimes in which bosonic quantum communication is possible.
We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the 'data-processing bound,' is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the 'ϵ-degradable bound' and the 'ϵ-close-degradable bound,' are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.
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\PhiΦ with finitely many input and output modes. Signal energy is measured by a positive quadratic Hamiltonian HG=12RTGRH_G=\tfrac12 R^{\mathsf T}GRHG​=21​RTGR, with G>0G>0G>0 and ground energy E0E_0E0​; RRR collects the canonical quadratures. The budgets satisfy E≥E0E\ge E_0E≥E0​ and P≥0P\ge0P≥0. For an nnn-use code, the mean total signal energy is at most nEnEnE, while the same initially shared maximally entangled state has Schmidt rank KnK_nKn​ satisfying log⁡Kn≤nP\log K_n\le nPlogKn​≤nP. Thus PPP 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)=max⁡U≥Vphysical12Tr⁡(GU)≤E,SG(V)≤P[SG(ΦU)+SG(V)−SG(ΦcV)]. C(E,P)=\max_{\substack{U\ge V\;\mathrm{physical}\\ \frac12\operatorname{Tr}(GU)\le E,\;S_G(V)\le P}}\bigl[S_G(\Phi U)+S_G(V)-S_G(\Phi^c V)\bigr]. C(E,P)=maxU≥Vphysical21​Tr(GU)≤E,SG​(V)≤P​​[SG​(ΦU)+SG​(V)−SG​(ΦcV)]. Here VVV is a Gaussian seed covariance, UUU is the average-input covariance, and U−VU-VU−V is the displacement-modulation covariance. A covariance is physical when V+iΩ/2≥0V+i\Omega/2\ge0V+iΩ/2≥0, where Ω\OmegaΩ is the canonical commutation matrix and the vacuum covariance is I/2I/2I/2. The notation SGS_GSG​ denotes Gaussian von Neumann entropy, and Φc\Phi^cΦc 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=0P=0P=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<10\le\varepsilon<10≤ε<1, the exact capacity under the mean-energy constraint is Cε(E,P)=C ⁣(E0+E−E01−ε, P). C_\varepsilon(E,P)=C\!\left(E_0+\frac{E-E_0}{1-\varepsilon},\,P\right). 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<10<\eta<10<η<1 be the transmissivity, b>0b>0b>0 the output mean photon number for vacuum input, and NNN the input mean photon number. Write its capacity as CN(P)C_N(P)CN​(P), and define g(N)=(N+1)log⁡(N+1)−Nlog⁡Ng(N)=(N+1)\log(N+1)-N\log Ng(N)=(N+1)log(N+1)−NlogN. The gain from entanglement obeys the sharp bound 0≤CN(P)−CN(0)≤η1+b P. 0\le C_N(P)-C_N(0)\le\frac{\eta}{1+b}\,P. 0≤CN​(P)−CN​(0)≤1+bη​P. For every fixed N>0N>0N>0, the coefficient η/(1+b)\eta/(1+b)η/(1+b) is attained as the initial right derivative at P=0P=0P=0. In the low-photon limit, the stronger resource law is CN(P)−CN(0)=η1+bmin⁡{P,g(N)}+O(N),N↓0. C_N(P)-C_N(0)=\frac{\eta}{1+b}\min\{P,g(N)\}+O(N),\qquad N\downarrow0. CN​(P)−CN​(0)=1+bη​min{P,g(N)}+O(N),N↓0. The remainder is uniform over P≥0P\ge0P≥0 at fixed η\etaη and positive bbb. Noise therefore determines the conversion efficiency, while g(N)∼Nlog⁡(1/N)g(N)\sim N\log(1/N)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 10−510^{-5}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.
We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein--Podolsky--Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. We further prove that the entanglement of formation is additive on arbitrary tensor products of two-mode Gaussian states, and hence coincides with the entanglement cost, settling the latter for all two-mode Gaussian states. As a consequence, we obtain in closed form the entanglement cost of thermal-loss, thermal-amplifier and additive-noise bosonic channels. Our results, which also extend to bisymmetric multimode states, establish that no two-mode state is less entangled than the Gaussian state with the same covariance matrix, yielding measurable lower bounds on both the entanglement of formation and the entanglement cost of arbitrary non-Gaussian states.
We find the maximum rate achievable in the private communication over a bosonic quantum channel with a fully Gaussian protocol based on optimal single-mode Gaussian measurements. This rate establishes a lower bound on the secret rate capacity of the channel. We focus on the class of phase-insensitive Gaussian channels. For the thermal-loss and thermal amplification channels, our results demonstrate the optimality, within the constraints of our analysis, of previously proposed protocols, while also providing a significantly simplified formula for their performance evaluation. For the added noise channel, our rate provides a better lower bound than any previously known.