A solution of the Gaussian optimizer conjecture

• Gaussian channels model thermal, attenuation, and amplification processes in common electromagnetic communication architectures, making their capacity an important theoretical and technical problem.Source: §1 · p. 2
• The optimal Gaussian ensemble conjecture proposes that Gaussian encodings optimize communication rates under energy constraints, while Gaussian additivity would make C(Φ) equal Cχ(Φ).Source: §1 · p. 3
• The minimum output entropy conjecture recasts Gaussian optimality as the claim that Gaussian input states minimize the channel's output entropy, but these conjectures largely remained open.Source: §1 · p. 3
• The paper reports that vacuum input minimizes output entropy for multimode gauge-covariant or contravariant Bosonic Gaussian channels, and Gaussian coherent-state ensembles attain energy-constrained capacity.Source: §1 · pp. 3–4
• The proof strategy decomposes Gaussian channels, uses entanglement-breaking additivity, reduces the entropy problem to a single-mode quantum-limited amplifier, and characterizes contravariant outputs.Source: §1 · pp. 3–4

• A multimode Bosonic Gaussian gauge-covariant channel is specified in the Heisenberg picture by a complex-linear operator K and Hermitian noise operator μ satisfying a complete-positivity inequality.Source: §2 · p. 5
• Gauge-covariant channels are quantum-limited when μ is a minimal solution, and they are diagonalizable when K and μ can be simultaneously diagonalized.Source: §2 · pp. 5–6
• Attenuators satisfy KK* ≤ I, whereas amplifier channels satisfy KK* ≥ I; singular-value decomposition reduces these quantum-limited channels to tensor products of one-mode channels up to passive transformations.Source: §2 · p. 6
• Gauge-contravariant channels implement phase inversion and admit analogous diagonalizable quantum-limited forms, with their environment modes in the vacuum state.Source: §2 · pp. 6–7
• Any bosonic Gaussian gauge-covariant channel decomposes into a quantum-limited attenuator followed by a diagonalizable quantum-limited amplifier.Source: §2 · p. 7
• Any bosonic Gaussian gauge-contravariant channel decomposes into a quantum-limited attenuator followed by a diagonalizable quantum-limited gauge-contravariant channel.Source: §2 · p. 7

Additivity

• For entanglement-breaking channels, superadditivity of the convex closure of output entropy implies additivity of minimal output entropy.The convex closure is optimized over probability distributions with a fixed barycenter, and the resulting relation holds for arbitrary bipartite input states.Source: §3 · p. 9

Capacity

• Additivity of the relevant χ-capacity yields additivity of constrained χ-capacity under mild regularity assumptions, with the constrained classical capacity identified accordingly.The setting uses an additive energy operator H^(n), and the Gaussian case satisfies the stated regularity conditions.Source: §3 · p. 9

Scope

• The results are restricted to finite-second-moment states, and remain valid for Gaussian channels with quadratic Hamiltonian constraints when optimizations are confined to this class.This state class is invariant under Gaussian channels, and its entropy quantities are finite.Source: §3 · pp. 9–10

Entanglement breaking

• Gaussian gauge-contravariant channels are entanglement-breaking, with a decomposition into Gaussian measurement followed by coherent-state preparation.The proof uses a noise decomposition, and the resulting representation measures a Gaussian observable before preparing outcome-dependent coherent states.Source: §3 · p. 10

Additivity

• This entanglement-breaking property implies additivity relations for a Gaussian gauge-contravariant channel combined with an arbitrary channel.The implication is stated for the additivity properties identified earlier in the paper.Source: §3 · p. 10

Quantum-limited amplifiers

• The same additivity properties extend to diagonal quantum-limited amplifiers by relating them to quantum-limited gauge-contravariant channels and complementary-channel output spectra.In the diagonal case, the channels decompose into tensor products of one-mode channels.Source: §3 · p. 10

Reductions of the Gaussian optimizer conjecture

• The minimal output entropy conjecture for arbitrary gauge-covariant channels reduces to the one-mode quantum-limited amplifier case, equivalently to a contravariant channel with vacuum minimizer.The reduction uses decompositions into quantum-limited attenuators and amplifiers, together with complementarity.Source: §4 · pp. 11–12
• For a quantum-limited attenuator, the vacuum is invariant and achieves zero minimal output entropy, providing the fixed-point base case for the reductions.The same argument transfers through the attenuator–amplifier decomposition and complementarity.Source: §4 · pp. 11–12
• Under hypothesis (A), the minimal output entropy is additive for diagonalizable quantum-limited amplifiers and their corresponding covariant or contravariant channels.The proof extends the one-mode statement to multimode diagonalizable channels using tensor-product structure and unitary diagonalization.Source: §4 · pp. 11–12
• The constrained classical capacity is additive, C(Φ⊗n,H(n),nE)=nC(Φ,H,E), and equals the χ-capacity achieved by Gaussian encodings.The capacity is expressed through a finite-dimensional covariance optimization under the energy constraint.Source: §4 · pp. 12–13
• The covariance optimization is a quantum analogue of waterfilling and is explicit only in special commuting cases.The admissible set is ΣE={ν:ν≥0,trνϵ≤E}.Source: §4 · p. 13

Some useful properties of contravariant channels

• A quantum-limited gauge-contravariant channel admits an explicit measure-reprepare representation on coherent states.The representation follows from the entanglement-breaking structure of these channels.Source: §5 · p. 14
• Skewed measure-reprepare channels prepare coherent states with transformed amplitudes after the measurement outcome.These channels are gauge-covariant but are not quantum-limited.Source: §5 · pp. 14–15
• The original contravariant output and both skewed-channel outputs have the same eigenvalues and therefore the same entropy.The spectral equivalence follows from transposition and a phase transformation.Source: §5 · p. 15
• For every input state, a unitary transformation relates the contravariant output to a corresponding state, preserving the relevant entropy.This state-dependent unitary relation supplies the spectral connection used later in the proof.Source: §5 · p. 15

Proof of the Gaussian optimizer conjecture

• Complementarity and spectral equivalence transfer the argument between covariant amplifiers, contravariant channels, and their skewed counterparts.The factorization Ψ+=Φ∘Φ1 connects the non-quantum-limited skewed channel to the original amplifier and attenuator.Source: §6 · pp. 15–16
• The minimal output entropy of the quantum-limited amplifier equals that of its composition with the associated attenuator, consistent with vacuum optimality.The attenuator has the vacuum as a fixed point.Source: §6 · p. 17
• Proposition 7 constructs iterative ensemble decompositions and connecting partial isometries for amplifier outputs composed with an associated attenuator.The construction proceeds by induction from the spectral decomposition of the attenuator output.Source: §6 · pp. 17–18
• The attenuator iteration converges in trace norm so that, in the limit, only the vacuum fixed point survives in the decomposition.This identifies the vacuum as the limiting state relevant to the entropy comparison.Source: §6 · p. 18
• Relative-entropy monotonicity with a Gibbs-state choice, together with lower semicontinuity, yields the entropy inequality comparing every finite-second-moment pure input with vacuum input.The Gibbs parameter γ>0 ensures the relevant term is well defined for states with finite second moments.Source: §6 · pp. 18–19

Implications and perspectives

• For multimode covariant or contravariant Bosonic Gaussian channels with finite second moments, vacuum input minimizes output entropy and implies additivity of minimal output entropy and constrained classical capacity.The capacity is achieved by Gaussian encodings.Source: §7 · pp. 19, 21
• For thermal-noise channels, the constrained capacity is C(Φ;H,E)=g(ηE+(1−η)N)−g((1−η)N).The parameters are K=√η and μ=(1−η)(N+1/2).Source: §7 · p. 20
• For noisy amplifier channels, the constrained capacity is C(Φ;H,E)=g(κE+(κ−1)(N+1))−g((κ−1)(N+1)).The parameters are K=√κ and μ=(κ−1)(N+1/2).Source: §7 · p. 20
• In single-mode settings, squeezed vacuum and corresponding coherent states minimize output entropy after metaplectic unitary equivalence.This covers nondegenerate and, more simply, degenerate channel cases.Source: §7 · p. 20
• The capacity problem remains when the channel and energy operator have incompatible complex structures, although the entropy argument extends to squeezed complex structures.The extension applies after reduction to the standard complex structure.Source: §7 · p. 21