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Basing on recently developed convex programming framework in the paper[arXiv:2204.10626], we provide a proof for a long-standing conjecture onoptimality of Gaussian encondings for the ultimate communication rate ofgeneralized heterodyne receivers under the oscillator energy constraint. Ourresults generalize previous ones (obtained under the assumption of validity ofthe energy threshold condition) and show a drastic difference in the structureof the optimal encoding within and beyond this condition. The core of the proofin the case beyond the threshold is a new log-Sobolev type inequality, whichrelates the generalized Wehrl entropy with the wavefunction gradient.

Proof of the Gaussian maximizers conjecture for the communication capacity of noisy heterodyne measurements