On quantum Gaussian optimizers conjecture in the case q=p

The quantum Gaussian optimizers conjecture says that q-p norm of a Bosonic Gaussian channel is attained on “Gaussian” operators. Recently R.L. Frank and E.H. Lieb confirmed the hypothesis in the case q=p for gauge-covariant channels with arbitrary number of modes s [3]. In the present note we remark that in the case q=p our results in [6] and [9] in fact allow to prove the hypothesis for all Gaussian channels. Thus the condition of gauge covariance, rather crucial in the case q=1,s>1, plays no role in the case q=p.