We develop a method for deriving Gaussian optimality results for bosonic quantum systems from inequalities for qubit systems. Inspired by Gross's derivation of the Gaussian logarithmic Sobolev inequality from an inequality on the two-point space, we use qubits as quantum two-point spaces and lift bosonic Weyl operators to pseudo-Weyl operators on large qubit systems. By controlling weighted norms and relative entropy in the fluctuation limit, we establish two transfer theorems that convert tensorized qubit inequalities into inequalities for bosonic Gaussian channels. As a first application, we prove that the sharp
2-logarithmic Sobolev inequality for the bosonic Ornstein--Uhlenbeck semigroup tensorizes without loss in the constant. We also establish a sharp output entropy inequality for tensor powers of a class of qubit channels. This includes a quantum Mrs.~Gerber's lemma determining the minimum output entropy of tensor powers of the qubit depolarizing channel at fixed input entropy. Applying our relative entropy transfer theorem, we obtain multimode constrained minimum output entropy bounds for Gaussian additive-noise channels and thermal attenuators in the heating range.
In this paper a number of generalizations of the classical Heisenberg‐Weyl uncertainty inequality are given. We prove the n‐dimensional Hirschman entropy inequality (Theorem 2.1) from the optimal form of the Hausdorff‐Young theorem and deduce a higher dimensional uncertainty inequality (Theorem 2.2). From a general weighted form of the Hausdorff‐Young theorem, a one‐dimensional weighted entropy inequality is proved and some weighted forms of the Heisenberg‐Weyl inequalities are given.