We prove the sharp Hausdorff--Young inequality on the quantum Euclidean space. Meanwhile, our result implies the sharp Hausdorff-Young constants for the Weyl transform, as well as that for the Heisenberg group. The key ingredient is a novel operator-valued flow related to the Gabor transform. Our method is mainly based on operator calculus and hence generally applicable to similar problems. Finally, we give the definition of noncommutative convolution and include the partial results for the sharp Young's inequality.
Comments
A rough manuscript including key ingredients. A polished version will be updated quickly
We prove the sharp Hausdorff--Young inequality on the quantum Euclidean space. Meanwhile, our result implies the sharp Hausdorff-Young constants for the Weyl transform, as well as that for the Heisenberg group. The key ingredient is a novel operator-valued flow related to the Gabor transform. Our method is mainly based on operator calculus and hence generally applicable to similar problems. Finally, we give the definition of noncommutative convolution and include the partial results for the sharp Young's inequality.
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.
To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every z∈Mn⊗Mm, we prove ∥z∥1≤2min{n,m}∥z∥ε, where ∥⋅∥1 is the trace norm and ∥⋅∥ε is the injective tensor norm associated with the trace norms on Mn and Mm. To prove the upper bound, we establish an L1 noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient 2 is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean.
Previous noncommutative Bohnenblust--Hille (BH) inequalities addressed operator decompositions in the tensor-product space M2(C)⊗n; \emph{i.e.,} for systems of qubits \cite{HCP22,VZ23}. Here we prove noncommutative BH inequalities for operators decomposed in tensor-product spaces of arbitrary local dimension, \emph{i.e.,} MK(C)⊗n for any K≥2 or on systems of K-level qudits. We treat operator decompositions in both the Gell-Mann and Heisenberg--Weyl basis, reducing to the recently-proved commutative hypercube BH \cite{DMP} and cyclic group BH \cite{SVZ} inequalities respectively. As an application we discuss learning qudit quantum observables.
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.
Entanglement criteria for an n-partite quantum system with continuous variables are formulated in terms of R\'enyi entropies. R\'enyi entropies are widely used as a good information measure due to many nice properties. Derived entanglement criteria are based on several mathematical results such as the Hausdorff-Young inequality, Young's inequality for convolution and its converse. From the historical viewpoint, the formulations of these results with sharp constants were obtained comparatively recently. Using the position and momentum observables of subsystems, one can build two total-system measurements with the following property. For product states, the final density in each global measurement appears as a convolution of n local densities. Hence, restrictions in terms of two R\'enyi entropies with constrained entropic indices are formulated for n-separable states of an n-partite quantum system with continuous variables. Experimental results are typically sampled into bins between prescribed discrete points. For these aims, we give appropriate reformulations of the derived entanglement criteria.
For a channel Φ on Md written in the generalised Bloch parameterisation r↦Ar+c, we prove that entanglement breaking implies ∥A∥∗2+2d(d−1)∣c∣2≤(d−1)2, where ∥⋅∥∗ denotes the nuclear norm. Read on the Choi state of the channel, the inequality is an instance of the covariance-matrix criterion of Gittsovich et al.; what is added here is an elementary proof in Holevo form and a complete description of the equality cases. Apart from the constant channel onto a pure state, which shows that the coefficient d(d−1)/2 cannot be increased, equality holds exactly for the measure-and-prepare channels with rank-one effects αk∣ek⟩⟨ek∣ and pure outputs ∣ψk⟩⟨ψk∣ whose overlaps are an affine image of those of the measurement directions, ∣⟨ψj∣ψk⟩∣2=P+(1−P)(d∣⟨ej∣ek⟩∣2−1)/(d−1) for a common P=TrΦ(1/d)2. These include the completely dephasing channel, the re-preparation channel of every rank-one POVM, a one-parameter family interpolating between dephasing and the constant channel, and non-unital non-projective examples already at d=2. The qubit case reads ∥A∥∗2+∣c∣2≤1 and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound ∥A∥∗≤1 by retaining the POVM completeness relation ∑kfk=0, which her argument discards; this recentres A from a second moment into a cross-covariance, and is the entire content of the improvement.