Single-shot quantum information theory is governed not only by entropy exponents, but also by the finite-resource constants that multiply them. These constants directly affect the quantitative performance of decoupling, covering, convex-splitting, position-based decoding, and one-shot communication protocols, yet they are often inherited from nonoptimal scalar estimates or from classical-to-quantum lifting arguments that introduce additional losses. In this work we show that the operator layer-cake representation provides a mechanism for lifting sharp scalar inequalities to the noncommutative setting without loss. Using an iterative Riemann--Stieltjes integration-by-parts method, we derive sharp quantum trace inequalities that tighten several standard single-shot bounds. For a logarithmic trace inequality recently introduced by Cheng \emph{et al.}\ and subsequently used in quantum covering and decoupling problems, we determine the exact optimal prefactor, replacing the previously known constant by a smaller Lambert-
W constant and proving universal optimality for positive operators. We also completely characterize the threshold behavior that appears under normalization to quantum states. In addition, we establish optimal two-sided collision-divergence inequalities, which lead to improved position-based decoding and single-shot classical communication bounds. These results show that several finite-resource bounds in single-shot quantum information can be tightened, and that within the layer-cake Rényi-divergence framework the resulting constants are genuine optimality barriers rather than artifacts of the proof.
We prove the sharp Hausdorff--Young inequality on the quantum Euclidean space. Our result implies the sharp Hausdorff--Young constants for the Weyl transform, as well as that for Heisenberg groups. The key ingredient is a novel flow related to the mixed-norm of noncommutative Gabor transform. This, meanwhile, implies a new proof of the classical sharp Hausdorff--Young inequality.
We then apply the sharp Hausdorff--Young inequality to establish the sharp Young inequality for noncommutative convolution in the range
1≤p,q≤2≤r≤∞, with
1/p+1/q=1+1/r. After appropriate rescaling and trace normalization, this convolution coincides with beam-splitter convolution for bosonic systems, yielding the corresponding Young inequalities with optimal constants in the same range.