Optimal Trace Inequalities for Single-Shot Quantum Information

TL;DR

This paper develops sharp quantum trace inequalities using the operator layer-cake representation, improving finite-resource bounds in single-shot quantum information and establishing their optimality.

Contribution

It introduces a novel method for lifting scalar inequalities to the quantum setting, deriving tight bounds with optimal constants for key quantum information tasks.

Findings

Determined the exact optimal prefactor for a logarithmic trace inequality.

Established universal optimality for positive operators in quantum bounds.

Improved bounds in quantum covering, decoupling, and classical communication protocols.

Abstract

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 R\'enyi-divergence framework the resulting constants are genuine optimality barriers rather than artifacts of the proof.