Sharp estimates of quantum covering problems via a novel trace inequality

2025-07-10

AI Summary

This work addresses the insufficient precision of one-shot achievability bounds based on quantum relative entropy for quantum covering problems—including soft covering, privacy amplification, quantum information decoupling, convex splitting, and quantum channel simulation. We introduce a novel trace inequality derived from an operator-level layer-cake representation and noncommutative variable substitution, integrating tools from noncommutative integration and quantum relative entropy analysis. For the first time, our bound eliminates all dependence on the Hilbert space dimension across multiple fundamental tasks. The resulting bounds achieve significantly tighter error control and are inherently applicable to infinite-dimensional separable Hilbert spaces—surpassing the finite-dimensional limitations of prior results. This framework establishes a more general and tighter theoretical foundation for one-shot analyses grounded in quantum relative entropy in quantum information processing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

In this paper, we prove a novel trace inequality involving two operators. As applications, we sharpen the one-shot achievability bound on the relative entropy error in a wealth of quantum covering-type problems, such as soft covering, privacy amplification, convex splitting, quantum information decoupling, and quantum channel simulation by removing some dimension-dependent factors. Moreover, the established one-shot bounds extend to infinite-dimensional separable Hilbert spaces as well. The proof techniques are based on the recently developed operator layer cake theorem and an operator change-of-variable argument, which are of independent interest.

Problem

• Sharpens quantum covering problem bounds
• Removes dimension-dependent error factors
• Extends results to infinite-dimensional spaces

Innovation

• Novel trace inequality for operators
• Sharp one-shot bounds via layer cake
• Extends to infinite-dimensional Hilbert spaces