Convex Split Lemma without Inequalities

2025-02-10

AI Summary

This work addresses the looseness of the convex-split lemma’s bounds in quantum source coding. Methodologically, it integrates quantum information theory, smooth entropy techniques, and Rényi entropy analysis. First, it replaces the max-mutual information with the collision mutual information, thereby tightening the original inequality into an exact equality—significantly improving achievability bounds for quantum state merging and state splitting. Second, it derives a dimension-independent universal upper bound on the smooth max-mutual information, expressed solely in terms of Rényi entropies. These contributions provide novel analytical tools for foundational problems—including the reverse quantum Shannon theorem—and enhance both the theoretical precision and practical applicability of quantum communication and compression protocols.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

We introduce a refinement to the convex split lemma by replacing the max mutual information with the collision mutual information, transforming the inequality into an equality. This refinement yields tighter achievability bounds for quantum source coding tasks, including state merging and state splitting. Furthermore, we derive a universal upper bound on the smoothed max mutual information, where"universal"signifies that the bound depends exclusively on R'enyi entropies and is independent of the system's dimensions. This result has significant implications for quantum information processing, particularly in applications such as the reverse quantum Shannon theorem.

Problem

• Refining convex split lemma
• Improving quantum source coding
• Deriving universal upper bound

Innovation

• Uses collision mutual information
• Transforms inequality into equality
• Derives universal upper bound