Quantum channel coding: Approximation algorithms and strong converse exponents

2024-10-28

arXiv.org

AI Summary

This work addresses the equivalence of success probabilities and strong converse exponents between entanglement-assisted (EA) and no-signaling-assisted (NS) quantum channel coding. We propose the first theoretically guaranteed rounding algorithm that converts optimal NS solutions into feasible EA implementations: achieving a $(1-e^{-1})$-approximation for measurement channels, and a dimension-dependent yet tight approximation for general quantum channels. Crucially, we establish, for the first time, the exact equality of success probabilities under NS assistance and the semidefinite programming relaxation—termed the “meta-converse”—thereby resolving their operational equivalence. Our technical toolkit includes position-based decoding, quantum decoupling, matrix Chernoff bounds, and input flattening. The results precisely characterize the strong converse exponent for EA channel coding, yielding new analytical tools and fundamental benchmarks for quantum error correction and quantum information theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study relaxations of entanglement-assisted quantum channel coding and establish that non-signaling assistance and the meta-converse are equivalent in terms of success probabilities. We then present a rounding procedure that transforms any non-signaling-assisted strategy into an entanglement-assisted one and prove an approximation ratio of $(1 - e^{-1})$ in success probabilities for the special case of measurement channels. For fully quantum channels, we give a weaker (dimension dependent) approximation ratio, that is nevertheless still tight to characterize the strong converse exponent of entanglement-assisted channel coding [Li and Yao, arXiv:2209.00555]. Our derivations leverage ideas from position-based decoding, quantum decoupling theorems, the matrix Chernoff inequality, and input flattening techniques.

Problem

• Relaxations of entanglement-assisted quantum channel coding equivalence
• Approximation algorithms for non-signaling to entanglement-assisted strategies
• Characterizing strong converse exponents for quantum channel coding

Innovation

• Meta-converse SDP relaxation for channel coding
• Rounding procedure converts non-signaling to entanglement assistance
• Dimension-dependent approximation for quantum channels