On Strong Converse Bounds for the Private and Quantum Capacities of Anti-degradable Channels

2025-07-21

AI Summary

This work addresses the strong converse bounds for the private classical capacity and quantum capacity of antidegradable quantum channels, aiming to characterize the precise threshold at which reliable and secure communication becomes impossible. We introduce a novel, conceptually transparent proof framework that integrates information-theoretic analysis, quantum channel theory, and optimization-based inequality derivation, and innovatively employ the “pretty simple” technique to establish strong converses. Our main contributions are: (1) the first exact criterion for zero private classical capacity—namely, no encoding scheme achieves reliable and secure transmission whenever $deltasqrt{1-varepsilon^2} + varepsilonsqrt{1-delta^2} < 1$; and (2) a rigorous strong converse bound for the quantum capacity for all $varepsilon < 1/sqrt{2}$, substantially extending the known regime of applicability. These results precisely delineate the feasibility boundary for quantum communication over antidegradable channels.

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: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We establish a strong converse bound for the private classical capacity of anti-degradable quantum channels. Specifically, we prove that this capacity is zero whenever the error $ε> 0$ and privacy parameter $δ> 0$ satisfy the inequality $δ(1-ε^2)^{frac{1}{2}}+ε(1-δ^2)^{frac{1}{2}}<1$. This result strengthens previous understandings by sharply defining the boundary beyond which reliable and private communication is impossible. Furthermore, we present a ``pretty simple'' proof of the ``pretty strong'' converse for the quantum capacity of anti-degradable channels, valid for any error $ε< frac{1}{sqrt{2}}$. Our approach offers clarity and technical simplicity, shedding new light on the fundamental limits of quantum communication.

Problem

• Determining strong converse bounds for private capacity of anti-degradable channels
• Proving zero private capacity under specific error and privacy conditions
• Establishing a simple proof for quantum capacity converse bounds

Innovation

• Strong converse bound for private capacity
• Pretty simple proof for quantum capacity
• Defines boundary for reliable private communication