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

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 $\epsilon >0$ and privacy parameter $\delta >0$ satisfy the inequality $\delta (1-{\epsilon }^2{)}^{\frac{1}{2}}+\epsilon (1-{\delta }^2{)}^{\frac{1}{2}}<1$. 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 $\epsilon <\frac{1}{\sqrt{2}}$. Our approach offers clarity and technical simplicity, shedding new light on the fundamental limits of quantum communication.