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 $ε> 0$ 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.