Strong Converse for Privacy Amplification against Quantum Side Information

TL;DR

This paper proves a strong converse bound for privacy amplification against quantum side information, showing exponential convergence of trace distance in finite blocklength scenarios and applications to quantum communication security.

Contribution

It establishes a one-shot strong converse bound for privacy amplification with quantum side information, extending to entropy accumulation and moderate deviation regimes.

Findings

Trace distance exponentially converges to one when rate exceeds quantum conditional entropy.

Trace distance vanishes when rate exceeds quantum mutual information, ensuring security.

Results apply to classical-quantum wiretap channels and private quantum communication.

Abstract

We establish a one-shot strong converse bound for privacy amplification against quantum side information using trace distance as a security criterion. This strong converse bound implies that in the independent and identical scenario, the trace distance exponentially converges to one in every finite blocklength when the rate of the extracted randomness exceeds the quantum conditional entropy. The established one-shot bound has an application to bounding the information leakage of classical-quantum wiretap channel coding and private communication over quantum channels. That is, the trace distance between Alice and Eavesdropper's joint state and its decoupled state vanishes as the rate of randomness used in hashing exceeds the quantum mutual information. On the other hand, the trace distance converges to one when the rate is below the quantum mutual information, resulting in an exponential strong converse. Our result also leads to an exponential strong converse for entropy accumulation, which complements a recent result by Dupuis [arXiv:2105.05342]. Lastly, our result and its applications apply to the moderate deviation regime. Namely, we characterize the asymptotic behaviors of the trace distances when the associated rates approach the fundamental thresholds with speeds slower than $O(1/\sqrt{n})$.