Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing

In this paper, we study the tradeoffs between the error probabilities of classical-quantum channels and the blocklength <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> when the transmission rates approach the channel capacity at a rate lower than <inline-formula> <tex-math notation="LaTeX">$1/\sqrt {n}$ </tex-math></inline-formula>, a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight quantum sphere-packing bound, obtained via Chaganty and Sethuraman’s concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation analysis for quantum hypothesis testing is also established. The converse directly follows from our channel coding result, while the achievability relies on a martingale inequality.

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