The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the asymptotic decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the asymptotic decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding. Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched R\'enyi divergence of order $s\in (1,2]$, which has not been addressed previously in the quantum setting.

翻译：最大相对熵及其平滑版本是量子信息理论中的基本工具。本文中，我们基于纯化距离推导了在平滑最大相对熵过程中量子态微小修正的渐近衰减的精确指数。随后，我们将此结果应用于存在量子侧信息的隐私放大问题，并得到了不安全性（以纯化距离或相对熵度量）渐近衰减指数的一个上界。该上界补充了Hayashi先前建立的下界，且当随机提取速率高于临界值时，两个界相匹配。因此，对于高速率情形，我们确定了精确的安全指数。随后，我们给出实例并表明在低速率情形下，上界和下界通常都不是紧致的。这呈现出与信道编码中误差指数相似的现象。最后，我们研究了在阶数$s\in (1,2]$的三明治Rényi散度安全度量下，等价密钥率及其指数的渐近行为，该问题在量子设置中此前尚未被探讨过。