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 decay of the small modification of the quantum state in smoothing the max-relative entropy. 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 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.

翻译：最大反射酶及其平滑版本是量子信息理论的一个基本工具。 在本文中, 我们得出了量子状态小变换在平滑最大反射酶中衰减的确切缩记。 然后我们用这个结果对量子侧信息的隐私放大问题应用这个结果, 并且我们用纯度距离或相对加密测量的不安全感下降的缩略图获得一个上方框。 我们的上边框补充了Hayashi 建立的较早的较低约束值, 而当随机提取率超过临界值时, 两个边框相匹配。 因此, 在高率的情况下, 我们确定了精确的安全缩写。 在此之后, 我们举个例子并显示, 在低率的情况下, 上边或下边的界限一般都是不紧的。 这展示了类似于频道编码中错误的缩写图 。