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:在本文中,我们得出了量信息理论中量子信息理论中一个基本工具,即最大反动酶,以及其平滑的版本。我们得出了量子信息理论中一个基本工具。在净化距离基础上平滑最大反动酶的量子状态微小修改在平滑最大反动酶的细微衰减的精确指数。然后,我们将这一结果应用于针对量子侧信息的隐私放大问题,我们获得了对无序减少不安全状况指数的上端线,通过纯化距离或相对加密测量。最后,我们的上界补充了Hayashi建立的较早期较低约束,而当随机提取速度超过一个关键值时,两个界限匹配了。因此,对于高率的情况,我们确定了确切的安全指数。在此之后,我们举例并表明,在低率案例中,上界和下界一般而言,我们得到的界限与频道编码中误差指数相似的图象。最后,我们调查的是,在安全等级差异度和美元指数(1个)下,在之前的变差状态下,在安全等级设置了1个(Ry)的变差。