We have obtained the optimal upper bound of entropic uncertainty relation for $N$ Mutually Unbiased Bases (MUBs). We have used the methods of variational calculus for the states that can be written in terms of $N$ MUBs. Our result is valid for any state when $N$ is $d+1$, where $d$ is the dimension of the related system. We provide a quantitative criterion for the extendibility of MUBs based on entropic certainty relation. In addition, we have applied our result to information exclusion relation of $(d+1)$ observables conditioned with a classical memory which can be used to detect the compatibility of the observables.

中文翻译：

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互无偏基的熵不确定关系的最优上界

我们已经获得了熵不确定关系的最优上限 $N$相互无偏的基础 (MUB)。对于可以写成以下形式的状态，我们已经使用了变分微积分的方法$N$MUB。我们的结果对任何状态都有效$N$ 是 $d+1$， 在哪里 $d$是相关系统的维度。我们为基于熵确定性关系的 MUB 的可扩展性提供了一个定量标准。此外，我们将我们的结果应用于信息排除关系$(d+1)$ 以经典记忆为条件的 observables，可用于检测 observables 的兼容性。