Entropy is a key concept of quantum information theory. The entropy of a quantum system is a measure of its randomness and has many applications in quantum communication protocols, quantum coherence, and so on. In this paper, based on the Rényi entropy and Tsallis entropy, we derive the bounds of the expectation value and variance of a quantum observable. By the maximal value of Rényi entropy, we show an upper bound on the product of variance and entropy. Furthermore, we obtain the reverse uncertainty relation for the product and sum of the variances for n observables respectively.

中文翻译：

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通过 Rényi 熵和 Tsallis 熵对量子可观测量的期望值和方差进行界

熵是量子信息论的一个关键概念。量子系统的熵是其随机性的度量，在量子通信协议、量子相干等方面有很多应用。在本文中，基于 Rényi 熵和 Tsallis 熵，我们推导了一个量子可观测量的期望值和方差的界限。通过 Rényi 熵的最大值，我们展示了方差和熵乘积的上限。此外，我们获得了乘积的反向不确定关系和方差之和n分别是可观察的。