We investigate the extremality of stabilizer states to reveal their exceptional role in the space of all $n$-qubit/qudit states. We establish uncertainty principles for the characteristic function and the Wigner function of states, respectively. We find that only stabilizer states achieve saturation in these principles. Furthermore, we prove a general theorem that stabilizer states are extremal for convex information measures invariant under local unitaries. We explore this extremality in the context of various quantum information and correlation measures, including entanglement entropy, conditional entropy and other entanglement measures. Additionally, leveraging the recent discovery that stabilizer states are the limit states under quantum convolution, we establish the monotonicity of the entanglement entropy and conditional entropy under quantum convolution. These results highlight the remarkable information-theoretic properties of stabilizer states. Their extremality provides valuable insights into their ability to capture information content and correlations, paving the way for further exploration of their potential in quantum information processing.

翻译：我们研究稳定子态的极值性，以揭示其在所有$n$量子比特/qudit态空间中的特殊作用。我们分别建立了特征函数和维格纳函数的不确定性原理，发现只有稳定子态能实现这些原理的饱和。进一步地，我们证明了一个一般性定理：稳定子态是局部酉变换下凸信息测度的极值点。我们探讨了这种极值性在各种量子信息与关联测度中的体现，包括纠缠熵、条件熵及其他纠缠测度。此外，基于量子卷积下稳定子态作为极限态的最新发现，我们建立了纠缠熵和条件熵在量子卷积下的单调性。这些结果凸显了稳定子态卓越的信息论特性，其极值性为理解它们捕获信息内容与关联的能力提供了深刻见解，并为进一步探索其在量子信息处理中的潜力开辟了道路。