We investigate the extremality of stabilizer states to reveal their exceptional role in the space of all $n$-qubit and $n$-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 Gaussian properties of stabilizer states in qudit and qubit systems.

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• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism
• Quantum foundations
• Quantum information theory
• Resource theories
• Uncertainty relation