We initiate an investigation into a notion of state complexity for discrete-variable quantum systems. Specifically, we propose an information-theoretic quantifier for the complexity of quantum states within the stabilizer formalism of quantum computation. This is achieved by leveraging the symmetric Jordan product (associated with classicality) and the skew-symmetric Lie product (linked to quantumness) between the square root of the quantum state and the Heisenberg-Weyl displacement operators. We reveal fundamental properties of this quantifier and demonstrate that state complexity is closely related to the nonstabilizerness of quantum states via the $L^4$-norm of their characteristic functions. By the way, we characterize the maximally mixed states, the stabilizer states, and the symmetric informationally complete positive operator valued measure fiducial states from the perspective of state complexity.

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• Quantum Fisher information
• Quantum computation
• Quantum foundations
• Quantum information theory
• Resource theories