Extremizing Measures of Magic on Pure States by Clifford-stabilizer States

Abstract: Magic states are essential resources enabling universal, fault-tolerant quantum computation within the stabilizer framework. Their non-stabilizerness provides the additional resource required to overcome the constraints of stabilizer codes, as formalized by the Eastin-Knill theorem, while still permitting fault-tolerant distillation. Although numerous measures of magic have been introduced, not every state with nonzero magic has been shown to be distillable by a stabilizer code, and all currently known distillable states arise as special cases of Clifford-stabilizer states, defined as pure states uniquely stabilized by finite subgroups of the Clifford group. In this work, we develop a general framework for group-covariant functionals on the real manifold of Hermitian operators. We formalize the notions of $G$-stabilizer spaces, states, and codes for arbitrary finite subgroups $G \subset \mathrm{U}(\mathcal{H})$, and introduce analytic families of $G$-covariant functionals. Our main theorem shows that any $G$-invariant pure state is an extremal point of a broad class of derived functionals, including symmetric, max-type, and Rényi-type functionals, provided the underlying family is $G$-covariant. This extremality holds for variations restricted to directions orthogonal to the stabilized subspace while preserving purity. Specializing to the Pauli and Clifford groups, our framework unifies the extremality structure of several canonical magic measures, including mana, stabilizer Rényi entropies, and stabilizer fidelity. In particular, Clifford-stabilizer states extremize these measures. We classify such states for qubits, qutrits, ququints, and two-qubit systems, identifying new candidates for magic distillation protocols. We further propose an inefficient distillation protocol for a two-qubit magic state with stabilizer fidelity exceeding that of standard benchmark states.