Thermal entanglement on a frustrated tetrahedron: Probing quantum resources through concurrence and stabilizer structure
In a frustrated spin system, highly entangled eigenstates can form a separable thermal mixture. We study this distinction for four spin-$1/2$ moments on a tetrahedron, the elementary unit of the pyrochlore lattice, with the general four-parameter exchange Hamiltonian. Tetrahedral symmetry allows a restricted search over thermal-state decompositions to be carried out by linear optimization, yielding an upper bound on multipartite concurrence and explicit separable decompositions where this bound vanishes. At low temperature, the concurrence maps show extended separable regions near the all-in--all-out limit, whereas suppression near the spin-ice point is narrowly localized. Heating broadens the latter region as nearby multiplets are repopulated. The optimization identifies fully separable thermal states even when the eigenstate-averaged concurrence remains large. Negativity independently confirms finite-temperature entanglement in selected coupling regions. A mixed-state stabilizer Rényi diagnostic, used to explore quantum magic, favours different couplings and can increase locally with temperature. These results connect thermal entanglement to the splitting and population of symmetry multiplets, and provide a practical basis for identifying frustrated spin configurations for studies of quantum resources beyond the ground-state limit.