Typical Superpositions are Magical

The non-stabilizerness, or "magic", of a quantum state is a common metric for the hardness of simulation on a classical computer, and is often quantified by the stabilizer Rényi entropy (SRE). We show how random superpositions of classically simulable states saturate the SRE to the symmetry-restricted Haar-random maximum. For any large- N random pure-state ensemble satisfying the conditions of Schur's lemma, the average SRE of the m-component superposition M 2(m)=N- 2[W(m)] is governed by an average witness W(m) that universally decomposes into a Haar floor and an ensemble-specific excess. As the number of components m increases, the excess is suppressed, resulting in a logarithmic scaling of the SRE before plateauing at the Haar-valued maximum. We derive a general closed form for W(m) in the thermodynamic limit, and numerically verify our result for random stabilizer states and random Gaussian states.