Geometric Analysis of the Stabilizer Polytope for Few-Qubit Systems

Abstract: Non-stabilizerness, or magic, is a fundamental resource for quantum computational advantage, differentiating classically simulatable circuits from those capable of universal quantum computation. In this work, we investigate the geometry of the stabilizer polytope in few-qubit quantum systems, using the trace distance to the stabilizer set to quantify magic. By randomly sampling quantum states, we analyze the distribution of magic for both pure and mixed states and compare the trace distance with other magic measures, as well as entanglement. Additionally, we classify Bell-like inequalities corresponding to the facets of the stabilizer polytope and establish a general concentration result connecting magic and entanglement via Fannes' inequality. Our findings provide new insights into the geometric structure of magic and its role in small-scale quantum systems, offering a deeper understanding of the interplay between quantum resources.