A trace distance-based geometric analysis of the stabilizer polytope for few-qubit systems

Non-stabilizerness is a fundamental resource for quantum computational advantage, differentiating classically simulable circuits from those capable of universal quantum computation. Recently, non-stabilizerness has been shown to be relevant for a few qubit systems. 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 non-stabilizerness. By randomly sampling quantum states, we analyze the distribution of non-stabilizerness for both pure and mixed states and compare the trace distance with other non-stabilizerness measures, as well as entanglement. Additionally, we give an analytical expression for the introduced quantifier, classify Bell-like inequalities corresponding to the facets of the stabilizer polytope, and establish a general concentration result connecting non-stabilizerness and entanglement via Fannes’ inequality. Our findings provide new insights into the geometric structure of non-stabilizerness and its role in small-scale quantum systems, offering a deeper understanding of the interplay between quantum resources

💡 Research Summary

This paper investigates the geometry of the stabilizer polytope for few‑qubit quantum systems and introduces a trace‑distance‑based quantifier of non‑stabilizerness (often called “magic”). The authors define the non‑stabilizerness measure NTD(ρ) as the minimum trace distance between a quantum state ρ and any point σ inside the stabilizer polytope, i.e.
 NTD(ρ)=½ min_{σ∈P_STAB}‖ρ−σ‖₁.
Because the trace distance has a clear operational meaning—distinguishability under optimal measurements—NTD applies to both pure and mixed states, unlike many earlier magic monotones that are defined only for pure states or require costly quasiprobability calculations.

The paper first reviews the stabilizer formalism, the Clifford group, and the convex hull of all pure stabilizer states, which defines the stabilizer polytope P_STAB. For one‑qubit, one‑qutrit (d=3), and two‑qubit systems the authors explicitly enumerate the polytope’s facets (inequalities) and vertices. In the one‑qubit case the eight facets form an octahedron inscribed in the Bloch sphere; a single representative inequality I₂=1+⟨X⟩+⟨Y⟩+⟨Z⟩≥0 generates all others via Clifford conjugation. The T‑state ρ_T=½