Curator's Take
AI Commentary
This article pinpoints a precise purity threshold— Tr(ρ²)≤1/(2ⁿ‑a*) with a*≈0.4583—below which any n‑qubit state is guaranteed to be magic‑free, giving the first rigorous geometric boundary for the stabilizer polytope in high dimensions. By combining that bound with volume‑radius estimates, the authors reveal a sharp phase transition in the likelihood of “magic” appearing when random pure states are partially traced out, and they show that describing the magic‑free region exactly would require doubly‑exponential numbers of linear constraints. The results tighten our understanding of how rare genuine quantum resources are in generic states, sharpening benchmarks for magic‑state distillation and informing hardware designers about the intrinsic difficulty of generating useful non‑stabilizer resources.
— Mark Eatherly
Summary
We prove that, for an $n$-qubit system of dimension $d=2^n$, every state satisfying $\operatorname{Tr}(ρ^2)\le 1/(d-a_\ast)$, with $a_\ast=0.458327\cdots$, lies inside the stabilizer polytope and is therefore magic-free. Combining this result with general geometric properties of high-dimensional polytopes, we establish quantitative estimates for the Hilbert--Schmidt inradius and volume radius of the stabilizer polytope, and use them to characterize the typicality of magic in random induced states obtained by tracing out a $k$-dimensional subsystem from a $d\times k$-dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension $k_\star$ is bounded between $Ω(d^2/\log^2d)$ and $\mathcal{O}(d^2)$. We further prove that the number of facets of the stabilizer polytope lies between $\exp[Ω(d^2/\log^2 d)]$ and $\exp[\mathcal{O}(d^2\log^2 d)]$, substantially improving upon the previous quasipolynomial lower bound and implying that any exact description of the magic-free region requires a doubly exponential number of linear inequalities in the number of qubits. Overall, our results show that the stabilizer polytope exhibits near-maximal geometric complexity allowed for a high-dimensional polytope with a certain number of vertices.