On the geometry and typicality of quantum magic

Zhenhuan Liu, Z-Wen Liu (Sep 04 2026).

Abstract: We prove that, for an $n$-qubit system of dimension $d=2^n$, every state satisfying $\operatorname{Tr}(\rho^2)\le 1/(d-a_\ast)$, 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 $\Omega(d^2/\log^2d)$ and $\mathcal{O}(d^2)$. We further prove that the number of facets of the stabilizer polytope lies between $\exp[\Omega(d^2/\log^2 d)]$ 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.

Arxiv: https://arxiv.org/abs/2609.03944