On the geometry and typicality of quantum magic

• URL: http://arxiv.org/abs/2609.03944v2
• Date: Tue, 08 Sep 2026 16:32:44 GMT
• Title: On the geometry and typicality of quantum magic
• Abstract summary: We prove that for an $n$-qubit system of dimension $d=2n$, every state satisfying $operatornameTr(2)le 1/(d-a_ast)$ lies inside the stabilizer polytope and is therefore magic-free.
• Score: 2.9443300988715944
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\exp[\mathcal{O}(d^2\log^2 d)]$ employing a result of Bourgain and Milman in convex geometry, substantially improving upon the previous quasipolynomial lower bound and implying that doubly-exponentially many linear inequalities in the number of qubits are required for an exact description of the magic-free region. Overall, our results reveal the near-extremal geometry of the stabilizer polytope and provide a quantitative foundation for understanding the typicality, robustness, and detectability of magic.

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