On the geometry and typicality of quantum magic
Abstract
We prove that, for an n-qubit system of dimension d=2n, every state satisfying Tr(ρ2)≤ 1/(d−a∗), with a∗=0.458327⋯, 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× k-dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension k⋆ is bounded between Ω(d2/log2d) and 𝒪(d2). We further prove that the number of facets of the stabilizer polytope lies between exp[Ω(d2/log2 d)] and exp[𝒪(d2log2 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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BibTeX
@misc{liu2026geometry,
title = {{On the geometry and typicality of quantum magic}},
author = {Zhenhuan Liu and Zi-Wen Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03944},
eprint = {2609.03944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03944},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the geometry and typicality of quantum magic
AU - Zhenhuan Liu
AU - Zi-Wen Liu
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.03944
UR - https://arxiv.org/abs/2609.03944
AB - We prove that, for an n-qubit system of dimension d=2^n, every state satisfying Tr(ρ^2)≤ 1/(d−a_∗), with a_∗=0.458327⋯, 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× k-dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension k_⋆ is bounded between Ω(d^2/log^2d) and 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[𝒪(d^2log^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.
ER -