bibkey: cepollaro2025stabilizer authors: S. Cepollaro, G. Cuffaro, M. B. Weiss, S. Cusumano, A. Hamma, S. Lloyd year: 2025 title: “Stabilizer Entropy of Subspaces” doi: 10.48550/arXiv.2512.23013 url: https://arxiv.org/abs/2512.23013v1 claim: “Numerically, when the single-qudit dimension d_B is a power of 2, a three-dimensional subspace with zero average stabilizer entropy gap is always achievable; the authors state that it is unclear why.” strata_touched:

• D5/S3/Quantum/Magic/CepollaroPowerOfTwoZeroMagicGap license: citation-only triage: anchor

Cepollaro, Cuffaro, Weiss, Cusumano, Hamma and Lloyd, stabilizer entropy of subspaces

S. Cepollaro, G. Cuffaro, M. B. Weiss, S. Cusumano, A. Hamma and S. Lloyd, Stabilizer Entropy of Subspaces, arXiv:2512.23013v1 (28 December 2025, quant-ph).

Verified locator

DOI: 10.48550/arXiv.2512.23013. Primary version: https://arxiv.org/abs/2512.23013v1 (the only arXiv version). The TeX source subspaceSRE_arxiv.tex of v1 supplies §II (Weyl–Heisenberg group and linear stabilizer entropy), §III (the average stabilizer entropy gap and its closed form), §V.A and the Conclusion.

Source statements

Weyl–Heisenberg group (§II.A): “$Z|k⟩\equiv {\omega }^k|k⟩$, $X|k⟩\equiv |k+1\text{ mod }d⟩$, where $\omega =e^{2\pi i/d}$”, and “$D_{\text{a}}\equiv {\tau }^{a_1{a}_2}{X}^{a_1}{Z}^{a_2}$, where $\tau =-e^{i\pi /d}$”.

Linear stabilizer entropy (§II.C): “M(\psi)\equiv M_{\lin}^{(2)}(\psi)=1-\frac{1}{d^{n}}\sum_{\B{a}}|\trace(D_{\B{a}}^\dagger \psi)|^4=1-d^n\trace\left(Q\,\psi^{\otimes4}\right)”, where “$Q_{\alpha }=\frac{1}{(d^n{)}^{\alpha }}{\sum }_{\text{a}}(D_{\text{a}}\otimes D_{\text{a}}^{†}{)}^{\otimes \alpha }$” and $Q\equiv Q_2$; on one qudit (§III) “$Q_S=\frac{1}{d_S^2}{\sum }_{\text{a}}(D_{\text{a}}^S\otimes D_{\text{a}}^{S†}{)}^{\otimes 2}$”.

Average stabilizer entropy gap (§III): “\Delta M(\mathcal{E}) \equiv \mathbb{E}_{\psi}\big[\Delta M(\psi)\big] = d_S\trace\left(Q_S \mathcal{A}^S_4\right)-d_B\trace\left(Q_B \mathcal{E}^{\otimes 4}(\mathcal{A}^S_4)\right)”, with “${\mathcal{A}}_4^S={\mathbb{E}}_U[(U\psi U^{†}{)}^{\otimes 4}]={\left(\genfrac{}{}{0ex}{}{d_S+3}{4}\right)}^{-1}{\Pi }_{{\text{sym}}^4}^S$” and “${\Pi }_{{\text{sym}}^4}^S=\frac{1}{24}{\sum }_{\sigma \in S_4}{T}_{\sigma }$”, and the intrinsic average “$1-\frac{3}{d_S+2}$” for odd $d_S$.

§V.A: “But this is not the only case: evidently, when $d_B$ is a power of 2, a zero ASE subspace of dimension three is always achievable.”

Conclusion: “It is unclear why, in the single-qudit case, when the global Hilbert space has dimension a power of two, zero magic gap subspaces of dimension three consistently emerge.”

Scope

The observation comes from numerical optimization over three-dimensional subspaces for $d_B=3,\dots ,16$. The paper proves zero or negative gap for stabilizer codespaces (its Theorem 2); a three-dimensional subspace of ${\mathbb{C}}^{2^m}$ is not such a codespace.