AI 中文总结
本文刻画了随机费米子高斯态的魔法性，推导了其Pauli谱与稳定化熵的精确表达式，发现冻结现象，并证明典型态在丢弃一定比例模式后仍保持魔法性。
AI 中文摘要
我们通过随机费米子高斯态的Pauli谱及其稳定化熵来刻画其魔法性（magic）。对于马约拉纳高斯态的Haar系综，我们推导了平均稳定化纯度（stabilizer purities）的有限$N$闭合表达式，并精确重构了Pauli谱，将其表示为按马约拉纳权重分解的β分布变量乘积的混合。稳定化熵及其过滤版本在Rényi指数$q_c=2$处冻结：对于$q>2$，魔法密度为$1/(q-1)$，由罕见的权重为2的马约拉纳弦控制。解析这些贡献需要指数数量的特征样本，使得直接采样估计稳定化熵在该冻结区域效率低下。对于固定填充下的Haar随机高斯态，我们将正整数$q$处的平均稳定化纯度表示为$\u00a0mathrm{SU}(2q)$上的系数积分，其维度与$N$无关，并精确计算了$q=2$的情形。我们确立了冻结密度描述了马约拉纳系综和半填充数守恒系综中的典型态，并将这些系综与相应的$\mathrm{SYK}_2$基态联系起来。最后，我们证明在热力学极限下，典型费米子高斯态在丢弃任意固定比例$\kappa<\kappa_\star^{\rm G}\simeq0.7613$的模式后仍保留可认证的魔法性，即使在超过全希尔伯特空间中Haar随机态的$2/3$阈值后仍保持魔法性。
英文摘要
We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. For the Haar ensemble of Majorana Gaussian states we derive closed finite-$N$ expressions for the average stabilizer purities and reconstruct the Pauli spectrum exactly, as a mixture of products of beta-distributed variables resolved by Majorana weight. The stabilizer entropies, and their filtered versions, freeze at Rényi index $q_c=2$: for $q>2$ the magic density is $1/(q-1)$, controlled by rare weight-two Majorana strings. Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer $q$ as coefficient integrals over $\mathrm{SU}(2q)$, with dimension independent of $N$, and evaluate the $q=2$ case exactly. We establish that the frozen density describes typical states in the Majorana and half-filled number-conserving ensembles and relate these ensembles to the corresponding $\mathrm{SYK}_2$ ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction $κ<κ_\star^{\rm G}\simeq0.7613$ of their modes in the thermodynamic limit, remaining magical even beyond the $2/3$ threshold for Haar-random states in the full Hilbert space.
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