AI 中文总结
本研究拓展量子极值数为鲁棒版本，定义约化态混合缺陷与带误差界的极值计数，通过局部稳定性不等式推导超图禁戒条件，得到多量子比特系统的稳定区间与上界，将精确Turán障碍转化为定量鲁棒性结果。
AI 中文摘要
绝对最大纠缠态要求至多半数参与者的每个约化态都为最大混合态，这一条件不仅严格，对量子比特系统而言通常还无法实现。此前的研究引入了量子极值数，用于最大化恰好为最大混合态的半体约化态的数量，并确定了精确值Qex(8,4)=56。本研究对这一极值问题进行了鲁棒性拓展。对于子系统$A$，我们将约化态最大混合缺陷定义为：$D_A=2^{|A|}\text{Tr}(ρ_A^2)-1 =2^{|A|}\\|ρ_A-\frac{I_A}{2^{|A|}}\\|_2^2$，并将$Q_{\mathrm{ex},\varepsilon}^{D}(n,k)$定义为n量子比特纯态中满足$D_A\leq\varepsilon$的k体约化态的最大数量。这一计数问题不同于近似k均匀性，后者要求所有k体约化态都满足统一的误差界。\n 对于4m个量子比特的纯态，我们建立了如下局部稳定性不等式：$\sum_{i\in T}D_{T\setminus\{i\}}\geq1$（其中$|T|=2m+1$）。由此可得，当$\varepsilon<1/(2m+1)$时，由ε优的2m子集构成的超图不含$K_{2m+1}^{(2m)}$子结构。结合已知的精确八量子比特构造，我们得到了稳定性平台：当$0\leq\varepsilon<\frac15$时，$Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56$。对于2k+1个量子比特的奇数系统，我们利用精确禁戒超图$H_k$推导出了显式的有限误差稳定半径。具体而言，当$0\leq\varepsilon<1/17$时，$Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120$。这些结果将精确量子Turán障碍转化为定量的鲁棒性表述，并确定了量子极值数在约化态混合度不完美时仍保持稳定的区间。
英文摘要
Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem $A$, the marginal maximal-mixing defect is defined by \[ D_A=2^{|A|}\operatorname{Tr}(ρ_A^2)-1 =2^{|A|}\left\|ρ_A-\frac{I_A}{2^{|A|}}\right\|_2^2, \] and $Q_{\mathrm{ex},\varepsilon}^{D}(n,k)$ is defined as the maximum number of $k$-body marginals satisfying $D_A\leq\varepsilon$ in an $n$-qubit pure state. This counting problem differs from approximate $k$-uniformity, which requires all $k$-body marginals to obey a common error bound. For pure states on $4m$ qubits, the following local stability inequality is established: \[ \sum_{i\in T}D_{T\setminus\{i\}}\geq1 \qquad (|T|=2m+1). \] It follows that, whenever $\varepsilon<1/(2m+1)$, the hypergraph of $\varepsilon$-good $2m$-subsets is $K_{2m+1}^{(2m)}$-free. Combined with the known exact eight-qubit construction, this yields the stability plateau \[ Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15. \] For odd systems of $2k+1$ qubits, the exact forbidden hypergraph $H_k$ is used to derive explicit finite-error stability radii. In particular, $Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120$ for $0\leq\varepsilon<1/17$. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.
