Robust Quantum Extremal Numbers

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 $$D_A=2^{\mathrm{\mid }A\mathrm{\mid }}\mathrm{Tr}({\rho }_A^2)-1=2^{\mathrm{\mid }A\mathrm{\mid }}{\parallel {\rho }_A-\frac{I_A}{2^{\mathrm{\mid }A\mathrm{\mid }}}\parallel }_2^2,$$ and $Q_{\mathrm{e}\mathrm{x},\epsilon }^D(n,k)$ is defined as the maximum number of $k$-body marginals satisfying $D_A\le \epsilon$ 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: $$\sum _{i\in T}{D}_{T\setminus \{i\}}\ge 1(\mathrm{\mid }T\mathrm{\mid }=2m+1)\mathrm{.}$$ It follows that, whenever $\epsilon <1\mathrm{/}(2m+1)$, the hypergraph of $\epsilon$-good $2m$-subsets is $K_{2m+1}^{(2m)}$-free. Combined with the known exact eight-qubit construction, this yields the stability plateau $$Q_{\mathrm{e}\mathrm{x},\epsilon }^D(8,4)=56,0\le \epsilon <\frac{1}{5}\mathrm{.}$$ For odd systems of $2k+1$ qubits, the exact forbidden hypergraph $H_k$ is used to derive explicit finite-error stability radii. In particular, $Q_{\mathrm{e}\mathrm{x},\epsilon }^D(9,4)\le 120$ for $0\le \epsilon <1\mathrm{/}17$. 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.