Robust Quantum Extremal Numbers

Abstract

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 \[ D_A=2^{|A|}\operatorname{Tr}(\rho_A^2)-1 =2^{|A|}\left\|\rho_A-\frac{I_A}{2^{|A|}}\right\|_2^2, \] 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 Tur\'an obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.