Multiqubit monogamy relations beyond shadow inequalities

TL;DR

This work characterizes multipartite quantum correlations in pure N-qubit states through sector lengths $S_m$, establishing monogamy inequalities that complement shadow inequalities. By combining reduced-density-matrix-based bounds with shadow-enumerator constraints, the authors prove that the achievable sector-length vectors form a convex polytope for $N\le 5$, enabling efficient optimization of linear-entanglement and shadow-based quantities via polytope vertices. They identify extremal pure states (e.g., GHZ, AME, and Dicke variants) that saturate the polytope's boundaries and provide detailed results for $N=2$–$6$, including explicit constructions and boundary states. For larger $N$, the structure becomes substantially more intricate, with numerical evidence suggesting that $\mathcal{S}$ is strictly contained in the larger region $\mathcal{R}$ and may no longer be a polytope; the work also connects sector lengths to linear entropies, shadow enumerators, and quantum coding theory, revealing practical implications for entanglement measures and error-correcting codes.

Abstract

Multipartite quantum systems are subject to monogamy relations that impose fundamental constraints on the distribution of quantum correlations between subsystems. These constraints can be studied quantitatively through sector lengths, defined as the average value of $m$-body correlations, which have applications in quantum information theory and coding theory. In this work, we derive a set of monogamy inequalities that complement the shadow inequalities, enabling a complete characterization of the numerical range of sector lengths for systems with $N\leq 5$ qubits in a pure state. This range forms a convex polytope, facilitating the efficient extremization of key physical quantities, such as the linear entropy of entanglement and the quantum shadow enumerators, by a simple evaluation at the polytope vertices. For larger systems ($N\geq 6$), we highlight a significant increase in complexity that neither our inequalities nor the shadow inequalities can fully capture.

Figures (3)

• Figure 1: Projection of the sets $\mathcal{R}_3 \subset \mathcal{R}_2 \subset \mathcal{R}_1$ onto the $(S_1 ,S_2)$ plane for $N=4$ (top) and $N=5$ qubits (bottom), colored in purple, red and green, respectively. For these particular numbers of qubits, $\mathcal{R} = \mathcal{R}_3$ and every point in $\mathcal{R}_3$ corresponds to a physically realizable quantum state.
• Figure 2: Projection of the sets $\mathcal{R}_1$, $\mathcal{R}_2 \subset \mathcal{R}_1$, $\mathcal{R}_3 \subset \mathcal{R}_1$, and $\mathcal{R}=\mathcal{R}_2\cap \mathcal{R}_3$ onto the $(S_1 ,S_2, S_3)$ subspace for $N=6$. Note that $\mathcal{R}_3$ is not contained in $\mathcal{R}_2$ and vice versa (see the differences in the faces of the polyhedra in the planes $S_2=0$ and $S_1=0$).
• Figure 3: Top: the polytope $\mathcal{R}=\mathcal{R}_2 \cap \mathcal{R}_3$ for $N=6$ in the $(S_1,S_2,S_3)$ space. Bottom: smaller polytope obtained by including additional linear constraints given in Eq. \ref{['reducedpolytopeN6']}, derived by numerical optimization. States corresponding to vertices or points along the edges are indicated. The vertex $P=(0,7,8)$, shown in grey, is one for which we have not identified any state realizing these sector lengths.

Theorems & Definitions (9)

• Proposition 1
• Theorem 1
• Conjecture 1
• Proposition 2
• Lemma 1
• Theorem 2
• Theorem 3
• Conjecture 2
• Proposition 3