Multiqubit monogamy relations beyond shadow inequalities

TL;DR

This paper develops new monogamy inequalities for multiqubit systems that extend shadow inequalities, enabling a complete characterization of sector lengths for systems with up to 5 qubits, and discusses the increased complexity for larger systems.

Contribution

It introduces a set of monogamy inequalities that complement shadow inequalities, allowing for a full characterization of sector lengths in small multiqubit systems.

Findings

Complete characterization of sector lengths for N≤5 qubits

The sector length range forms a convex polytope

Complexity increases significantly for N≥6 qubits

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.