UNIFIED MONOGAMY AND POLYGAMY RELATIONS FOR MULTIPARTITE SYSTEMS

Quantum entanglement is inherently limited in how it can be shared among multiple parties, a property known as monogamy. Conversely, assisted entanglement describes how correlations can be spread with the help of an external party, known as polygamy. Existing mathematical bounds for these phenomena are often loose, failing to capture the complex, asymmetric ways entanglement is distributed in many-body systems. This paper seeks to develop a more flexible, hierarchical framework to provide tighter, more physically nuanced constraints on entanglement sharing.

The authors propose a unified framework indexed by a hierarchy parameter (m) and state-dependent asymmetry parameters (mu and nu). By increasing the integer parameter m, the framework activates higher-order correction terms, constructing a tower of inequalities that become progressively tighter. The asymmetry parameters allow the bounds to adapt to the specific geometry of the quantum state, accounting for imbalances in how entanglement is partitioned across different subsystems. The researchers derive these refined inequalities for both the alpha-power of monogamy and the beta-power of polygamy, validating them through analytical comparisons and numerical evaluations using concurrence and concurrence of assistance.

The study demonstrates that the proposed hierarchical approach significantly outperforms existing, standard inequalities. The optimal monogamy bound emerges as a piecewise function of the power parameter alpha, where additional correction terms are activated as alpha crosses successive integer thresholds. This creates a staircase-like refinement that offers a strictly sharper characterization of entanglement distribution than previous continuous formulations. Numerical examples using three-qubit states show that these new bounds provide the tightest known estimates for entanglement, particularly in systems where correlations are highly asymmetric.