Unified monogamy and polygamy relations for multipartite systems

For a bipartite entanglement measure mathcal{E} that satisfies the γth-power monogamy inequality Eq. eqref{e:chap1-ineq1}, and for its assisted counterpart mathcal{E}_a that obeys the δth-power polygamy inequality Eq. eqref{e:chap1-ineq2}, we introduce a unified, tunable framework indexed by a parameter mgeq1. Within this framework, we derive two hierarchical families of refined inequalities: a tightened α-power monogamy relation for mathcal{E}, valid for all αgeq mγ; a tightened β-power polygamy relation for mathcal{E}_a, applicable for (m-1)δ< βleq mδ. As m increases, the bounds become progressively tighter, recovering known results at m=1. Notably, the optimal monogamy bound emerges as a piecewise function of α, with additional correction terms activated as α crosses successive integer thresholds, thereby offering a sharper characterization of entanglement distribution. We demonstrate that our results generalize and strengthen existing monogamy and polygamy relations through analytical comparisons and numerical evaluations using concurrence and concurrence of assistance. This hierarchical, parameterized approach offers enhanced and flexible tools for applications in quantum communication, quantum networks, and multipartite quantum information processing.