Maximum residual strong monogamy inequality for multiqubit entanglement

TL;DR

This work advances the quantitative understanding of multipartite entanglement monogamy by introducing two new inequalities: the weighted strong monogamy (WSM) and the maximum residual strong monogamy (MRSM). WSM uses coefficient weights over $m$-partite contributions, while MRSM relies on the maximum $m$-partite term, both proven for the squared concurrence in arbitrary $n$-qubit states and extendable to other convex entanglement measures. Through analytical proofs and comparisons, the authors show WSM and MRSM sharpen prior CKW-type bounds and demonstrate MRSM's ability to capture genuine residual entanglement that vanishes for separable states, with concrete four- and five-qubit examples illustrating trade-offs among entanglement components of differing party counts. The results provide a rigorous framework for quantifying multipartite entanglement distribution and suggest directions for sharper bounds and higher-dimensional extensions with potential operational relevance.

Abstract

We establish two new inequalities, the weighted strong monogamy (WSM) and the maximum residual strong monogamy (MRSM), which sharpen the generalized Coffman-Kundu-Wootters inequity for multiqubit states. The WSM inequality distinguishes itself from the strong monogamy (SM) conjecture [Phys. Rev. Lett. 113, 110501 (2014)] by using coefficients rather than exponents to modulate the weight allocated to various m-partite contributions. In contrast, the MRSM inequality is formulated using only the maximum m-partite entanglement. We find that the residual entanglement of the MRSM inequality can effectively distinguish the separable states. We also compare the tightness of various SM inequalities and provide examples using a four-qubit mixed state and a five-qubit pure state to illustrate the MRSM inequality. These examples characterize the trade-off relations among entanglement components involving varying numbers of qubits. Our results provide a rigorous framework to characterize and quantify the monogamy of multipartite entanglement.

Figures (3)

• Figure 1: (Color online) Dependence of $T_1-T_2$ on the parameter $a$ for the states: $|G_{abc}^{2}\rangle$ with $c=b+1=a$ and $b \ge 0$ (red solid line), $|G_{ab}^{3}\rangle$ with $b=a/4$ (green dashed line), $|G_{ab}^{4}\rangle$ with $b=a/2$ (blue dotted line), $|G_{a}^{5}\rangle$ (magenta dot-dashed line).
• Figure 2: (Color online) Dependence of the four-partite residual entanglement $\tau _{ABCD}$ (red solid line) and the maximum reduced three-tangle $\tau _{ABC}$ (blue dashed line) on the parameter $p$ for the mixed state ${\rho_{p}}$ [Eq. \ref{['Eq.b25']}].
• Figure 3: (Color online) Dependence of the five-partite entanglement $\tau_{12345}$ (red solid line), the maximum four-partite entanglement $\tau^{(4)}$ (green dotted line) and the tripartite entanglement $\tau_{123}$ (blue dashed line) on the parameter $p$ for the state $\left | \Psi \right \rangle _{p}$ [Eq. \ref{['Eq.c26']}].