Generalized product-form monogamy relations in multi-qubit systems
Wen Zhou, Zhong-Xi Shen, Hong-Xing Wu, Zhi-Xi Wang, Shao-Ming Fei
TL;DR
This work tackles MOE in multi-qubit and higher-dimensional systems by deriving tighter product-form inequalities for the $ν$-th power of concurrence ($ν\ge2$) and for CREN-based negativity. Central to the method are two lemmas that relate the global A|rest entanglement to products and sums of bipartite entanglements, yielding bounds such as $\mathcal{C}_{A|B_1\cdots B_{N-1}}^ν \ge [4(\mathcal{C}^2_{AB_1}+\frac{\kappa}{2})((N-2)(\prod_{i=1}^{N-2}\mathcal{C}^2_{AB_{i+1}})^{1/(N-2)}+\frac{\kappa}{2})]^{ν/4}$ for $ν\ge2$, and an analogous CREN-based bound with $\epsilon$. The results improve upon prior summation- and product-form MOE bounds and are supported by examples, including a three-qubit $W$ state and CKW counterexamples in higher dimensions, where the new bounds remain valid. These tighter MOE relations offer finer constraints on entanglement sharability and may inform analyses of entanglement distribution in quantum networks, as well as extend to other quantum correlations.
Abstract
Monogamy of entanglement essentially characterizes the entanglement distributions among the subsystems. Generally it is given by summation-form monogamy inequalities. In this paper, we present the product-form monogamy inequalities satisfied by the $ν$-th ($ν\geq2$) power of the concurrence. We show that they are tighter than the existing ones by detailed example. We then establish tighter product-form monogamy inequalities based on the negativity. We show that they are valid even for high dimensional states to which the well-known CKW inequality is violated.