Characterizing entanglement shareability and distribution in $N$-partite systems

TL;DR

The paper addresses how entanglement can be shared across multipartite quantum systems by establishing hierarchical monogamy relations for the squared G_q-concurrence (S$G_q$C) in $N$-qubit states and extending the framework to multilevel systems. It develops two classes of hierarchical entanglement indicators from S$G_q$C and proves an analytical link between G_q-concurrence and concurrence in $2\otimes d$ systems. The results show that the monogamy property of S$G_q$C is stronger than that of squared concurrence (SC) in certain multipartite and multilevel settings, with concrete examples such as W states demonstrating improved entanglement witnessing. Together, these insights enhance understanding of entanglement distribution and provide practical tools for detecting multipartite entanglement in complex quantum systems.

Abstract

Exploring the shareability and distribution of entanglement possesses fundamental significance in quantum information tasks. In this paper, we demonstrate that the square of bipartite entanglement measures $G_q$-concurrence, which is the generalization of concurrence, follows a set of hierarchical monogamy relations for any $N$-qubit quantum state. On the basis of these monogamy inequalities, we render two kinds of hierarchical indicators that exhibit evident advantages in the capacity of witnessing entanglement. Moreover, we show an analytical relation between $G_q$-concurrence and concurrence in $2\otimes d$ systems. Furthermore, we rigorously prove that the monogamy property of squared $G_q$-concurrence is superior to that of squared concurrence in $2\otimes d_2\otimes d_3\otimes\cdots\otimes d_N$ systems. In addition, several concrete examples are provided to illustrate that for multilevel systems, the squared $G_q$-concurrence satisfies the monogamy relation, even if the squared concurrence does not. These results better reveal the intriguing characteristic of multilevel entanglement and provide critical insights into the entanglement distribution within multipartite quantum systems.

Figures (5)

• Figure 1: The blue line $l_q$ is a lower bound of $\tau_{q3}(|\psi\rangle_{A|BC})$, the red line denotes $\tau(|\psi\rangle_{A|BC})=0$, the intersection point of these two lines is denoted as $q_0$.
• Figure 2: The entanglement distribution $M_q(|\varphi\rangle_{A|BC})=\mathscr{C}_q^2(|\varphi\rangle_{A|BC})-\mathscr{C}_q^2(\rho_{AB})-\mathscr{C}_q^2(\rho_{AC})$ as a function of $\theta~(0\leq\theta\leq\frac{\pi}{2})$ and $q~(1\leq q\leq1.5)$ is nonnegative, which indicates S$G_q$C is monogamous.
• Figure 3: The red line denotes the solutions of $\frac{\partial M(t,q)}{\partial t}=0$.
• Figure 4: The blue line illustrates that $\lim\limits_{t\rightarrow1}g_q=\lim\limits_{t\rightarrow1}\frac{\partial^2h_q}{\partial t^2}$ is non-positive for $1<q\leq2$.
• Figure 5: The blue line states that $\lim\limits_{t\rightarrow1}\widetilde{g}_q=\lim\limits_{t\rightarrow1}\frac{\partial^2{h}_q^2}{\partial t^2}\geq0$ for $1<q\leq2$.

Theorems & Definitions (4)

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