Multipartite Monogamy of Entanglement for Three Qubit States

TL;DR

This work investigates how entanglement can be distributed in pure three-qubit states under monogamy constraints, using the multipartite source entanglement $E_s$ and accessible entanglement $E_a$ alongside bipartite entanglement of formation $E_f$. It analyzes GHZ and W class states, deriving that for GHZ generic states the upper bound $E_s^2\ge E_{12}^2+E_{13}^2+E_{23}^2$ holds (with non-generic GHZ exceptions) and that $E_a^2$-based monogamy can fail in many cases; for W class states, $E_s^2$-monogamy remains valid while $E_a^2$-monogamy is typically violated when reduced subsystems are entangled. The study also discusses special MES$_3$-class states where $E_s=1$ but $E_a$ varies, and provides numerical evidence supporting the analytic GHZ results and the general trend across classes. Overall, the paper reveals distinct monogamy behavior depending on whether a bound is provided by source or accessible entanglement, offering insights into entanglement distribution with potential extensions to higher dimensions and mixed states and implications for quantum cryptography.

Abstract

The distribution of entanglement in a multiparty system can be described through the principles of monogamy or polygamy. Monogamy is a fundamental characteristic of entanglement that restricts its distribution among several number of parties(more than two). In this work, our aim is to explore how quantum entanglement can be distributed in accordance with monogamy relations by utilizing both the genuine multipartite entanglement measures and bipartite entanglement measures. Specifically, we treat source entanglement as the genuine multipartite entanglement measure and use the entanglement of formation specifically for bipartite cases. For GHZ class states, we analytically demonstrate that the square of the source entanglement serves as an upper bound for the sum of the squares of the entanglement of formation of the reduced subsystems, with some exceptions for specific non-generic GHZ states. We also present numerical evidence supporting this result for W class states. Additionally, we explore the monogamy relation by using accessible entanglement as an upper bound.

Figures (18)

• Figure 1: Graph of parameters $g_1,r$ vs monogamy score $M_1$ for non generic GHZ class with $g_1\neq 0$,$g_2=0$,$g_3=0$,$r\in(0,1).$
• Figure 3: Graph of parameter $g_1$ vs monogamy score $M_1(M_2)$ for non generic GHZ class with $g_1\neq 0$,$g_2=0$,$g_3=0$,$r=1$.
• Figure 4: Monogamy score $M_1$ for random states belonging to non generic GHZ class with $g_1\neq0$,$g_2\neq0$,$g_3=0$,$r\in(0,1)$.
• Figure 6: Graph of parameters $g_1$,$g_2$ vs monogamy score $M_1$ for non generic GHZ class with $g_1\neq0$,$g_2\neq0$,$g_3=0$,$r=1$.
• Figure 7: Graph of parameters $g_1$,$g_2$ vs monogamy score $M_2$ for non generic GHZ class with $g_1\neq0$,$g_2\neq0$,$g_3=0$,$r=1$.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4