Monogamy equality in $2\otimes 2 \otimes d$ quantum systems

TL;DR

This paper investigates the monogamy equality of entanglement in $2\otimes 2 \otimes d$ quantum systems, establishing conditions under which certain equalities hold and providing examples of states with specific entanglement properties.

Contribution

It extends the understanding of monogamy equality from three-qubit systems to $2\otimes 2 \otimes d$ systems, identifying key conditions and examples.

Findings

$\,\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$ iff $\mathcal{C}_{AC}^a=0$

If $\mathcal{C}_{A(BC)}=\mathcal{C}_{AC}^a$, then $\mathcal{C}_{AB}=0$

Existence of states with $\mathcal{C}_{AB}=0$ but $\mathcal{C}_{A(BC)} > \mathcal{C}_{AC}^a$

Abstract

There is an interesting property about multipartite entanglement, called the monogamy of entanglement. The property can be shown by the monogamy inequality, called the Coffman-Kundu-Wootters inequality [Phys. Rev. A {\bf 61}, 052306 (2000); Phys. Rev. Lett. {\bf 96}, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, $\mathcal{C}_{A(BC)}^2=\mathcal{C}_{AB}^2+(\mathcal{C}_{AC}^a)^2$, in the three-qubit system. In this paper, we consider the monogamy equality in $2\otimes 2 \otimes d$ quantum systems. We show that $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$ if and only if $\mathcal{C}_{AC}^a=0$, and also show that if $\mathcal{C}_{A(BC)}=\mathcal{C}_{AC}^a$ then $\mathcal{C}_{AB}=0$, while there exists a state in a $2\otimes 2 \otimes d$ system such that $\mathcal{C}_{AB}=0$ but $\mathcal{C}_{A(BC)}>\mathcal{C}_{AC}^a$.