Monogamy equality in 2⊗2⊗d quantum systems

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 61, 052306 (2000); Coffman-Kundu-WoottersPhys. Rev. Lett. 96, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, C_{A(BC⁾²=CAB²+(^CACa)2, in the three-qubit system. In this paper, we consider the monogamy equality in 2⊗2⊗d quantum systems. We show that CA(BC)=CAB if and only if CAC^a=0 and also show that if CA(BC)=CAC^a, then CAB=0, while there exists a state in a 2⊗2⊗d system such that CAB=0 but CA(BC)}>CA_C^a.