Abstract
For any three-qubit quantum systems ABC, Oliveira et al. numerically found that both the concurrence and the entanglement of formation (EoF) obey the linear monogamy relations in pure states. They also conjectured that the linear monogamy relations can be saturated when the focus qubit A is maximally entangled with the joint qubits BC. In this work, we prove analytically that both the concurrence and EoF obey linear monogamy relations in an arbitrary three-qubit state. Furthermore, we verify that all three-qubit pure states are maximally entangled in the bipartition A|BC when they saturate the linear monogamy relations. We also study the distribution of the concurrence and EoF. More specifically, when the amount of entanglement between A and B equals to that of A and C, we show that the sum of EoF itself saturates the linear monogamy relation, while the sum of the squared EoF is minimum. Different from EoF, the concurrence and the squared concurrence both saturate the linear monogamy relations when the entanglement between A and B equals to that of A and C.
Highlights
For any three-qubit quantum systems ABC, Oliveira et al numerically found that both the concurrence and the entanglement of formation (EoF) obey the linear monogamy relations in pure states
We verify that the three-qubit pure states must be maximally entangled between qubit A and the joint qubits BC when they saturate the linear monogamy relation
We prove exactly that both the concurrence and EoF are linearly monogamous, verify that maximally entangled three-qubit states saturating the linear monogamy relations, and study the distribution of the concurrence and EoF in three-qubit states
Summary
EF(ρAB) + EF(ρAC) ≈ 1.2018 which shows that Ψ ABC attains the upper bound 1.2018 of the monogamous inequality in Eq (3) for EoF when the focus qubit A is maximally entangled with the joint qubits BC They numerically pointed out that the concurrence is linearly monogamous. We have max f(x) = f(1/2) ≈ 1.2018 and derive the monogamy inequality of Eq (4), such that we have completed the whole proof showing that EoF is linearly monogamous in three-qubit mixed states. These results can be intuitively observed from Fig. 1(a).
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