Linear monogamy of entanglement in three-qubit systems

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.

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.
In its original sense 14 , the monogamy relation gives insight into the way that quantum correlation exists across the three qubits, so it is not accessible if only pairs of qubits are considered. It relates a bipartite entanglement measure E between bipartitions as follows: where A, B and C are the respective particles of a tripartite state ρ ABC , each pair ρ Ai denotes the reduced state of the focus particle A and the particle i = {B, C}, and the vertical bar is the notation for the bipartite partition. The original monogamy inequality has been generalized to n-qubit systems for the squared concurrence by Osborne and Verstraete 15 . The squared entanglement of formation (SEoF) is also a monogamous entanglement measure for qubits which has been proved by Bai et al. 16,17 . However, the concurrence and the entanglement of formation (EoF) themselves do not satisfy the monogamy relation. Therefore, it is usually said that the concurrence and EoF are not monogamous. Here, EoF in a two-qubit state ρ AB is defined as 1 : is the binary Shannon entropy and 4 is the concurrence with the decreasing nonnegative λ i being the eigenvalues of the matrix ρ σ σ ρ σ σ ( ⊗ ) ( ⊗ ) ⁎ AB y y AB y y . Recently, Oliveira et al. 18 claimed that violating Eq. (1) does not mean that the concurrence and EoF can be freely shared. In fact, they numerically found that both the concurrence and EoF are linearly monogamous in three-qubit pure states, which means that either of the two entanglement measures satisfies the following inequality where the upper bound λ < 2 is a constant. They conjectured that λ = 1.2018 for EoF and the linear monogamy relations can be saturated only when the focus qubit A is maximally entangled with the joint qubits BC. Based on the above numerical results, it is natural to ask whether the concurrence and EoF obey the linear monogamy relation for an arbitrary three-qubit (pure or mixed) state, whether there exist three-qubit states which saturate these upper bounds, and whether they must be maximally entangled states between the focus qubit A and the joint qubits BC. In this paper, we prove analytically that both the concurrence and EoF are linearly monogamous in three-qubit states. We also find that the upper bound for E F (ρ AB ) + E F (ρ AC ) can be attained when two entangled pairs E F (ρ AB ) and E F (ρ AC ) are equal, while in the same case is minimum. Moreover, 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. For the concurrence, we prove analytically that C(ρ AB ) + C(ρ AC ) and C 2 (ρ AB ) + C 2 (ρ AC ) are maximum when C(ρ AB ) = C(ρ AC ). Here, E F (ρ AB ) is EoF of a two-qubit state ρ AB , and C(ρ AB ) is the concurrence between the qubits A and B.

Results
This section is organized as follows. In the first subsection, we give a brief review on the linear monogamy conjectures from Oliveira et al. 18 in detail. In the other subsections, 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.
The linear monogamy conjecture from Oliveira et al. The original monogamy relation 14 gives much insight on the manner in which entanglement is shared across three parties. Then it can be used to characterize genuine tripartite entanglement 17 . However, the linear monogamy relation can only be used to indicate the restrictions for entanglement distribution between AB and AC. Nonetheless, the linear monogamy relations show that there exist upper bounds on the sum of the two entangled pairs, E(ρ AB ) + E(ρ AC ), for the concurrence and EoF themselves, and then the two entanglement measures cannot be freely shared.
For EoF itself, Oliveira et al. numerically found the upper bound 1.1882 for E F (ρ AB ) + E F (ρ AC ), which is considerably smaller than 2. The upper bound is obtained by considering a sampling of 10 6 random pure states for three-qubit systems. Thus they claimed that it is at least misleading to say that EoF can be freely shared. So, they conjectured that EoF obeys the linear monogamy relation in Eq. For the state, In the following subsections, we will prove these numerical conjectures.

Linear monogamy of EoF.
A key result of this subsection is to prove analytically that EoF obeys a linear monogamy inequality in an arbitrary three-qubit mixed state, i.e., For proving the general inequality, we first give the following expressions: . For any three-qubit state ρ ABC , the total amount of entanglement that can be shared is restricted by Eq. (1): After some deduction, we have On the other hand, we have For any x ∈ [0, c], the first-order derivative is positive. We can deduce that f is a monotonically increasing function of c. The details for illustrating the above results are presented in Methods. Because (2), and then we have 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). Therefore, we draw the conclusion that the conjecture on the linear monogamy from Oliveira et al. is true, and the saturation of the upper bound 1.2018 comes from two equal pairs, i.e., E F (ρ AB ) = E F (ρ AC ). In the following two paragraphs, we will prove that the conjecture (that the saturated states must be maximally entangled states 19,20 According to Eq. (2) and the result that we have r 2 = r 3 and = / r r   It can be verified that the first-order derivative dh(x)/dx = 0 when x = c/2. So x = c/2 is a stationary point of h(x). The details for illustrating the results have been presented in Methods, and they can also be intuitively found out from Fig. 1 Then, there exist some three-qubit pure states in Eq. (10) satisfying the above specified conditions. From the viewpoint of quantum informational theory, the phenomenon can be interpreted as follows: the more closing to each other of EoF itself in two pairs AB and AC, the less amount of entanglement in the sum of SEoF.
In the next subsection, we similarly study the linear monogamy of the concurrence and the properties of the concurrence and its squared version.

Linear monogamy of the concurrence.
A key result of this subsection is to prove analytically that the concurrence obeys a linear monogamy inequality in an arbitrary three-qubit mixed state, i.e., ( ) Because SEoF satisfies the monogamy relation for three-qubit states, we find that e ∈ [0, 1].

Figure 1. f(x), the sum of EoF, is a concave function of x, while h(x), the sum of squared EoF, is a convex function of x.
Their function curves translate upwards as a whole with the growth of c.   and Eq. (2). These results can be easily verified by a Mathematica program for the binary function p, and they can also be intuitively observed from Fig. 2(a). Thus we obtain the conclusion that the saturation of the upper bound 1.4142 also comes from both entangled pairs AB and AC with equal intensity, i.e., C(ρ AB ) = C(ρ AC ).
Moreover, we verify that the conjecture (that the saturated states must being maximally entangled states in the bipartition A|BC) from Oliveira et al. is also ture in general for the concurrence.
Similarly to the example in Eq. . So ϕ ABC is a maximally entangled state in the bipartition A|BC, and then the maximum value of C(ρ AB ) + C(ρ AC ) can be attained when the focus qubit A is maximally entangled with the other two qubits BC for any three-qubit pure states.
Finally, we study the properties of the squared concurrence, and point out that the concurrence and its squared version are always similar. The phenomenon is completely different from EoF and SEoF. For any e ∈ [0, 1], it is not hard to determine that q(y) is a concave function of y, 2y is its maximum value, and y = q(y)/2 is a stationary point of q(y). So the saturation of the upper bound comes similarly from both pairs AB and AC. More specifically, max[C 2 (ρ AB ) + C 2 (ρ AC )] = 1 if and only if C(ρ AB ) = C(ρ AC ) = 0.7071. These results can be proved as the processing of h(x), and can be intuitively visualized from Fig. 2(b) in a similar way. From the viewpoint of quantum informational theory, the phenomenon can be interpreted as follows: the more closing to each other of the two pairs C(ρ AB ) and C(ρ AC ), the more value of entanglement for the concurrence and its squared version exists.

Discussion and Summary
Different from the original monogamy relation, the linear monogamy relation can only be used to indicate the restrictions for entanglement distribution. In this work, we respectively investigate the linear monogamy relation for the concurrence and EoF. For three-qubit states, we provide analytical proofs that both the concurrence and EoF obey the linear monogamy relations respectively. We also 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. Finally, we find there are the following different phenomena in the distribution of the concurrence and EoF: when the entanglement between A and B equals to that of A and C, the sum of EoF itself saturates Eq. (4), while the sum of SEoF is minimum. Different from EoF, the sum of the concurrence itself saturates Eq. (13) when the entanglement between A and B equals to that of A and C, and the sum of the squared concurrence is maximum at the same condition.
For future work, there are several open questions. Firstly, our results cannot be used to restrict the sharing entanglement in multiqubit states. Then it is interesting to consider whether our method can be modified to facilitate more generalized n-qubit states. Secondly, Zhu and Fei 23 presented the αth power monogamy, where the sum of all bipartite αth power entanglement may change with different α. Therefore, another interesting open question is to study relations between the upper bound of the sum and α (particularly for α ∈ ( , ) 0 2 ). Thirdly, quantum correlations, such as quantum discord [24][25][26] , generally do not possess the property of the original monogamy relation [27][28][29] . Our approach may be used to study the linear monogamy properties of quantum correlations. It is easy to verify that = ( )

Methods
when c ∈ (0, 1), and then x = c/2 is a stationary point of can also be obtained if the first-order derivative dh(x)/dx = 0. According to Eq. (5), we have    Therefore, dp/de ≥ 0 and then it is a monotonically increasing function of e.
The derivative functions of p(x). According to Eqs. (2) and (14), E F (ρ AC ) has the form ( )  From Eq. (9) in ref. 30, we find that the second-order derivative d 2 E F (ρ AC )/dx 2 ≥ 0 and similarly d 2 E F (ρ AB )/dp 2 (x) ≥ 0 in the region x ∈ [0, 1]. So the second-order derivative d 2 E F (ρ AB )/dx 2 ≤ 0 in the same region. Combining with the chain rule, the second-order derivative d 2 E F (ρ AB )/dx 2 can be written as Thus, we prove that the second-order derivative d 2 p(x)/dx 2 ≤ 0 in the whole region x ∈ [0, 1], and then complete the proof of the results in the main text.