Monogamy relation of multi-qubit systems for squared Tsallis-q entanglement

Tsallis-q entanglement is a bipartite entanglement measure which is the generalization of entanglement of formation for q tending to 1. We first expand the range of q for the analytic formula of Tsallis-q entanglement. For , we prove the monogamy relation in terms of the squared Tsallis-q entanglement for an arbitrary multi-qubit systems. It is shown that the multipartite entanglement indicator based on squared Tsallis-q entanglement still works well even when the indicator based on the squared concurrence loses its efficacy. We also show that the μ-th power of Tsallis-q entanglement satisfies the monogamy or polygamy inequalities for any three-qubit state.

On the other hand, Tsallis-q entanglement is also a well-defined entanglement measure which is the generalization of EOF. For q tending to 1, the Tsallis-q entanglement converges to the EOF. A natural question is whether the monogamy relation can be generalized to Tsallis-q entanglement. In fact, Kim has derived a monogamy relation in terms of Tsallis-q entanglement 22 . However, the result in ref. 22 fails in including EOF as a special case and only holds for 2 ≤ q ≤ 3. In this paper we further consider the monogamy relation in terms of the squared Tsallis-q entanglement(STqE). Firstly we expand the range of q for the analytic formula of Tsallis-q entanglement. Then we prove a monogamy inequality of multi-qubit systems in terms of STqE in an arbitrary N-qubit mixed state for ≤ ≤ − + q 5 13 2 5 13 2 , which covers the case of EOF as a special case. Finally, we show that the μ-th power of the Tsallis-q entanglement satisfies the monogamy inequalities for three-qubit state.

Results
Analytic formula of Tsallis-q entanglement. Firstly we recall the definition of Tsallis-q entanglement introduced in ref. 22. For a bipartite pure state |ψ〉 AB , the Tsallis-q entanglement is defined as for any q > 0 and q ≠ 1, where ρ A = tr B |ψ〉 AB 〈 ψ| is the reduced density matrix by tracing over the subsystem B. For the case when q tends to 1, T q (ρ) converges to the von Neumann entropy, that is For a bipartite mixed state ρ AB , Tsallis-q entanglement is defined via the convex-roof extension where the minimum is taken over all possible pure state decompositions of ρ ψ ψ = ∑ p AB i i i AB i . In ref. 22, Kim has proved an analytic relationship between Tsallis-q entanglement and concurrence for 1 ≤ q ≤ 4 as follows where the function g q (x) is defined as According to the results in ref. 22, the analytic formula in Eq. (5) holds for any q such that g q (x) in Eq. (6) is monotonically increasing and convex. Next we shall generalize the range of q when the function g q (x) is convex and monotonically increasing with respect to x. The monotonicity and convexity of g q (x) follow from the nonnegativity of its first and second derivatives. After a direct calculation, we find that the first derivative of g q (x) with respect to x is always nonnegative for q ≥ 0 22 . Kim has also proved the nonnegative of the second-order derivative g q (x) for 1 ≤ q ≤ 4. We can further consider the second-order derivative of g q (x) beyond the region 1 ≤ q ≤ 4. We first analyze the nonnegative region for the second-order derivative g q (x) for q ∈ (0, 1). Numerical calculation shows that under the condition ∂ 2 T q (C)/∂ x 2 = 0, the critical value of x increases monotonically with the parameter q. In Fig. 1(a), we plot the solution (x, q) to this critical condition, where for each fixed x there exists a value of q such that the second-order derivative of T q (C) is zero. Because x varying monotonically with q, we should only consider the condition ∂ 2 T q (C)/∂ x 2 = 0 in the limit x → 1. When x = 1, we have x q q . When q > q c1 , the second-order ∂ 2 T q /∂ x 2 is always nonnegative. For q ∈ (4, 5), we find that the value of x decreases monotonically with respect to q as shown in Fig. 1(b). In order to determine the critical point we should only consider the condition ∂ 2 T q /∂ x 2 = 0 in the limit x → 1. After direct calculation, we can obtain that the critical point = ≈ .
. When q < q c2 , the second-order ∂ 2 T q /∂ x 2 is always nonnegative. Combining with the previous results in ref. 22, we get that the second derivative of g q (x) is always a nonnegative function for ≤ ≤ − + Monogamy inequalities for STqE in N-qubit systems. In the following we consider the monogamy properties of STqE. Using the results presented in Methods, we can prove the main result of this paper.
For an arbitrary N-qubit mixed state ρ , the squared Tsallis-q entanglement satisfies the monogamy relation quantifies the Tsallis-q entanglement in the partition A 1 |A 2 ···A n and ρ T ( ) For proving the above inequality, we first analyze an N-qubit pure state ψ | 〉 where in the first inequality we have used the monogamy relation of squared concurrence and the monotonically increasing property of T C ( ) q 2 2 which has been proved in Methods, and the second inequality is due to the convex property of T C ( ) q 2 2 (The details for proving the convexity property can be seen from Methods).
Next, we prove the monogamy relation for an N-qubit mixed state ρ . In this case, the formula of Tsallis-q entanglement cannot be applied to ρ since the subsystem A 2 ···A n is not a logic qubit in general. But we can still use the definition of Tsallis-q entanglement in Eq. (4). Thus, we have where the minimum is taken over all possible pure state decompositions {p i , |ψ i 〉 } of the mixed state ρ ; the third inequality is due to that T q is a monotonically increasing and convex function of the concurrence for (The details for illustrating the property of STqE can be seen from Methods). Thus we have completed the proof of the monogamy inequalities for STqE in N-qubit systems.
As an application of the established monogamy relation in Eq. (8), we can construct the multipartite entangle- to detect the genuine multipartite entanglement. We 2 . For the quantum state |ψ(p)〉 , its three-tangle 8 6 (1 ) /9 2 3 which has two zero points at p 1 = 0 and p 2 ≈ 0.627. On the other hand, we can directly calculate the value of τ q (|ψ(p)〉 ) since the Tsallis-q entanglement has an analytical formula for two-qubit quantum states. In Fig. 2 we plot the three-tangle and the indicator τ q for the order q = 0.8, 1.1, 1.4. It is shown that the indicator τ q is always positive for the different order q in contrast to the three-tangle τ having two zero points. Thus we have shown that the indicator in terms of Tsallis-q entanglement could detect the genuine entanglement in |ψ(p)〉 better than SC.

Monogamy relation of the μ-th power of Tsallis-q entanglement. Finally, besides the squared
Tsallis-q entanglement, we can further consider the monogamy relation of the μ-th power of Tsallis-q entanglement.
For any three-qubit state ρ A A A 1 2 3 , we can obtain , μ ≥ 2. For proving Eq. (12), we consider the three-qubit case, according to the monogamy relation (8), we have , we can obtain where the second inequality comes from the property (1 + , the inequality obviously holds. Similarly, we have the following polygamy inequalities. For any three-qubit ρ A A A 1 2 3 , we have   where in the second inequality we have used the inequality (1 + x) t < 1 + x t for x > 0, t ≤ 0.

Discussion
In this paper we have generalized the analytic formula of Tsallis-q entanglement to the region ≤ ≤ − + q 5 13 2 5 13 2 . Then we proved the monogamy relation in terms of STqE for an arbitrary multi-qubit systems, which include previous result in terms of EOF as a special case. Based on the monogamy properties of Tsallis-q entanglement, we have shown that the corresponding indicator can work well even when the indicator based on the squared concurrence loses its efficacy. In addition, we considered the monogamy or polygamy relation of the μ-th power of Tsallis-q entanglement. One distinct advantage of our result is that infinitely many inequalities parameterized by q provides greater flexibility than previous monogamy relation in terms of EOF.

T C
( ) q 2 2 is a monotonically-increasing function of the squared concurrence C 2 for all q ≥ 0. Notice that Eq. (5) can also be written as where the function f q (x) is defined as The squared Tsallis-q entanglement is a monotonically increasing function of C 2 if the first-order derivative with x = C 2 . By direct calculation, we have, with x = C 2 . We first define a function , then the nonnegativity of the second-order derivative T q 2 can be guaranteed by the nonnegativity of F q since it varies with ∂ ∂ T C x ( )/ q 2 2 2 2 by a positive constant. After some deduction, we have In order to prove the nonnegativity of F q , it is suffice to consider its maximum or minimum values on the domain D. The critical points of F q satisfy the condition q q q In Fig. 3(a,b), we have plotted the value of x and q which satisfies the equation ∂ F q /∂ q = 0 and ∂ F q /∂ x = 0 respectively. Combining the results in Fig. 3(a,b), we find that the solution of the above equation is q = 1 which is one of the boundary of domain D. To ensure the nonnegative of F q , we should only consider the other two cases on the boundary of F q , i.e., x = 0 and x = 1. For the case x = 0, x q q q 0 1 2 which is always nonnegative in the region q ∈ (1, 4). For the case when x = 1, x q q q q q 1 2 where Eq. (23) is always nonnegative for q = 1 and q = 4, and the first-order derivative of Eq. (23) increases first and then decreases for 1 ≤ q ≤ 4. Thus we prove that Eq. (23) is nonnegative in the region 1 ≤ q ≤ 4. Notice that F q has no critical points in the interior of D, we conclude that F q is always nonnegative for 1 ≤ q ≤ 4. The nonnegative of the F q is also plotted in Fig. 4. Furthermore, we can consider the nonnegative region for the second-order derivative ∂ ∂ T x / q 2 2 2 when q ranges in (0, 1). Under the condition ∂ ∂ = T x / 0 q 2 2 2 , we find that the critical value of x increases monotonically with the parameter q ∈ (0, 1). In Fig. 5(a), we plot the solution (x, q) to the critical condition where for each fixed x there exists a value of q such that the second-order derivative of T q 2 is zero. We should only consider the condition ∂ ∂ ≥ T x / 0 q 2 2 2 in the limit x → 1. In this case, we have x q q q q q 1 2 2 2 2 which gives the critical point q c3 ≈ 0.65. When q ≥ q c3 , the second-order ∂ ∂ T x / q 2 2 2 is always positive. Similarly, we can also analyze the nonnegative region for the second-order derivative ∂ ∂ T x / q 2 2 2 when q ranges in (4,5). In Fig. 5(b), it is shown that the critical value of x decreases monotonically along with the parameter q ∈ (4, 5), and the critical point q c4 ≈ 4.65. When q ≤ q c4 , the second-order ∂ ∂ T x / for (a) q ∈ (0, 1) and (b) q ∈ (4, 5) respectively.