Introduction

Quantum entanglement as a physics resource for quantum communication and quantum information processing has been the subject of many recent studies in recent years1,2,3,4,5,6,7. The study of quantum entanglement from various view points has been a very active area and has led to many interesting results. Monogamy of entanglement(MOE)8 is an interesting property discovered recently in the context of multi-qubit entanglement, which means that quantum entanglement cannot be shared freely in multi-qubit quantum systems. The bipartite monogamy inequality was first proposed and proved by Coffman, Kundu and Wootters(CKW) in a three-qubit system9, and it is also named as CKW inequality:

where is the squared of concurrence between the pair i and j10. Later, the monogamy inequality was generalized into various entanglement measures such as continuous-variable entanglement11,12,13, squashed entanglement14,15,16, entanglement negativity17,18,19,20,21, Tsallis-q entanglement22,23, and Rényi-α entanglement24,25,26. The applications of monogamy relation include many fields of physics such as characterizing the entanglement structure in multipartite quantum systems27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, the security proof in quantum cryptography42, the frustration effects observed in condensed matter physics43, and even black hole physics43,44,45,46,47,48. Originally, MOE was established in terms of the squared concurrence(SC). Analogously, Bai et al.49,50 have proved that the squared entanglement of formation(SEF) obeys the monogamy relation in arbitrary N-qubit mixed state. It should be noted that the entanglement of formation(EOF) itself does not satisfy the monogamy relation even for three-qubit pure states. The new monogamy relation in terms of SEF overcomes some flaws of the SC and can be used to detect all genuine multipartite entanglement for N-qubit systems.

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 entanglement22. 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 , 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 ρA = trB|ψ〉AB〈ψ| is the reduced density matrix by tracing over the subsystem B. For the case when q tends to 1, Tq(ρ) converges to the von Neumann entropy, that is

For a bipartite mixed state ρAB, Tsallis-q entanglement is defined via the convex-roof extension

where the minimum is taken over all possible pure state decompositions of .

In ref. 22, Kim has proved an analytic relationship between Tsallis-q entanglement and concurrence for 1 ≤ q ≤ 4 as follows

where the function gq(x) is defined as

According to the results in ref. 22, the analytic formula in Eq. (5) holds for any q such that gq(x) in Eq. (6) is monotonically increasing and convex. Next we shall generalize the range of q when the function gq(x) is convex and monotonically increasing with respect to x. The monotonicity and convexity of gq(x) follow from the nonnegativity of its first and second derivatives. After a direct calculation, we find that the first derivative of gq(x) with respect to x is always nonnegative for q ≥ 022. Kim has also proved the nonnegative of the second-order derivative gq(x) for 1 ≤ q ≤ 4. We can further consider the second-order derivative of gq(x) beyond the region 1 ≤ q ≤ 4. We first analyze the nonnegative region for the second-order derivative gq(x) for q ∈ (0, 1). Numerical calculation shows that under the condition ∂2Tq(C)/∂x2 = 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 Tq(C) is zero. Because x varying monotonically with q, we should only consider the condition ∂2Tq(C)/∂x2 = 0 in the limit x → 1. When x = 1, we have

Figure 1
figure 1

The plot of the dependence of x with q which satisfies the equation for (a) q ∈ (0, 1) and (b) q ∈ (4, 5) respectively.

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which gives the critical point . When q > qc1, the second-order ∂2Tq/∂x2 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 ∂2Tq/∂x2 = 0 in the limit x → 1. After direct calculation, we can obtain that the critical point . When q < qc2, the second-order ∂2Tq/∂x2 is always nonnegative. Combining with the previous results in ref. 22, we get that the second derivative of gq(x) is always a nonnegative function for . Thus we have shown that the analytic formula of Tsallis-q entanglement in Eq. (5) holds 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 , the squared Tsallis-q entanglement satisfies the monogamy relation

where quantifies the Tsallis-q entanglement in the partition A1|A2 ···An and quantifies the one in two-qubit subsystem A1Ai with the parameter .

For proving the above inequality, we first analyze an N-qubit pure state . Under the partition A1|A2 ···An, we have

where in the first inequality we have used the monogamy relation of squared concurrence and the monotonically increasing property of which has been proved in Methods, and the second inequality is due to the convex property of (The details for proving the convexity property can be seen from Methods).

Next, we prove the monogamy relation for an N-qubit mixed state . In this case, the formula of Tsallis-q entanglement cannot be applied to since the subsystem A2 ···An 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 {pi, |ψi〉} of the mixed state . Under the optimal decomposition , we have

where in the second equality we have used the pure state formula of the Tsallis-q entanglement and taken the Tq(C) as a function of the concurrence C for ; the third inequality is due to that Tq is a monotonically increasing and convex function of the concurrence for ; the forth inequality is due to the convex property of concurrence for mixed state; and in the sixth and seventh inequalities we used the monotonically increasing and convex properties of as a function of the squared 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 entanglement indicator to detect the genuine multipartite entanglement. We consider a three-qubit pure state , which is the superposition of a GHZ state and a W state with and . The three-tangle τ introduced in ref. 9 is defined as . For the quantum state |ψ(p)〉, its three-tangle is which has two zero points at p1 = 0 and p2 ≈ 0.627. On the other hand, we can directly calculate the value of τq(|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 τq for the order q = 0.8, 1.1, 1.4. It is shown that the indicator τq 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.

Figure 2: The indicator τq for the superposition state |ψ(p)〉 with q = 0.8 (red line), q = 1.1 (blue line), and q = 1.4 (green line).
figure 2

We also plot the three-tangle of |ψ(p)〉 with a black line.

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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 , we can obtain

for all , μ ≥ 2.

For proving Eq. (12), we consider the three-qubit case, according to the monogamy relation (8), we have

for any three-qubit state with . Without loss of generality, assuming , we can obtain

where the second inequality comes from the property (1 + x)t ≥ 1 + xt for x ≤ 1, t ≥ 1. If or , the inequality obviously holds.

Similarly, we have the following polygamy inequalities. For any three-qubit , we have

for all , μ ≤ 0.

For any three-qubit state with , we have

where in the second inequality we have used the inequality (1 + x)t < 1 + xt for x > 0, t ≤ 0.

Discussion

In this paper we have generalized the analytic formula of Tsallis-q entanglement to the region . 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.

Methods

is a monotonically-increasing function of the squared concurrence C2 for all q ≥ 0

Notice that Eq. (5) can also be written as

where the function fq(x) is defined as

The squared Tsallis-q entanglement is a monotonically increasing function of C2 if the first-order derivative with x = C2. By direct calculation, we have,

which is always nonnegative on 0 ≤ x ≤ 1 for all q ≥ 0, where L = 1/(q − 1)2, , , and the equality holds only at the boundary. Thus we get that is a monotonically increasing function of x with x = C2.

is a convex function of the squared concurrence C2 for

The convex property of the squared concurrence is satisfied if the second-order derivative with x = C2. We first define a function on the domain D = {(x, q)|0 ≤ x ≤ 1, 1 ≤ q ≤ 4}, then the nonnegativity of the second-order derivative can be guaranteed by the nonnegativity of Fq since it varies with by a positive constant. After some deduction, we have

In order to prove the nonnegativity of Fq, it is suffice to consider its maximum or minimum values on the domain D. The critical points of Fq satisfy the condition

In Fig. 3(a,b), we have plotted the value of x and q which satisfies the equation ∂Fq/∂q = 0 and ∂Fq/∂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 Fq, we should only consider the other two cases on the boundary of Fq, i.e., x = 0 and x = 1.

Figure 3
figure 3

The plot of the dependence of x with q which satisfies the equation (a) and (b) respectively.

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For the case x = 0,

which is always nonnegative in the region q ∈ (1, 4).

For the case when x = 1,

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 Fq has no critical points in the interior of D, we conclude that Fq is always nonnegative for 1 ≤ q ≤ 4. The nonnegative of the Fq is also plotted in Fig. 4.

Figure 4
figure 4

Fq is plotted as a function of x and q for 0 ≤ x ≤ 1, 1 ≤ q ≤ 4.

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Furthermore, we can consider the nonnegative region for the second-order derivative when q ranges in (0, 1). Under the condition , 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 is zero. We should only consider the condition in the limit x → 1. In this case, we have

Figure 5
figure 5

The plot of the dependence of x with q using the equation for (a) q ∈ (0, 1) and (b) q ∈ (4, 5) respectively.

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which gives the critical point qc3 ≈ 0.65. When q ≥ qc3, the second-order is always positive. Similarly, we can also analyze the nonnegative region for the second-order derivative 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 qc4 ≈ 4.65. When q ≤ qc4, the second-order is always positive. Notice that the analytical formula of Tq is established only for , we conclude that the second-order derivative is positive for which completes the proof of the convexity property of with the squared concurrence C2 for .

Additional Information

How to cite this article: Yuan, G.-M. et al. Monogamy relation of multi-qubit systems for squared Tsallis-q entanglement. Sci. Rep. 6, 28719; doi: 10.1038/srep28719 (2016).