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P J Geetha,Sudha,K S Mallesh. Comparative analysis of entanglement measures based on monogamy inequality. Chinese Physics B, 2017, 26(5): 050301  
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Comparative analysis of entanglement measures based on monogamy inequality
P J Geetha1, Sudha1,2, K S Mallesh3
Department of Physics, Kuvempu University, Shankaraghatta, Shimoga-577 451, India
Inspire Institute Inc., Alexandria, Virginia 22303, USA Department of Studies in Physics, University of Mysore, Mysore-570 006, India

 

† Corresponding author. E-mail: lyl_nuist@nuist.edu.cn

Abstract

We evaluate the monogamy inequality for symmetric, non-symmetric pure states of importance in terms of squared concurrence, squared entanglement of formation, squared negativity of partial transpose and compare the corresponding tangles. We show that though concurrence and concurrence tangle are zero for two special classes of mixed entangled states, both negativity tangle and entanglement of formation (EOF) tangle turn out to be non-zero. A comparison of different tangles is carried out in each case and it is shown that while the concurrence tangle captures the genuine multiqubit entanglement in N-qubit pure states with N distinct spinors (containing GHZ and superposition of W-, obverse W states) either negativity tangle or EOF tangle is to be used as a better measure of entanglement in the W-class of states with two distinct spinors and in the special classes of mixed multiqubit states.

PACS: 03.65.Ud;03.67.Bg
Keyword:monogamy of quantum entanglement;concurrence tangle;negativity tangle;Majorana representation of symmetric pure states
1. Introduction

Characterizing and quantifying entanglement of multipartite states are fundamental issues in quantum information theory. One of the most important properties of multipartite quantum systems, the limited shareability of quantum entanglement/correlation, has evoked great interest in recent years.[1–38] While the entanglement measure called concurrence-tangle was obtained via the monogamy relation using squared concurrence in N-qubit pure states,[1,7] the measure called negativity-tangle was obtained through the formulation of monogamy inequality using squared negativity of partial transpose in Ref. [10] for N-qubit states. Though entanglement of formation does not obey the monogamy relation, recently it was shown[37] that squared entanglement of formation satisfies the monogamy inequality and hence the tangle in terms of entanglement of formation is shown to quantify genuine multiqubit entanglement in N-qubit pure states.[37] We observe here that, by definition, the multiqubit entanglement measures such as concurrence tangle, negativity tangle, and entanglement of formation (EOF) tangle formulated using monogamy inequality, quantify the three-party or residual entanglement[1] which is the entanglement not accounted for by the two-qubit entanglements in a multiqubit state.[38] In fact, the generalized monogamy inequalities and the associated partition-dependent residual entanglement quantifiers accounting for arbitrary partitions of a multiqubit system have also been proposed[21] and were used in studying entanglement dynamics of N-qubit GHZ, W states interacting with N independent reservoirs. It is important to acknowledge the fact that a partition dependent residual entanglement gives a better characterization of multiqubit entanglement as it accounts for the entanglement between any two blocks which is over and above the sum of all two-qubit entanglements across the bipartition between blocks. While we intend to use partition dependent residual entanglement quantifiers in future works, we have restricted ourselves to a comparative analysis of the measures for usual residual entanglement in this article.

The motivation for the present work lies in the fact that there are different residual entanglement quantifiers such as concurrence tangle,[1,7] negativity tangle[10] (tangle in terms of squared negativity of partial transpose[39–41]), and EOF tangle[37] (tangle in terms of squared entanglement of formation[42,43]). The suitability of a measure to quantify residual or three-party entanglement in SLOCC inequivalent classes of pure states and special classes of mixed states is an issue that is worth examining. In fact, a comparitive analysis of different measures of residual entanglement in 3-qubit, 3-qudit states[20] and in N-qubit W-class of states[38] have been carried out respectively in Refs. [20] and [38]. We have also observed that for non-symmetric multiqubit states (states that do not remain invariant under interchange of qubits), the definitions of concurrence tangle[1,7] and EOF tangle[37] are to be slightly modified to account for the average entanglement in all tripartitions. While the definition of negativity tangle[10] is suitable for both symmetric and non-symmetric states, the need for an analogous definition for concurrence tangle, EOF tangle and comparison of these tangles for non-symmetric states are other motivations behind this work.

It can be recalled here that by using the Majorana representation[44–46] of pure symmetric N-qubit states, it was shown in Ref. [38] that the concurrence tangle vanishes for the W-class of states (N-qubit states with two distinct Majorana spinors) whereas the negativity tangle, being non-zero, properly quantifies the residual entanglement in this family. Here we evaluate the value of EOF tangle for the W-class of states to check whether it is non-vanishing and hence can quantify the three-party entanglement in this class. We also consider superposition of W and obverse W states which belong to the family of pure symmetric N-qubit states with N distinct spinors and evaluate their concurrence tangle, negativity tangle, and EOF tangle. This is done in order to examine whether measures of entanglement suitable to quantify the three-party entanglement in W-class of states are also suitable for states belonging to an SLOCC inequivalent class, the class of states with N-distinct spinors. We will show here that unlike the W-class where concurrence tangle fails to quantify the residual entanglement, all the three tangles are able to capture the three-party entanglement in the family of pure states with N distinct spinors. Quite similar to the W states, it is shown that their non-symmetric counterparts, the generalized W states, have vanishing concurrence tangle and both negativity tangle and EOF tangle serve as good measures of three-party entanglement for these states.

For mixed states, the evaluation of each tangle requires the knowledge of an optimal pure state decomposition[1,10,37] (that results in a minimum value of the tangle) for the state and hence all the three tangles cannot be readily evaluated for the mixed states. Efforts are underway to find optimal pure state decompositions of mixed states of interest and it is seen that there exists a special class of mixed entangled states having zero concurrence and concurrence tangle.[35,36] In Ref. [35], the authors showed that the concurrence tangle fails to identify genuine entanglement in a class of entangled three-qubit mixed states with no bipartite entanglement. Another class of N-qubit mixed entangled states having no bipartite entanglement, its genuine multiqubit entanglement not recognized by the concurrence tangle, has been identified in Ref. [36]. But the formulation of monogamy inequality in terms of squared entanglement of formation has led to the revelation that the genuine tripartite entanglement in the three-qubit mixed states in Ref. [35] can be effectively captured by the EOF tangle.[37] In this work, we examine whether the EOF tangle identifies the genuine multiqubit entanglement in the N qubit mixed entangled states without two-qubit entanglement and vanishing concurrence tangle.[36] We also evaluate the negativity tangle for these special states and examine whether, along with the EOF tangle, it serves the purpose of quantifying genuine multiqubit entanglement in mixed states.

The article is divided into five sections. In Section 2, we evaluate and compare the tangles corresponding to pure symmetric N-qubit states belonging to two inequivalent SLOCC classes. In Section 3, on defining concurrence tangle, EOF tangle to suit their use for non-symmetric states, we evaluate and compare all the three tangles for generalized W states. Section 4 gives the evaluation and comparison of the tangles in the two special classes of mixed entangled states. Section 5 provides a concise summary of the results.

2. Monogamy of pure symmetric states
2.1. qubit states belonging to W-class: symmetric states with 2 distinct spinors

The W-class or the Dicke class of states are symmetric (invariant under interchange of qubits) N-qubit states with 2 distinct spinors.[44–46] An arbitrary N-qubit pure symmetric state belonging to the W class is written[38] using Majorana representation[44–46] as

with a single parameter θ, describing the state.

It has been shown in Ref. [38] that the monogamy inequality with squared concurrence holds good with equality for the states , in other words, the N-concurrence tangle given by

vanishes for the family of states . The fact that N-concurrence tangle underestimates the genuine multiqubit entanglement in N-qubit pure states has been illustrated in Ref. [38] through the evaluation of non-zero negativity-tangle
for the W-class of states , with being the negativity of partial transpose between the qubits A1 and Ak. Continuing on this work,[38] we will show here that tangle in terms of squared EOF is non-zero for the family of states .

For N-qubit pure symmetric states, tangle in terms of squared EOF is given by

where the entanglement of formation in the bipartitions[42,43] , and in the bipartition are respectively given by
with being the binary entropy. The squared concurrence between any two qubits and that between a single qubit and the remaining qubits of the state are shown to be
On explicit evaluation of , using the concurrences in Eq. (4), the EOF tangle of the state is obtained as a function of the parameters θ and N, the number of qubits. Figure 1 shows the variations of negativity tangle and EOF tangle with respect to θ and N.

Fig. 1. (color online) Negativity tangle and EOF tangle versus θ and N for the state belonging to the W-class.

Though the concurrence tangle is zero for the W-class of states, from Fig. 1, it is readily seen that both EOF tangle and negativity tangle are non-zero, indicating genuine multiqubit entanglement in this class of states. It can also be seen that the negativity tangle is greater than the EOF tangle.

When in , one obtains the N-qubit W-state

By substituting in Eq. (4), one has
leading to the vanishing concurrence tangle for N qubit W states . In Ref. [38], the negativity-tangle for N-qubit W states is shown to be
Using the concurrences in Eq. (5), we calculate the EOF tangle for N-qubit W-states and tabulate the values of these tangles for different numbers N of qubits. It can be readily seen that both negativity tangle and EOF tangle can be used as multiqubit entanglement measures for N-qubit W states.

2.2. Superposition of W and obverse W states: symmetric states with N distinct spinors

We recall here that the so-called obverse W states are also states with 2-distinct spinors similar to the W states.[46] The N-qubit obverse W state is given by

Though both W and obverse W states belong to the SLOCC family of 2-distinct spinors, their equal superposition given by
belongs to the family of N distinct spinors.[46] It is to be recalled here that N-qubit GHZ states are symmetric states with N-distinct spinors[45,46] and hence both and belong to the same SLOCC class, the class of N distinct spinors. It has been shown in Ref. [46] that, inspite of belonging to the same SLOCC class of 3 distinct spinors, the entanglement properties of are quite different from those of 3 qubit GHZ states.[46,47] Here, we examine the monogamous nature of the states and show that except for , the states possess the maximal monogamous behavior of states. In the following, we evaluate the tangles with respect to squared concurrence, negativity, and squared entanglement of formation for the states .

On evaluating the two-qubit and single-qubit density matrices of the 3-qubit state , we obtain

The evaluation of negativity tangle of leads us to as
For the 4-qubit state , it can be seen that
For N-qubit states , , while the single qubit density matrix turns out be the maximally mixed state , the two-qubit density matrix is given by
The concurrence as well as negativity of the state are readily seen to be zero. The entanglement between the first qubit (or any other) and the remaining qubits being , we obtain as , with being a maximally mixed state. We therefore have all the three tangles equal and equal to 1 for , . On explicit evaluation, we obtain and can conclude that
In fact, the residual entanglement of the state with matches exactly with that of the N-qubit GHZ state for which[38]
It can thus be concluded that, except for , the two states and belonging to the SLOCC family of N distinct spinors have the same monogamous behavior and maximal residual entanglement.

2.3. Generalized GHZ State

Consider the N-qubit generalized GHZ state[25] given by

The two-qubit and single-qubit density matricesz of are readily seen to be
implying , , and . The concurrence tangle for the generalized GHZ state thus turns out to be
As , and , the negativity tangle of the state is the same as that of concurrence tangle . Owing to the vanishing , we have , , but is non-zero and given by
As
hence the EOF tangle for is obtained as
with , and being the binary entropy.

A comparison of all the three tangles for the generalized GHZ state is shown in Fig. 2.

Fig. 2. (color online) Concurrence tangle , EOF tangle , and negativity tangle versus r1 for the generalized GHZ-state.

It can be readily seen from Fig. 2 that the maximal monogamous behavior with is realized when , i.e., for the GHZ state .

3. Non-symmetric pure states

Before examining the monogamy inequality and entanglement in non-symmetric pure states, we observe that only the tangle in terms of squared negativity of partial transpose defined in Ref. [10] is suitable for quantifying entanglement in non-symmetric states and both concurrence tangle[1,7] and EOF tangle[37] are to be redefined so that different partitions of the non-symmetric state are taken into account.

The negativity tangle for N-qubit pure states is given by[10,38]

where
It is observed here that though leading to , for symmetric states, one has to take into account all possible bipartitions as given in Eq. (20) for non-symmetric states. Analogous to the definition of negativity tangle in Eq. (20), we redefine the concurrence tangle and EOF tangle as
where
and
where
so that they properly quantify the entanglement in non-symmetric states.

In the following, we make use of Eqs. (20)–(25) to evaluate the three different measures of three-party entanglement in the generalized W state.

3.1. Generalized W state

The 3-qubit generalized W state[25] is of the form

and is readily seen to be non-symmetric under interchange of qubits.

The two-qubit and single-qubit density matrices of the state are given by

The evaluation of concurrences in different bipartitions of the state leads us to
Now it can be readily seen that
implying that the concurrence tangle is quite similar to the case of their symmetric counterparts .

The two-qubit and single qubit-density matrices in Eq. (27) allow us to evaluate the negativity of partial transpose in different bipartitions of the state and we have

The negativity tangles for each focus qubit can now be readily evaluated and we obtain negativity tangle for the state as a function of the parameters r1 and r2 by using .

Though the concurrence tangle vanishes for , the EOF tangle for each focus qubit is seen to be non-zero. In fact, and the EOF tangle depends on the parameters . Figure 3 gives the comparison of negativity tangle with EOF tangle for .

Fig. 3. (color online) EOF tangle and negativity tangle with respect to the parameters r1 and r2 for the generalized W-state. Observe that is quite similar to the W states.
4. Monogamy of mixed states

In this section, we are going to evaluate the different tangles for two special families of mixed states. Here, we recall that

where , , are the tangles of the pure states appearing in an arbitrary pure state decomposition ( , of the mixed state ρ). The minimum in Eq. (29) is to be taken over all possible pure state decompositions of ρ. If
is the optimal pure state decomposition that gives the minimum tangle for ρ, then one can readily have
as the concurrence-tangle, EOF tangle, negativity tangle, respectively of the mixed state ρ. It is thus evident that the evaluation of the tangles for a mixed state requires the knowledge of its optimal pure state decomposition.

4.1. N-qubit mixed state without two-qubit entanglement

Consider an N-qubit mixed entangled state[36] given by

with and being N-qubit spin-down and W-states, respectively.

This is an entangled state but without two-qubit entanglement, and its genuine multiqubit entanglement not detected by concurrence tangle as vanishes for any .[36] Here, we evaluate the EOF tangle and negativity tangle for and check whether they can quantify the entanglement in . We also compare negativity tangle with EOF tangle for all N.

The state is readily seen to be symmetric and its two qubit density matrices , are given by

This gives , , indicating no two-qubit entanglement in the state. As the pure state decomposition of the state in Eq. (32) is shown to be optimal in Ref. [35], one can readily evaluate the tangles for using
with being a separable state, . By using the tangles for the N-qubit W state obtained in subSection 2.1, one has
As , the concurrence tangle of is zero for all N, but the EOF tangle is nonzero and detects the genuine multiqubit entanglement in the state as observed in Ref. [36]. We calculate the negativity tangle and compared it with the other two tangles in Fig. 4.

Fig. 4. (color online) The comparison of the tangles , , for the mixed symmetric state . Notice that for all N.

It can be readily seen from Fig. 4 that negativity tangle also helps to quantify the genuine multiqubit entanglement in the mixed state . In fact, both negativity tangle and EOF tangle serve as good measures of residual entanglement in the state not having two-qubit entanglement and with vanishing concurrence tangle.

4.2. Mixed state containing GHZ and W states

Consider the mixed 3-qubit state [35] composed of GHZ and W states

It has been shown in Ref. [35] that when , the state corresponds to an entangled state without two-qubit concurrence and with vanishing concurrence tangle.

The optimal pure state decomposition for the state when is given by[35,37]

where the pure state component and the parameter varies from 0 to 1 as , .

The two-qubit and single-qubit density matrices of the symmetric pure states , are respectively given by

where . The concurrences , are obtained as functions of p0 and it is found that , independent of j. By substituting , we obtain
As , and the tangle for the W state is zero, the concurrence tangle of the state given by
turns out to be zero.

The negativity of the partial transpose of the states , is also found to be independent of j and is obtained as a function of p0. Substituting , we readily obtain . As , the negativity tangle of the symmetric states turns out to be

The negativity of the state is thus given by
An evaluation of the EOF tangle for the state using the concurrences CAB, gives us . Using the EOf tangle (see Table 1), we have

Table 1.

Concurrence tangle, negativity tangle, and EOF tangle of for different N.

.

In Fig. 5, we plot the tangles , , . It is seen that although the concurrence-tangle is zero, the non-zero values of EOF tangle and negativity tangle are helpful in quantifying the genuine three-qubit entanglement in the state .

Fig. 5. (color online) The nonzero and detect the genuine three-qubit entanglement in the mixed state when .

It is to be noted that, in Ref. [37], EOF tangle ξ1 with the first qubit A as the focus qubit has been evaluated and it was indicated that the EOF tangle is able to quantify the tripartite entanglement in . Here, we have shown that due to the permutation symmetry of , and is the actual EOF tangle of the state .

5. Conclusion

A comparison of different three-party entanglement measures formulated using monogamy inequality of pure symmetric/non-symmetric and mixed multiqubit states has been carried out. Two inequivalent SLOCC families of pure symmetric states are considered and it is shown that though the concurrence tangle is zero, both negativity tangle and tangle in terms of squared entanglement of formation can be used to quantify the three party entanglement in the W-class of states with 2 distinct spinors. For the equal superposition of N-qubit W, obverse W states characterized by N distinct spinors, all the three tangles are shown to be equal and capable of quantifying the maximal entanglement when . By redefining the concurrence tangle, tangle in terms of squared entanglement of formation to account for all bipartitions of a non-symmetric state, we have evaluated these tangles for the 3-qubit generalized W state. For two special classes of mixed entangled states with zero concurrence and zero concurrence tangle, we have evaluated the negativity tangle as well as the tangle in terms of squared entanglement of formation. A comparison of tangles reveals the fact that the negativity tangle and the tangle in terms of squared entanglement of formation are non-zero for these mixed states, indicating genuine multiqubit entanglement in them. It is observed that except for superposition of 3-qubit W, obverse W states, the negativity tangle is greater than the tangles in terms of squared concurrence and squared entanglement of formation. Through our evaluation, comparison of tangles in different classes of pure, mixed states, it is illustrated that whereas the concurrence tangle is a good measure of multipartite entanglement in pure symmetric states with N-distinct spinors, one needs to use the negativity tangle and the tangle in terms of squared entanglement of formation for pure symmetric states with 2 distinct spinors and other mixed multiqubit states with vanishing concurrence tangle. It would be of interest to formulate different tangles using the generalized monogamy relations[21] and evaluate their values to obtain suitable entanglement quantifiers for multiqubit states of importance.

Acknowledgment

P J Geetha acknowledges the support of the Department of Science and Technology (DST), Govt. of India through the award of the INSPIRE fellowship.

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