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Zha Xin-Wei, Miao Ning, Li Ke. Relations between tangle and I concurrence for even n-qubit states. Chinese Physics B, 2019, 28(12): 120304  
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Relations between tangle and I concurrence for even n-qubit states
Zha Xin-Wei, Miao Ning †, Li Ke
Xi’an University of Posts and Telecommunications, Xi’an 710121, China

 

† Corresponding author. E-mail: 673431488@qq.com

Abstract

Gilad Gour and Nolan R Wallach [J. Math. Phys. 51 112201 (2010)] have proposed the 4-tangle and the square of the I concurrence. They also gave the relationship between the 4-tangle and the square of the I concurrence. In this paper, we give the expression of the square of the I concurrence and the n-tangle for six-qubit and eight-qubit by some local unitary transformation invariant. We prove that in six-qubit and eight-qubit states there exist strict monogamy laws for quantum correlations. We elucidate the relations between the square of the I concurrence and the n-tangle for six-qubit and eight-qubits. Especially, using this conclusion, we can show that 4-uniform states do not exist for eight-qubit states.

PACS: 03.67.Mn;03.65.Ud
Keyword:quantum correlations;entangled relations;n-tangle;six-qubit and eight-qubit
1. Introduction

Entanglement is considered as the central resource for quantum information and computation,[1–4] and numerous theoretical and experimental researches have been conducted in this field.[5–8] In particular, the search for maximally entangled states has received a great deal of attention.[9–24] Then there comes a fundamental question which states are maximally entangled. In the case of 2 qubits, it is known that Bell states are maximally entangled with respect to any measures of entanglement.[1] The concurrence has been shown to be a useful entanglement measure for pure two qubits. In 2000, Coffman, Kundu, and Wootters[9] used concurrence to examine three-qubit quantum systems and introduced the concept of “residual entanglement”, or the 3-tangle. In 2001, Wong and Christensen[20] gave the definition of n-tangle for even n qubits. The n-tangle of even n qubits is invariant under permutations of the qubits, which is an entanglement monotone. An expression for 4-tangle was obtained by examining the negativity fonts present in a four-way partial transpose under local unitary operations in 2019.[21]

For higher number of qubits, the problem is no longer simple and depends in general on the entanglement measure. In 2010, Gilad Gour and Nolan R Wallach[10] proposed the tangle or the square of the I concurrence. In four qubits there are four bipartite cuts consisting of one qubit versus the remaining three quibts and three bipartite cuts consisting of two qubits versus the remaining two qubits. Denoting the four qubits by A, B, C, and D, they are defined as

where with ρA = trρBCD (|ψ⟩ ⟨ψ|) being the reduced density matrix and with ρAB = trρCD (|ψ⟩ ⟨ψ|) being the reduced density matrix. Also they are related by where τABCD is the 4-tangle, which is defined by Wong and Christensen[20] as follows: In this paper, we investigate the relation between the n-tangle with τ1, τ2, … for a six-qubit system and those for an eight-qubit system.

2. Relation between n-tangle and square of I concurrence
2.1. Relation between n-tangle and square of I concurrence of six-qubit

In the following, we show how specific monogamy relations can be deduced from local unitary transformation invariant.

For a six-qubit pure state, Like Gilad Gour and Nolan R Wallach’s definition, for six-qubit pure states, the square of the I concurrence can be given as where , , , etc.

And the 6-tangle is In Ref. [25] it has been shown that where It is obvious that such invariants satisfy Fi ⩾ 0, Fij ⩾ 0, and Fijk ⩾ 0.

From Eqs. (6)–(10), we can obtain From Eqs. (6)–(12), we can obtain It is easy to show that 0 ⩽ τ1 ⩽ 1.

Similarly, from Eqs. (6)–(12), we can obtain Therefore, one can show that 0 ⩽ τ2 ⩽ 3/2.

Also, from Eqs. (8)–(12), we can obtain Thus, we have 0 ⩽ τ3 ⩽ 7/4.

On the other hand, we can show From Eqs. (11)–(14), we have We can measure the entanglements of the product, GHZ, and maximally entangled state by this relation.

For the product state, we have FA = FB = FC = FD = FE = 1, FAB = FAC = ··· = FD E = 1, FABC = FABD = ··· = FD E F = 1.

From Eqs. (9)–(15), we can obtain τABCDEF = 0, τ1 = τ2 = τ3 = 0, obviously, and it satisfies Eq. (17). For the GHZ state, one can have From Eqs. (9)–(14), we can obtain τABCDEF = 1, τ1 = τ2 = τ3 = 1, and it also satisfies Eq. (17).

For the maximally six-qubit entangled state[26] we can show that From Eqs. (9)–(15), we can obtain τ1 = 1, τ2 = 3/2, τ3 = 7/4, τABCDEF = 1, and it also satisfies Eq. (17).

2.2. Relation between n-tangle and square of I concurrence of eight-qubit

For an eight-qubit state Similarly, we can have Similarly, we can obtain From Eqs. (21)–(25), we have For the product state, we have From Eqs. (23)–(27), we can obtain τABCDEFGH = 0, τ1 = τ2 = τ3 = τ4 = 0, obviously, it satisfies Eq. (29).

Similarly, for the GHZ state, , one can have From Eqs. (23)–(27), we can obtain τ1 = τ2 = τ3 = τ4 = 1, τABCDEFGH = 1. Obviously, it satisfies Eq. (29).

3. Discussion

For a six-qubit pure state, from Eq. (17), we know that For the 3-uniform state, it must have then, we have Therefore, we obtain τABCDEF = 1. This result is just consistent with maximally six-qubit entangled state. Therefore, we can say that maximally six-qubit entangled state is a 3-uniform state, and its 6-tangle is 1.

For the eight-qubit state, from Eq. (29), we know that Assuming that the eight-qubit state is a 4-uniform states, we have From Eq. (29), we obtain τABCDEFGH = −41/8. Because τABCDEFGH ⩾ 0, therefore, it is impossible have a 4-uniform state for the eight-qubit state. This conclusion is consistent with the result in Ref. [27].

4. Conclusions

We have derived two relations of pure-state six-qubit and eight-qubit entanglement, the central results are Eqs. (17) and (29). We first introduce a relationship between the n-tangle and the multipartite entanglement of even n qubits. For a maximally six-qubit entangled state, we can find that it is a 3-uniform state with the 6-tangle equal to 1. Therefore, we can say that n-tangle equal to 1 is the necessary condition of maximally six-qubit entangled state. But for an eight-qubit state, using this relation, we find that it is impossible to have a 4-uniform state. Therefore, maximally entangled eight-qubit is also a 3-uniform state with the 8-tangle equal to 0.

Quantum metrology is the most promising technology originating from quantum information,[28,29] and already shows a shining future. Later, we will calculate the quantum capture information in the case, and try to find the relationship between these relevant information and quantum capture information.

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