1 Introduction

Entanglement of multipartite pure states has been object of many studies both theoretical and experimental ^[1,3]. The reason for the above is that multipartite entanglement is a basic ingredient for Quantum Information Protocols (QIP). Although certainly there have been advances in the study of multipartite entanglement ^[4,11], it is not yet understood the time evolution of the initial entanglement of a system of several qubits. In particular, it arises the question about the characteristics of the time evolution of the 3-tangle of a system of 3-qubit interacting mutually through a XY Hamiltonian.

As it has been pointed out in Ref. ^[4] the 3-tangle can be an important quantity for measuring the entanglement of a 3-qubit system. In the present paper we study the time evolution of the 3-tangle associated to a 3-qubit system in a pure state. In order to do the above we employ the 3-tangle introduced in Ref. ^[4] and also the quantum Heisenberg XY-Hamiltonian ^[12] for a system of 3-qubit.

Thus, given an initial 3-qubit state |ψt=0=c0t=0|000+c1t=0|001+c2t=0|010+c3t=0|011+c4t=0|100+c5t=0|101+c6t=0|110+c7t=0|111, the time evolution of such a state is given by the Heisenberg operator i.e. |ψt=e-iHt|ψt=0=c0t|000+c1t|001+c2t|010+c3t|011+c4t|100+c5t|101+c6t|110+c7t|111 where H is the XY-Hamiltonian of the 3-qubit system. In our approach, we derive an analytic expression for the Heisenberg operator e-iHt with which if the initial 3-tangle (𝒯(t = 0)) is known in terms of the initial coefficients cit=0 i=0,1,…,7 then the final tangle 𝒯(t) will be known in terms of the final coefficients cit i=0,1,…,7, the value of J and the time t.

As a result we find noticeable harmonic-like time behavior for the 3-tangle. The later seemingly suggests that the entanglement of a 3-qubit system interacting through a XY Hamiltonian is a quantized quantity. The paper is organized as follows: in Section 2 we derive the formalism for a 3-qubit system interacting through a XY-Hamiltonian. In Section 3 we find an expression for the 3-tangle as a function of time. Finally, we conclude the work by giving a discussion of our results in a section of Conclusions.

2 3-qubits XY Hamiltonian

In order to facilitate our calculations it is employed the decimal notation, which is defined as follows:

Then, a general pure 3-qubits state can be defined in terms of a superposition of the above basis as follows:

where:

With the decimal notation it is possible to associate a matrix with a Hamiltonian operator. The respective associated matrix elements to the Hamiltonian operator H become:

The so called XY-Hamiltonian for n qubits is: ^[12]

where N = 2ⁿ, J is the coupling constant, and Sia is the a (a = x, y) component of the spin of the i - th qubit. In the present case we have n = 3 qubits (i.e. N = 8).

Let us observe that the states |0 and |7 are annihilated by the action of the operator H of Eq. (5), that is:

Furthermore, the action of the XY Hamiltonian H of Eq. (5) on the rest of the decimal states is:

Through the use of the Eqs. (4)-(7) and the orthonormality of the decimal basis, the construction of the matrix associated to H yields:

On the other hand, the time evolution operator can be expanded in powers of H as follows:

We observe that the several different powers of H of Eq. (8) behave peculiarly. For instance the quadratic power is:

In a similar way, for the other powers we obtain that:

ewhere I2-7 has been defined in Eq. (11). In general for the n - th power we find that:

However, we can see that an=2bn-1 and bn=bn-1+an-1=bn-1+2bn-2, then the above equation can be expressed as:

We observe from the above equation that for n = 0, the second term will be equal to zero and that the first one is equal to 1. However, in this case, H0=I2-7 and this is not the identity I₈ as can be seen from Eq. (11). Such a problem can be solved as follows:

where:

From the above equation we find that the time evolution operator will always be linear on H, and the time evolution operator can be written as:

It is worth to observe that the last expression can be written in terms of exponentials with which the time evolution operator takes a simple form:

Let us note that according to Eqs. (9) and (10) the time evolution of the state |ψt=0 is given by:

It can be observed from the above equation that we can calculate the coefficients at any time cjtj=0,1,…,7 if the initial coefficients cjt=0j=0,1,…,7 are known and if it is also known the action of the time evolution operator on each of the decimal states, that is, Ut|i for i = 0, ...,7. Through the use of Eqs. (6), (7), (11), (16), and (18) it is found that:

To substitute Eqs. (20)-(27) into Eq. (19), we find the coefficients at any time cjtj=0,1,…,7 in terms of both the above exponentials and the initial coefficients cjt=0j=0,1,…,7 where ∑j=07cjt=02=1.

3 3-tangle as a Measure of Multipartite Entanglement of a 3-qubit System

The measure of entanglement for a 3-qubit system can be is obtained through the 3-tangle which is defined as ^[4]

with:

where c _i represents the coefficient of basic state |i. Thus, by calculating the coefficients c_i (i = 0, 1, ... , 7) as a function of time, in the way it was explained at the end of the above section, we shall be able of finding the 3-tangle of Eq. (28) as a function of time. That is to find T3t=4d1t-2d2t+4d3t providing the coefficients c_i (t) are known. It is worth to observe from Eqs. (18) and (19) that the coefficients c_i (t) (i = 0,1,..., 7) will depend on the initial coefficients c_j (t = 0) (j = 0,1,7), the antiferromagnetic constant J and the time t. By the way, in the present work the initial coefficients c_j (t=0) ∑j=07cj2=1 are found in a random way with which the coefficients c_i (t) (i = 0,1, ...,7) at time t will result a two variables function namely J and t.

Before of considering a general state we are focusing on the so called W and GHZ states which are defined as:

The respective initial 3-tangle for the GHZ-state is unit while for the W-state the initial 3-tangle is zero. Now, the W-state time evolution is only over the phase. Therefore the 3-tangle of the W-state does not change in time. Thus, the XY Hamiltonian keeps constant the entanglement of the W-state which is an important result. On the other hand, the GHZ-state also is not modified by the time evolution operator of Eq. (19) hence its associated 3-tangle keeps constant in time. We conclude that the XY Hamiltonian assures that the entanglement of the GHZ-state does not change in time.

Let us now consider an arbitrary initial 3-qubit state at t = 0 denoted by |ψt=0=c0t=0|000+c1t=0|001+c2t=0|010+c3t=0|011+c4t=0|100+c5t=0|101+c6t=0|110+c7t=0|111 where ∑i=07cit=02=1. In order to evaluate the 3-tangle at time t from Eqs. (28)-(31), we employ eqs. (19)-(27) where the initial coefficients c_i (t = 0) are found in a random way. We perform the above procedure in three different cases and calculate the respective 3-tangle in each one of the three different cases. In the Appendix we write the three different random initial 3-qubit states employed in the present work. In figure 6, we show the time evolution of the 3-tangle as a function of both j and t associated to each of the three different random initial 3-qubit states employed in the present work.

4 Relevance of Entanglement for Technological Applications

Quantum entanglement is essential not only for technological applications such as quantum computation ^[13], data base search algorithm ^[14] or quantum cryptography ^[15] and quantum secret sharing ^[16] but also for non-artificial systems. For instance for photosynthesis ^[17]-[18], navigational orientation of animals ^[19], the imbalance of matter and antimatter in the universe ^[20] and evolution itself ^[21].

5 Random Initial 3-qubit States

We write the three different random initial 3-qubit states that we have employed in the present work.

Such a states are the following:

We observe that all of the above three 3-qubit states are normalized to unit.

6 Conclusions

We have studied the behavior in time of the 3-tangle associated to a 3-qubit system interacting through the XY Hamiltonian given by Eqs. (5) and (8). The 3-tangle associated to the state |ψt=c0t|000+c1t|001+c2t|010+c3t|011+c4t|100+c5t|101+c6t|110+c7t|111 is given by Eqs. (28)-(31) where each one of the coefficients cit i=0,1,…,7 depend on the random initial coefficients cjt=0 j=0,1,…,7, J and the time t as it can be seen from Eqs. (18)-(27).

An important result obtained in the present work is that the entanglement of both the W-state and the GHZ-state keeps constant in time providing the three qubits interact through the XY Hamiltonian given by Eq. (5).

Such a result could have important experimental advantages whereas both the W-state and the GHZ-state can be used on solid basis for testing different QIP protocols.

In Figure we have plotted the 3-tangle of Eq. (28) as a function of both the time t and the antiferromagnetic factor J for three different random 3-qubit states. It is worth to point out that the 3-tangle shows a noticeable periodic behavior as it is appreciated from Figure being the respective period t=4π/J. Such a behavior in time is a consequence of the harmonic structure of the time evolution operator of Eq. (18).

Our results invoke to the present experimental facilities to measure the 3-tangle for a system of 3-qubits by taking into account that for certain times the entanglement disappears and that for other values of both the time and the antiferromagnetic constant J such a quantity is maximal. The maximal values of the 3-tangle can be used for implementing Quantum Information Processing protocols where entanglement is a resource. Our results might indicate that the 3-tangle associated to a 3-qubit system resembles to a quantized physical quantity providing the three qubits interact through a XY Hamiltonian.