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## On-line version ISSN 2007-9737Print version ISSN 1405-5546

### Comp. y Sist. vol.23 n.2 Ciudad de México Apr./Jun. 2019  Epub Mar 10, 2021

#### https://doi.org/10.13053/cys-23-2-2309

Articles

Time Evolution of the 3-Tangle of a System of 3-Qubit Interacting through a XY Hamiltonian

Abstract:

We consider a pure 3-qubits system interacting through a XY-Hamiltonian with antiferromagnetic constant J. We employ the 3-tangle as an efficient measure of the entanglement between such a 3-qubit system. The time evolution of such a 3-tangle is studied. In order to do the above, the 3-tangle associated to the pure 3-qubit state |ψt=c0t|000+c1t|001+c2t|010+c3t|011+c4t|100+c5t|101+c6t|110+c7t|111 is calculated as a function of the initial coefficients cit=0 i=0,1,,7, the time t and the antiferromagnetic constant J. We find that the 3-tangle of the 3-qubit system is periodic with period t=4π/J. Furthermore, we also find that the 3-tangle as a function of the time t and J has maximal and minimum values. The maximal values of the 3-tangle can be employed in Quantum Information Protocols (QIP) that use entanglement as a basic resource. The pattern found for the 3-tangle of the system of three qubits interacting through a XY Hamiltonian as a function of J and the time t resembles to a quantized physical quantity.

Keywords: 3-qubits; non-classical communications; quantum information processing; entanglement

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:

|0=|000,

|1=|001,

|2=|010,

|3=|011, (1)

|4=|100,

|5=|101,

|6=|110,

|7=|111,

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

ψ=i=07cii, (2)

where:

i=07ci2=1. (3)

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:

Hij=iHj. (4)

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

H=ji=0N-1SixSi+1x+SiySi+1y, (5)

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:

H0=0,H7=0. (6)

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

H1=J22+4,H2=J21+4,H3=J25+6,H4=J22+1,H5=J26+3,H6=J25+3. (7)

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

H=J20000001001000000011000010001000000001000100001100000001001000000. (8)

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

Ut=exp-iHt=1-iHt+-i22Ht2+-i33!Ht3. (9)

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

H2=J240000021001200002011000010001000000001000100001102000021001200000=J2420000010000100001000000000000000000001000000000001000010000100000+0000001001000000011000010001000000001000100001100000001001000000J222I2-7+J2H.

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

H3=J232I2-7+J223H,H4=J242*3I2-7+J233+2H,H5=J252*5I2-7+J245+6H,H6=J262*11I2-7+J2511+10H, (10)

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

Hn=J2nanI2-7+J2n-1bnH. (11)

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:

Hn=J2n23--1n-1+2n-1I2-7+J2n-1--1n+2n3H, n1. (12)

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:

Hn=I1,8δ0n+J2n23--1n-1+2n-1I2-7+J2n-1--1n+2n3H, n0. (13)

where:

I1,81000000000000000000000000000000000000000000000000000000000000001. (14)

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:

Ut=n=0-iHtnn!=n=0-itnn!I1,8δ0n+J2n23--1n-1+2n-1I2-7+j2n-1--1n+2n3H=I1,8+2I2-73n=01n!-itJ2n--1n-1+2n-1+2H3Jn=01n!-itJ2n--1n+2n. (15)

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:

Ut=I1,8+2I2-73eiJt2+12e-iJt+2H3Je-iJt-eiJt2. (16)

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

ψt=Uψt=0=Uc0t=00+c1t=01+c2t=02+c3t=03+c4t=04+c5t=05+c6t=06+c7t=07=c0t0+c1t1+c2t2+c3t3+c4t4+c5t5+c6t6+c7t7. (17)

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:

Ut0=0, (18)

Ut1=23eiJt2+12e-iJt1 (19)

+13e-iJt-eiJt22+4,

Ut2=23eiJt2+12e-iJt2 (20)

+13e-iJt-eiJt21+4,

Ut3=23eiJt2+12e-iJt3 (21)

+13e-iJt-eiJt25+6,

Ut4=23eiJt2+12e-iJt4 (22)

+13e-iJt-eiJt22+1,

Ut5=23eiJt2+12e-iJt5 (23)

+13e-iJt-eiJt26+3,

Ut6=23eiJt2+12e-iJt6 (24)

+13e-iJt-eiJt25+3,

Ut7=7. (25)

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]

T3=4d1-2d2+4d3, (26)

with:

d1=c02c72+c12c62+c22c52+c42c32, (27)

d2=c0c7c3c4+c0c7c5c2+c0c7c6c1+c3c4c5c2+c3c4c6c1+c5c2c6c1, (28)

d3=c7c6c5c3+c7c1c2c4, (29)

where c i represents the coefficient of basic state |i. Thus, by calculating the coefficients ci (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 ci (t) are known. It is worth to observe from Eqs. (18) and (19) that the coefficients ci (t) (i = 0,1,..., 7) will depend on the initial coefficients cj (t = 0) (j = 0,1,7), the antiferromagnetic constant J and the time t. By the way, in the present work the initial coefficients cj (t=0) j=07cj2=1 are found in a random way with which the coefficients ci (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:

W=134+2+1, (30)

GHZ=120+7. (31)

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 ci (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:

ψ1t=00.0649682+0.480244i0+0.0820031+0.0744268i10.157695+0.567361i20.00990613+0.30057i30.159286+0.122371i40.136861+0.0406154i50.00576077+0.267818i60.424509+0.054595i7, (32)

ψ2t=00.254723+0.452791i0+0.205806+0.3656i10.119695+0.452655i20.10712+0.095714i30.000551918+0.408866i40.0713835+0.0732269i50.0279167+0.0993365i60.316043+0.161424i7, (33)

ψ3t=00.228717+0.66739i0+0.124412+0.62744i10.0241769+0.16416i20.00878132+0.0690814i30.0589419+0.165814i40.0255238+0.105097i50.0946251+0.0750734i60.00977502+0.0581965i7. (34)

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.

References

1. Werner, R. (1989). Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A, Vol. 40, 4277. DOI: 10.1103/PhysRevA.40.4277. [ Links ]

2. Bennett, C.H., Bernstein, H., Popescu, S., & Schumacher, B. (1996). Concentrating partial entanglement by local operations. Phys. Rev. A, Vol. 53, 2046. DOI: 10.1103/PhysRevA.53.2046. [ Links ]

3. Bennett, C.H. , DiVincenzo, D., Smolin, J.A., & Wootters, W.K. (1996). Mixed-state entanglement and quantum error correction. Phys. Rev. A, Vol. 54, 3824. DOI: 10.1103/PhysRevA.54.3824. [ Links ]

4. Coffman, V., Kundu J., & Wootters, W.K. (2000). Distributed entanglement. Phys. Rev. A, Vol. 61, 052306. DOI: 10.1103/PhysRevA.61.052306. [ Links ]

5. Dür, W., Vidal, G., & Cirac, J.J. (2000). Three qubits can be entangled in two inequivalent ways. Phys. Rev. A, Vol. 62, 062314. DOI:10.1103/PhysRevA.62.062314. [ Links ]

6. Acin, A., Costa, A.A.L., Jane, E., Latorre, J., & Tarrach, R. (2000). Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States. Phys. Rev. Lett. Vol. 85, 1560. DOI:10.1103/PhysRevLett.85.1560. [ Links ]

7. Carteret, H. & Sudbery, A. (2000). Local symmetry properties of pure 3-qubit states. J. Phys. A, Vol. 33, 4981. DOI: 10.1088/0305-4470/33/28/303. [ Links ]

8. Acin, A. , Adrianov, A., Jane, E. , & Tarrach, R. (2001). Three-qubit pure-state canonical forms. J. Phys. A, Vol. 31, 6725. DOI: 10.1088/0305-4470/34/35/301. [ Links ]

9. Acin, A. , Bruss, D., Lewenstein, M., & Sanpera, A. (2001). Classification of Mixed Three-Qubit States. Phys. Rev. Lett., Vol. 87, 040401. DOI: 10.1103/PhysRevLett.87.040401. [ Links ]

10. Wei, T.C. & Goldbart, P.M. (2003). Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A, Vol. 68, 042307. DOI: 10.1103/PhysRevA.68.042307. [ Links ]

11. Levay, P. (2005). Mott insulator to superfluid transition in the Bose-Hubbard model. Phys. Rev. A, Vol. 71, 012334. DOI: 10.1103/PhysRevA.71.033629. [ Links ]

12. Lieb, E., Schultz, T., & Mattis, D. (1961). Two soluble models of an antiferromagnetic chain. Ann. Phys. N.Y., Vol.16, 407. DOI: 10.1016/0003-4916(61)90115-4. [ Links ]

13. Deutsch, D. (1989). Quantum computational networks. Proc. R. Soc. London, Ser. A., Vol. 425, 73. DOI: 10.1098/rspa.1989.0099. [ Links ]

14. Avila, M. & Elizalde-Salas, J.B. (2017). Remedies for the Inconsistences in the Times of Execution of the Unsorted Database Search Algorithm within the Wave Approach. Computación y Sistemas, Vol. 21, 883. DOI: 10.13053/cys-21-4-2367. [ Links ]

15. Gisin, N., Ribordy, G., Tittel, W., & Zbinden, H. (2002). Quantum cryptography. Rev. Mod. Phys., Vol. 74, 145. [ Links ]

16. Hillery, M., Buzek, V., & Berthiaume, A. (1999). Quantum secret sharing. Phys. Rev. A, Vol. 59, 1829. DOI: 10.1103/PhysRevA.59.1829. [ Links ]

17. Caruso, F., Chin, A. W., Datta, A., Huelga, S.F., & Plenio, M.B. (2010). Entanglement and entangling power of the dynamics in light-harvesting complexes. Phys. Rev. A, Vol. 81, 062346. [ Links ]

18. Sarovar, M., Ishizaki, A., Fleming, G.R., & Whaley, K.B. (2010). Quantum entanglement in photosynthetic light-harvesting complexes. Nat. Phys., Vol. 6, 462. DOI: 10.1038/nphys1652. [ Links ]

19. Cai, J., Guerreschi, G.G., & Briegel, H.J. (2010). Quantum Control and Entanglement in a Chemical Compass. Phys.Rev. Lett., Vol. 104, 220502. DOI: 10.1103/PhysRevLett.104.220502. [ Links ]

20. Hiesmayr, B.C. (2007). Nonlocality and entanglement in a strange system. Eur. Phys. J. C, Vol. 50, 73. DOI: 10.1140/epjc/s10052-006-0199-x. [ Links ]

21. McFadden, J. & Al-Khalili, J. (1999). A quantum mechanical model of adaptive mutation. BioSystems, Vol. 50, 203. [ Links ]

Received: October 27, 2015; Accepted: May 20, 2016

* Corresponding author is Manuel Ávila Aoki. vlkmanuel@uaemex.mx