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Entanglement, phase correlation and dephasing of two-qubit states
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Optics Communications 281 (2008) 3943–3946 www.elsevier.com/locate/optcom
Entanglement, phase correlation and dephasing of two-qubit states Masashi Ban * Graduate School of Humanities and Sciences, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan CREST, Japan Science and Technology Agency, 1-1-9 Yaesu, Chuo-ku, Tokyo 103-0028, Japan Received 15 January 2008; received in revised form 21 February 2008; accepted 25 March 2008
Abstract The relation between entanglement and phase correlation of two-qubit states is studied. Decoherence of entanglement and phase correlation caused by correlated classical noises is also investigated. It is found that the decay of the phase correlation is quite different from that of the entanglement when the finite-time disentanglement occurs. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Entanglement; Phase correlation; Stochastic dephasing; Qubit
Quantum information processing has recently attracted very much attention in quantum physics and information science [1,2]. It provides the novel information technology such as quantum cryptography, quantum communication and quantum computation as well as new insights on the principles of quantum mechanics. One of the most important resources is undoubtedly entanglement of two-qubit states [1] which include the Bell state and the Werner state. Two-qubit states can be characterized by the several parameters such as purity (or mixedness), entanglement and correlation of two single qubit observables. The relation between the entanglement and mixedness of two-qubit states has been investigated in detail [3–5]. The maximally entangled states, the amount of entanglement of which cannot be increased by any unitary operation, have been found in these works. A phase plays an important role in quantum coherence. The phase correlation, however, has not been discussed. Therefore, in this paper, we consider the relation between the entanglement and phase correlation of twoqubit states. The phase correlation is investigated by means * Address: Graduate School of Humanities and Sciences, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan. Tel./fax: +81 3 5978 5326. E-mail address: [email protected]
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.058
of the Pegg–Barnett phase opetator defined on a twodimensional Hilbert space [6–8]. Furthermore we investigate the decoherence of the phase correlation and entanglement of two-qubit states. Yu and Eberly have shown that the decay of the entanglement is quite different from that of the quantum coherence or the off-diagonal elements of a single qubit density matrix [9–12]. They have found that the entanglement undergoes the finite-time disentanglement, called the entanglement sudden death. It has been investigated in detail for various system [13–19]. Furthermore the entanglement sidden death has been experimentally observed [20,21]. In this paper, we will find that the relaxation processes of the entanglement and phase correlation become quite different for two-qubit states in which the entanglement sudden death occurs, though they are equal for two-qubit states in which the finite-time disentanglement does not take place. The decoherence is assumed to be caused by classical noises subject to the stationary correlated Gauss-Markov process which includes the collective or global dephasing [10,22–24]. We consider the two-qubit state, called the X-state [11], which includes the Bell state [1], the Werner state [28] and the maximally entangled mixed state [3–5]. In the computa^z j1i ¼ j0i and tional base fj00i; j01i; j10i; j11ig, where r z z ^ j1i ¼ j1i with r ^ being the Pauli matrix, the density r matrix of the X-state is given by
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M. Ban / Optics Communications 281 (2008) 3943–3946
0 ^AB q
1
a
0
0
x
B0 B ¼B @0
b y
y c
0C C C: 0A
0
0
d
x
tion CðhA ; hB Þ with respect to the parameters hA and hB . Then we obtain ð1Þ
AB
The normalization and positivity, Tr^ q ¼ 1 and ^AB > 0, require that the parameters a; b; c; d are non-negq ative and a þ b þ c þ d ¼ 1 and the off-diagonal elements x and y satisfy pffiffiffiffiffiffi pffiffiffiffiffi jxj 6 ad ; jyj 6 bc: ð2Þ The entanglement of a two-qubit state can be measured by the concurrence C [29,30]. The entanglement of pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 formation [31] is given by E ¼ H ð2 ð1 þ 1 C ÞÞ with H ðxÞ ¼ x log x ð1 xÞ logð1 xÞ. The concurrence C ent ^AB is calculated to be [11] of the X-state q h pffiffiffiffiffi pffiffiffiffiffiffii C ent ¼ 2 max 0; jxj bc; jyj ad : ð3Þ pffiffiffiffiffi pffiffiffiffiffiffi Note that the inequalities jxj > bc and jyj > ad cannot be simultaneously fulfilled due to the condition given by Eq. (2). We apply the Pegg–Barnett phase operator method [6–8] to investigate the phase correlation of a two-qubit state. The Pegg–Barnett phase operator was originally introduced to define a quantum mechanical phase operator canonically conjugate to a number operator ^ ay ^ a of a hary monic oscillator, where ^ a and ^ a are bosonic annihilation and creation operators. Hence it may be suitable for describing a quantum coherence. Furthermore, as shown in Ref. [25], a finite dimensional system can be investigated by means of the Pegg–Barnett phase operator. Hence we use the Pegg–Barnett phase operator to investigate the phase correlation of two qubits. When we set a 2p-phase window h 6 / 6 h þ 2p with h being a real constant, the Pegg–Barnett phase operator defined on a two-dimensional Hilbert space is given by ^
ei/ ¼ j0ih1j þ e2ih j1ih0j;
ð4Þ
the eigenstates of which are given by j/0 i ¼ ðj0iþ pffiffiffi eih j1iÞ= 2 with eigenvalue eih and j/1 i ¼ ðj0i eih j pffiffiffi 1iÞ= 2 with eigenvalue eiðhþpÞ . We can express the Pegg– ^x Barnett phase operator in terms of the Pauli matrices r ^ y ið/hÞ x y ^ as e ^ cos h r ^ sin h. The two-dimensional and r ¼r Pegg–Barnett phase operator with h ¼ 0, which is equal to ^x , can be used to study complementarity the Pauli matrix r and visibility of a interferometer [26,27]. When we set the phase windows hA 6 / 6 h þ 2p and hB 6 / 6 h þ 2p for the two qubits, we obtain the correlation function of the ^AB , X-state q D E ^ ^ ^ ^ CðhA ; hB Þ ¼ ½eið/hA Þ heið/hA Þ i ½eið/hB Þ heið/hB Þ i ¼ 2Re½xe
iðhA þhB Þ
þ 2Re½ye
iðhA hB Þ
; ð5Þ
AB
^ . We define the phase correlation where h i ¼ Tr½ q ^AB by maximizing the correlation funcC ph of the X-state q
C ph ¼ 2ðjxj þ jyjÞ;
ð6Þ
where the maximum is attained at hA ¼ ðhx þ hy Þ=2 and hB ¼ ðhx hy Þ=2 with hx ¼ argðxÞ and hy ¼ argðyÞ. The condition given by Eq. (2) ensures the inequality C ph 6 1. Comparing Eq. (6) with Eq. (3) we find the inequality C ent 6 C ph 6 1, which implies that the phase correlation between the two qubits is indispensable for the X-state ^AB to be entangled. q We next consider the decoherence of the phase correlation and entanglement that is caused by classical noises subject to stochastic processes [10,22–24]. We suppose that ^AB are placed in an external the two qubits in the X-state q environment, the effect of which is described by stochastic processes. For example, when the system is two spin-1/2 particles, fluctuating magnetic fields cause this kind of decoherence. Furthermore, when a distance between the two qubits is very short, the fluctuating fields may not be independent. Then we assume that the time-evolution of the two-qubit system prepared is determined by the stochastic Hamiltonian, 1 b ðtÞ ¼ 1 hx1 ðtÞð^ ^z Þ; H rz ^1Þ þ hx2 ðtÞð^1 r 2 2
ð7Þ
where x1 ðtÞ and x2 ðtÞ are random variables. This is called the Kubo-Anderson model [32,33]. We further suppose that the random variables are subject to the stationary Gauss-Markov process with zero mean values hxk ðtÞis ¼ 0ðk ¼ 1; 2Þ. Here h is stands for the average over the stochastic process. Then the Doob theorem provides us the correlation function [34,35], X hxi ðtÞxk ð0Þis ¼ ðeMt Þij Gjk ; ð8Þ j¼1;2
where Gjk ¼ Gkj ¼ hxj ð0Þxk ð0Þis with G11 P jG12 j and G22 P jG12 j. In this equation, M is a non-negative 2 2 matrix. In the following, we assume that M is diagonal such that 1=s1 0 M¼ : ð9Þ 0 1=s2 When G12 ¼ 0, there is no correlation between the random variables and thus each qubit suffers from the independent classical noise. On the other hand, when the equalities G11 ¼ G22 ¼ G12 and s1 ¼ s2 hold, the Hamiltonian (7) is equivalent to b ðtÞ ¼ 1 hxðtÞð^ ^z Þ; H rz ^1 þ ^1 r 2
ð10Þ
which is called the collective or global dephasing of two qubits. The decoherence caused by the collective dephasing has been investigated in detail [10,22]. The time-evolution of the two-qubit system under the influence of the correlated dephasing can be obtained by ^l ^ solving the Heisenberg equations of motion for r 1,
M. Ban / Optics Communications 281 (2008) 3943–3946
^ ^l r ^m (l; m ¼ x; y; z) [24]. When the initial two^l and r 1r qubit state is given by Eq. (1) and the random variables x1 ðtÞ and x2 ðtÞ obeys the stationary Gauss-Markov process, we can obtain after some calculation, 0 1 a 0 0 xF ðtÞ B 0 b yF þ ðtÞ 0 C B C ^AB ðtÞ ¼ B ð11Þ q C; @ 0 y F þ ðtÞ c 0 A x F ðtÞ
0
0
d
where the functions F þ ðtÞ and F ðtÞ which characterize the qubit decoherence are given by F ðtÞ ¼ exp½/11 ðtÞ /22 ðtÞ /12 ðtÞ /21 ðtÞ; with /jk ðtÞ ¼
Gjk s2j
t t=sj 1þe : sj
ð12Þ
ð13Þ
When the random variables x1 ðtÞ and x2 ðtÞ has the positive or negative correlation (G12 > 0 or G12 < 0), the inequality F þ ðtÞ > F ðtÞ or F þ ðtÞ < F ðtÞ is satisfied. In the narrowing limit, we have F ðtÞ ¼ et=T , where the relaxation times T þ and T are given by T ¼
1 : ðG11 G12 Þs1 þ ðG22 G12 Þs2
ð14Þ
When there is any correlation between the random variables, the relaxation times T þ and T are different. In fact, we have T þ > T for G12 > 0 and T þ < T for G12 < 0.
3945
The phase correlation C ph ðtÞ and the concurrence C ent ðtÞ ^AB ðtÞ are calculated to be of the X-state q C ph ðtÞ ¼ 2ðjxjF ðtÞ þ jyjF ðtÞÞ; h pffiffiffiffiffi pffiffiffiffiffiffii C ent ðtÞ ¼ 2 max 0; jxjF ðtÞ bc; jyjF þ ðtÞ ad ;
ð15Þ ð16Þ
We now investigate the properties of the decoherence of the phase correlation C ph ðtÞ and the concurrence C ent ðtÞ of the ^AB ðtÞ. The finite-time disentanglement that C ent ðtÞ X-state q becomes zero at finite time is called the entanglement sudden death [9–11]. It is fond from Eq. (16) with Eq. (2) that the necessary and sufficient condition for the entanglement sudden death not to occur under the influence of the dephasing is given by ad ¼ 0 or bc ¼ 0. In this case, using Eqs. (15) and (16), we obtain C ph ðtÞ ¼ C ent ðtÞ ¼ 2jxjF ðtÞ or C ph ðtÞ ¼ C ent ðtÞ ¼ 2jyjF þ ðtÞ. Therefore when the entanglement sudden death does not occur, the entanglement ^AB ðtÞ are always equal and phase correlation of the X-state q under the influence of the stochastic dephasing. On the other hand, when the entanglement sudden death occurs, they are quite different (see below).pffiffiffiffiffi pffiffiffiffiffiffi Since either the inequality jxj > bc or jyj > ad must ^AB ðtÞ to be entangled, we be satisfied for the X-state q assume that the former is true. In this case, the entanglement pffiffiffiffiffi vanishes at the time tdis determined by F ðtdis Þ ¼ bc=jxj. Assuming the narrowing limit for the stochastic process,pffiffiffiffiffi we obtain the disentanglement time tdis ¼ T lnð bc=jxjÞ. Here we suppose that the random variables x1 ðtÞ and x2 ðtÞ have the positive correlation, that
^AB ðtÞ with the initial values a ¼ d ¼ x ¼ 1=2 b, Fig. 1. The time-evolutions of the phase correlation and entanglement, C ph ðtÞ and C ent ðtÞ, for the X-state q b ¼ c ¼ y, where we set s1 ¼ s2 s, G11 s2 ¼ G22 s2 ¼ 2:0 and G12 s2 ¼ 1:8.
^AB ðtÞ with the initial values a ¼ d ¼ x ¼ 0:4, b ¼ c ¼ y ¼ 0:2, Fig. 2. Decays of the phase correlation and entanglement, C ph ðtÞ and C ent ðtÞ, for the X-state q where we set s1 ¼ s2 s, G11 s2 ¼ G22 s2 ¼ 2:0 and G12 s2 ¼ 1:8.
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M. Ban / Optics Communications 281 (2008) 3943–3946
is, G12 > 0. Then the function F ðtÞ decays more rapidly than the function F þ ðtÞ. Note that the decay of the entanglement C ent ðtÞ is characterized by the function F ðtÞ. On the other hand, the decay of the phase correlation C ph ðtÞ with jxj > jyj is given by the function F ðtÞ in the short time region and by the function F þ ðtÞ in the long time region. Thus the time dependence of the phase correlation C ph ðtÞ is different from that of the entanglement C ent ðtÞ when the entanglement sudden death occurs. The time-evolutions of the phase correlation C ph ðtÞ and ^AB ðtÞ are plotted in the entanglement C ent ðtÞ of the X-state q Figs. 1 and 2. The figure clear shows the difference of their time-evolutions. In the narrowing limit, we have F ðtÞ ¼ et=T with T þ > T . In this case, when jxj > jyj, the phase correlation C ph ðtÞ decays with the relaxation time T þ in the short time region and with the relaxation time T in the long time region (see Fig. 2). On the other hand, the con^AB ðtÞ with jxj > jyj decay currence C ent ðtÞ of the X-state q t=T as e and becomes zero at the disentanglement time tdis . In summary, we have investigated the entanglement and phase correlation of the X-state given by Eq. (1), in which the phase correlation is always greater than the concurrence, namely, C ent 6 C ph . Furthermore we have investigated the decoherence of the phase correlation and entanglement that is caused by the classical noises subject to the correlated Gauss-Markov processes. When the entanglement sudden death does take place, the entanglement is always equal to the phase correlation under the influence of the dephasing. On the other hand, the timeevolution of the entanglement becomes quite different from that of the phase correlation when the entanglement sudden death occurs. References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
[2] S.L. Braunstein, A.K. Pati (Eds.), Quantum Information with Continuous Variables, Kluwer, Dordrecht, 2003. [3] S. Ishizaka, T. Hiroshima, Phys. Rev. A 62 (2000) 022310. [4] W.J. Munro, D.F. James, A.G. White, P.G. Kwiat, Phys. Rev. A 64 (2001) 030302. [5] T. Wei, K. Nemoto, P.M. Goldbart, P.G. Kwiat, W.J. Munro, F. Verstraete, Phys. Rev. A 67 (2003) 022110. [6] D.T. Pegg, S.M. Barnett, Europhys. Lett. 6 (1988) 483. [7] S.M. Barnett, D.T. Pegg, J. Mod. Opt. 36 (1989) 7. [8] D.T. Pegg, S.M. Barnett, Phys. Rev. A 39 (1989) 1665. [9] T. Yu, J.H. Eberly, Phys. Rev. Lett. 93 (2004) 140404. [10] T. Yu, J.H. Eberly, Opt. Commun. 264 (2006) 393. [11] T. Yu, J.H. Eberly, Quantum Inf. Comput. 7 (2007) 459. [12] T. Yu, J.H. Eberly, LANL quant-ph/0707.3215, 2007. [13] M. Yo¨nac, T. Yu, J.H. Eberly, J. Phys. B 39 (2006) S621. [14] K. Roszak, P. Machnikowski, Phys. Rev. A 73 (2006) 022313. [15] Z. Ficek, R. Tanas, Phys. Rev. A 74 (2006) 024304. [16] R. Liu, C. Chen, Phys. Rev. A 74 (2006) 024102. [17] M. Yo¨nac, T. Yu, J.H. Eberly, J. Phys. B 40 (2007) S45. [18] X. Su, A. Wang, X. Ma, L. Qiu, J. Phys. A 40 (2007) 11385. [19] A. Al-Qasimi, D. James, Phys. Rev. A 77 (2008) 012117. [20] M.P. Almeida, F. de Melo, M. Hor-Weyll, A. Salles, S.P. Walborn, P.H. Souto Ribeiro, L. Davidovich, Science 316 (2007) 579. [21] J. Laurat, K.S. Coi, H. Deng, C.W. Chou, H.J. Kimble, Phys. Rev. Lett. 99 (2007) 180504. [22] T. Yu, J.H. Eberly, Phys. Rev. B 68 (2003) 165322. [23] M. Ban, S. Kitajima, F. Shibata, Phys. Lett. A 349 (2006) 415. [24] M. Ban, F. Shibata, Phys. Lett. A 354 (2006) 35. [25] V. Buzˇek, A.D. Wilson-Gordon, P.L. Knight, W.K. Lai, Phys. Rev. A 45 (1992) 8079. [26] G. Bjo¨rk, J. So¨derholm, A. Trifonov, T. Tsegaye, A. Karlsson, Phys. Rev. A 60 (1999) 1874. [27] M. Jakob, J.A. Bergou, LANL quant-ph/0302075, 2003. [28] R. Werner, Phys. Rev. A 40 (1989) 4277. [29] S. Hill, W.K. Wootters, Phys. Rev. Lett. 78 (1997) 5022. [30] W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245. [31] C.H. Bennett, P. DiVincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. [32] R. Kubo, J. Phys. Soc. Jpn. 9 (1954) 935. [33] P. Anderson, J. Phys. Soc. Jpn. 9 (1954) 316. [34] R. Kubo, in: D. ter Haar (Ed.), In Fluctuation, Relaxation, and Resonance in Magnetic Systems, Oliver and Boyd, Edinburgh, 1962, p. 23. [35] R. Kubo, Adv. Chem. Phys. 15 (1969) 101.
Optics Communications 281 (2008) 3943–3946 www.elsevier.com/locate/optcom
Entanglement, phase correlation and dephasing of two-qubit states Masashi Ban * Graduate School of Humanities and Sciences, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan CREST, Japan Science and Technology Agency, 1-1-9 Yaesu, Chuo-ku, Tokyo 103-0028, Japan Received 15 January 2008; received in revised form 21 February 2008; accepted 25 March 2008
Abstract The relation between entanglement and phase correlation of two-qubit states is studied. Decoherence of entanglement and phase correlation caused by correlated classical noises is also investigated. It is found that the decay of the phase correlation is quite different from that of the entanglement when the finite-time disentanglement occurs. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Entanglement; Phase correlation; Stochastic dephasing; Qubit
Quantum information processing has recently attracted very much attention in quantum physics and information science [1,2]. It provides the novel information technology such as quantum cryptography, quantum communication and quantum computation as well as new insights on the principles of quantum mechanics. One of the most important resources is undoubtedly entanglement of two-qubit states [1] which include the Bell state and the Werner state. Two-qubit states can be characterized by the several parameters such as purity (or mixedness), entanglement and correlation of two single qubit observables. The relation between the entanglement and mixedness of two-qubit states has been investigated in detail [3–5]. The maximally entangled states, the amount of entanglement of which cannot be increased by any unitary operation, have been found in these works. A phase plays an important role in quantum coherence. The phase correlation, however, has not been discussed. Therefore, in this paper, we consider the relation between the entanglement and phase correlation of twoqubit states. The phase correlation is investigated by means * Address: Graduate School of Humanities and Sciences, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan. Tel./fax: +81 3 5978 5326. E-mail address: [email protected]
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.058
of the Pegg–Barnett phase opetator defined on a twodimensional Hilbert space [6–8]. Furthermore we investigate the decoherence of the phase correlation and entanglement of two-qubit states. Yu and Eberly have shown that the decay of the entanglement is quite different from that of the quantum coherence or the off-diagonal elements of a single qubit density matrix [9–12]. They have found that the entanglement undergoes the finite-time disentanglement, called the entanglement sudden death. It has been investigated in detail for various system [13–19]. Furthermore the entanglement sidden death has been experimentally observed [20,21]. In this paper, we will find that the relaxation processes of the entanglement and phase correlation become quite different for two-qubit states in which the entanglement sudden death occurs, though they are equal for two-qubit states in which the finite-time disentanglement does not take place. The decoherence is assumed to be caused by classical noises subject to the stationary correlated Gauss-Markov process which includes the collective or global dephasing [10,22–24]. We consider the two-qubit state, called the X-state [11], which includes the Bell state [1], the Werner state [28] and the maximally entangled mixed state [3–5]. In the computa^z j1i ¼ j0i and tional base fj00i; j01i; j10i; j11ig, where r z z ^ j1i ¼ j1i with r ^ being the Pauli matrix, the density r matrix of the X-state is given by
3944
M. Ban / Optics Communications 281 (2008) 3943–3946
0 ^AB q
1
a
0
0
x
B0 B ¼B @0
b y
y c
0C C C: 0A
0
0
d
x
tion CðhA ; hB Þ with respect to the parameters hA and hB . Then we obtain ð1Þ
AB
The normalization and positivity, Tr^ q ¼ 1 and ^AB > 0, require that the parameters a; b; c; d are non-negq ative and a þ b þ c þ d ¼ 1 and the off-diagonal elements x and y satisfy pffiffiffiffiffiffi pffiffiffiffiffi jxj 6 ad ; jyj 6 bc: ð2Þ The entanglement of a two-qubit state can be measured by the concurrence C [29,30]. The entanglement of pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 formation [31] is given by E ¼ H ð2 ð1 þ 1 C ÞÞ with H ðxÞ ¼ x log x ð1 xÞ logð1 xÞ. The concurrence C ent ^AB is calculated to be [11] of the X-state q h pffiffiffiffiffi pffiffiffiffiffiffii C ent ¼ 2 max 0; jxj bc; jyj ad : ð3Þ pffiffiffiffiffi pffiffiffiffiffiffi Note that the inequalities jxj > bc and jyj > ad cannot be simultaneously fulfilled due to the condition given by Eq. (2). We apply the Pegg–Barnett phase operator method [6–8] to investigate the phase correlation of a two-qubit state. The Pegg–Barnett phase operator was originally introduced to define a quantum mechanical phase operator canonically conjugate to a number operator ^ ay ^ a of a hary monic oscillator, where ^ a and ^ a are bosonic annihilation and creation operators. Hence it may be suitable for describing a quantum coherence. Furthermore, as shown in Ref. [25], a finite dimensional system can be investigated by means of the Pegg–Barnett phase operator. Hence we use the Pegg–Barnett phase operator to investigate the phase correlation of two qubits. When we set a 2p-phase window h 6 / 6 h þ 2p with h being a real constant, the Pegg–Barnett phase operator defined on a two-dimensional Hilbert space is given by ^
ei/ ¼ j0ih1j þ e2ih j1ih0j;
ð4Þ
the eigenstates of which are given by j/0 i ¼ ðj0iþ pffiffiffi eih j1iÞ= 2 with eigenvalue eih and j/1 i ¼ ðj0i eih j pffiffiffi 1iÞ= 2 with eigenvalue eiðhþpÞ . We can express the Pegg– ^x Barnett phase operator in terms of the Pauli matrices r ^ y ið/hÞ x y ^ as e ^ cos h r ^ sin h. The two-dimensional and r ¼r Pegg–Barnett phase operator with h ¼ 0, which is equal to ^x , can be used to study complementarity the Pauli matrix r and visibility of a interferometer [26,27]. When we set the phase windows hA 6 / 6 h þ 2p and hB 6 / 6 h þ 2p for the two qubits, we obtain the correlation function of the ^AB , X-state q D E ^ ^ ^ ^ CðhA ; hB Þ ¼ ½eið/hA Þ heið/hA Þ i ½eið/hB Þ heið/hB Þ i ¼ 2Re½xe
iðhA þhB Þ
þ 2Re½ye
iðhA hB Þ
; ð5Þ
AB
^ . We define the phase correlation where h i ¼ Tr½ q ^AB by maximizing the correlation funcC ph of the X-state q
C ph ¼ 2ðjxj þ jyjÞ;
ð6Þ
where the maximum is attained at hA ¼ ðhx þ hy Þ=2 and hB ¼ ðhx hy Þ=2 with hx ¼ argðxÞ and hy ¼ argðyÞ. The condition given by Eq. (2) ensures the inequality C ph 6 1. Comparing Eq. (6) with Eq. (3) we find the inequality C ent 6 C ph 6 1, which implies that the phase correlation between the two qubits is indispensable for the X-state ^AB to be entangled. q We next consider the decoherence of the phase correlation and entanglement that is caused by classical noises subject to stochastic processes [10,22–24]. We suppose that ^AB are placed in an external the two qubits in the X-state q environment, the effect of which is described by stochastic processes. For example, when the system is two spin-1/2 particles, fluctuating magnetic fields cause this kind of decoherence. Furthermore, when a distance between the two qubits is very short, the fluctuating fields may not be independent. Then we assume that the time-evolution of the two-qubit system prepared is determined by the stochastic Hamiltonian, 1 b ðtÞ ¼ 1 hx1 ðtÞð^ ^z Þ; H rz ^1Þ þ hx2 ðtÞð^1 r 2 2
ð7Þ
where x1 ðtÞ and x2 ðtÞ are random variables. This is called the Kubo-Anderson model [32,33]. We further suppose that the random variables are subject to the stationary Gauss-Markov process with zero mean values hxk ðtÞis ¼ 0ðk ¼ 1; 2Þ. Here h is stands for the average over the stochastic process. Then the Doob theorem provides us the correlation function [34,35], X hxi ðtÞxk ð0Þis ¼ ðeMt Þij Gjk ; ð8Þ j¼1;2
where Gjk ¼ Gkj ¼ hxj ð0Þxk ð0Þis with G11 P jG12 j and G22 P jG12 j. In this equation, M is a non-negative 2 2 matrix. In the following, we assume that M is diagonal such that 1=s1 0 M¼ : ð9Þ 0 1=s2 When G12 ¼ 0, there is no correlation between the random variables and thus each qubit suffers from the independent classical noise. On the other hand, when the equalities G11 ¼ G22 ¼ G12 and s1 ¼ s2 hold, the Hamiltonian (7) is equivalent to b ðtÞ ¼ 1 hxðtÞð^ ^z Þ; H rz ^1 þ ^1 r 2
ð10Þ
which is called the collective or global dephasing of two qubits. The decoherence caused by the collective dephasing has been investigated in detail [10,22]. The time-evolution of the two-qubit system under the influence of the correlated dephasing can be obtained by ^l ^ solving the Heisenberg equations of motion for r 1,
M. Ban / Optics Communications 281 (2008) 3943–3946
^ ^l r ^m (l; m ¼ x; y; z) [24]. When the initial two^l and r 1r qubit state is given by Eq. (1) and the random variables x1 ðtÞ and x2 ðtÞ obeys the stationary Gauss-Markov process, we can obtain after some calculation, 0 1 a 0 0 xF ðtÞ B 0 b yF þ ðtÞ 0 C B C ^AB ðtÞ ¼ B ð11Þ q C; @ 0 y F þ ðtÞ c 0 A x F ðtÞ
0
0
d
where the functions F þ ðtÞ and F ðtÞ which characterize the qubit decoherence are given by F ðtÞ ¼ exp½/11 ðtÞ /22 ðtÞ /12 ðtÞ /21 ðtÞ; with /jk ðtÞ ¼
Gjk s2j
t t=sj 1þe : sj
ð12Þ
ð13Þ
When the random variables x1 ðtÞ and x2 ðtÞ has the positive or negative correlation (G12 > 0 or G12 < 0), the inequality F þ ðtÞ > F ðtÞ or F þ ðtÞ < F ðtÞ is satisfied. In the narrowing limit, we have F ðtÞ ¼ et=T , where the relaxation times T þ and T are given by T ¼
1 : ðG11 G12 Þs1 þ ðG22 G12 Þs2
ð14Þ
When there is any correlation between the random variables, the relaxation times T þ and T are different. In fact, we have T þ > T for G12 > 0 and T þ < T for G12 < 0.
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The phase correlation C ph ðtÞ and the concurrence C ent ðtÞ ^AB ðtÞ are calculated to be of the X-state q C ph ðtÞ ¼ 2ðjxjF ðtÞ þ jyjF ðtÞÞ; h pffiffiffiffiffi pffiffiffiffiffiffii C ent ðtÞ ¼ 2 max 0; jxjF ðtÞ bc; jyjF þ ðtÞ ad ;
ð15Þ ð16Þ
We now investigate the properties of the decoherence of the phase correlation C ph ðtÞ and the concurrence C ent ðtÞ of the ^AB ðtÞ. The finite-time disentanglement that C ent ðtÞ X-state q becomes zero at finite time is called the entanglement sudden death [9–11]. It is fond from Eq. (16) with Eq. (2) that the necessary and sufficient condition for the entanglement sudden death not to occur under the influence of the dephasing is given by ad ¼ 0 or bc ¼ 0. In this case, using Eqs. (15) and (16), we obtain C ph ðtÞ ¼ C ent ðtÞ ¼ 2jxjF ðtÞ or C ph ðtÞ ¼ C ent ðtÞ ¼ 2jyjF þ ðtÞ. Therefore when the entanglement sudden death does not occur, the entanglement ^AB ðtÞ are always equal and phase correlation of the X-state q under the influence of the stochastic dephasing. On the other hand, when the entanglement sudden death occurs, they are quite different (see below).pffiffiffiffiffi pffiffiffiffiffiffi Since either the inequality jxj > bc or jyj > ad must ^AB ðtÞ to be entangled, we be satisfied for the X-state q assume that the former is true. In this case, the entanglement pffiffiffiffiffi vanishes at the time tdis determined by F ðtdis Þ ¼ bc=jxj. Assuming the narrowing limit for the stochastic process,pffiffiffiffiffi we obtain the disentanglement time tdis ¼ T lnð bc=jxjÞ. Here we suppose that the random variables x1 ðtÞ and x2 ðtÞ have the positive correlation, that
^AB ðtÞ with the initial values a ¼ d ¼ x ¼ 1=2 b, Fig. 1. The time-evolutions of the phase correlation and entanglement, C ph ðtÞ and C ent ðtÞ, for the X-state q b ¼ c ¼ y, where we set s1 ¼ s2 s, G11 s2 ¼ G22 s2 ¼ 2:0 and G12 s2 ¼ 1:8.
^AB ðtÞ with the initial values a ¼ d ¼ x ¼ 0:4, b ¼ c ¼ y ¼ 0:2, Fig. 2. Decays of the phase correlation and entanglement, C ph ðtÞ and C ent ðtÞ, for the X-state q where we set s1 ¼ s2 s, G11 s2 ¼ G22 s2 ¼ 2:0 and G12 s2 ¼ 1:8.
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M. Ban / Optics Communications 281 (2008) 3943–3946
is, G12 > 0. Then the function F ðtÞ decays more rapidly than the function F þ ðtÞ. Note that the decay of the entanglement C ent ðtÞ is characterized by the function F ðtÞ. On the other hand, the decay of the phase correlation C ph ðtÞ with jxj > jyj is given by the function F ðtÞ in the short time region and by the function F þ ðtÞ in the long time region. Thus the time dependence of the phase correlation C ph ðtÞ is different from that of the entanglement C ent ðtÞ when the entanglement sudden death occurs. The time-evolutions of the phase correlation C ph ðtÞ and ^AB ðtÞ are plotted in the entanglement C ent ðtÞ of the X-state q Figs. 1 and 2. The figure clear shows the difference of their time-evolutions. In the narrowing limit, we have F ðtÞ ¼ et=T with T þ > T . In this case, when jxj > jyj, the phase correlation C ph ðtÞ decays with the relaxation time T þ in the short time region and with the relaxation time T in the long time region (see Fig. 2). On the other hand, the con^AB ðtÞ with jxj > jyj decay currence C ent ðtÞ of the X-state q t=T as e and becomes zero at the disentanglement time tdis . In summary, we have investigated the entanglement and phase correlation of the X-state given by Eq. (1), in which the phase correlation is always greater than the concurrence, namely, C ent 6 C ph . Furthermore we have investigated the decoherence of the phase correlation and entanglement that is caused by the classical noises subject to the correlated Gauss-Markov processes. When the entanglement sudden death does take place, the entanglement is always equal to the phase correlation under the influence of the dephasing. On the other hand, the timeevolution of the entanglement becomes quite different from that of the phase correlation when the entanglement sudden death occurs. References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
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