- Email: [email protected]
Pairing entangled states
Physica A 275 (2000) 475–482
www.elsevier.com/locate/physa
Pairing entangled states Zheng Wei Zhou ∗ , Guang Can Guo Department of Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei 230026, People’s Republic of China Received 6 April 1999
Abstract On the basis of coherence preserving states a new concept called pairing entangled states is proposed. Pairing entangled states can e ctively avoid the decoherence in uence from the environment and realize the long time storage of entanglement. Some of their applications in quantum information are also discussed in this paper. We nd that the pairing entangled states c 2000 Elsevier Science B.V. All rights rehave more favorable features than Bell states. served. PACS: 03.67.-a; 03.65.Bz Keywords: Entangled state; Decoherence; Teleportation
1. Introduction Of late, quantum information science has been thriving. Great progress has been made in quantum computation [1–3], quantum cryptography [4 – 6] and quantum teleportation [7], etc. An apparent, good feature of quantum information, compared with classical information, lies in its coherence. The interaction between a quantum system and the environment makes the system transform its state from a pure state to a mixed state, which is called decoherence. As a major menace to quantum information, decoherence brings about quantum parallelism loss such that quantum computation loses ecacy [8]. To overcome this fragility of quantum information, some strategies, such as quantum error-correcting codes (QECCs) [9 –15] and quantum error-avoiding codes (QEACs) [16,17], have been proposed. QECCs are quantum counterparts of classical error-correcting codes. Since the rst scheme of QECCs, proposed by Shor [9], to date, only a general theory for quantum error correction has been presented [11,12]. ∗
Corresponding author. E-mail addresses: [email protected] (Z.W. Zhou), [email protected] (G.C. Guo)
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 3 9 9 - 4
476
Z.W. Zhou, G.C. Guo / Physica A 275 (2000) 475–482
Now, perfect QECCs can correct for any error incurred by any one of the ve qubits [13]. Another strategy to conquer decoherence is QEACs which build on the frame of the cooperative decoherence. The basic idea is described as the following. In the system Hilbert space H , under some special circumstances, there exists a subspace C in which all the quantum states will perceive the same environmental noise at any time. Thus, after taking the trace over the environment, the unitary evolution of the states in the subspace C will escape from the environmental destruction. We call all the states in the subspace C the coherence preserving states [16 –18]. The codes based on the coherence preserving states are QEACs. QECCs are capable of detecting and correcting quantum errors but not avoiding them. On the contrary, the QEACs can only avoid errors. To date, only a general theory of QEACs has been developed. Furthermore, the existence of the quantum error avoiding and correcting codes in a special circumstance has been found which has the ability to avoid and at the same time correct quantum errors [19]. Entangled states have been widely applied to many aspects of quantum information. But they also cannot escape from decoherence due to the inevitable interaction with the environment. In this paper we present pairing entangled states, which depend on the idea of QEACs to overcome decoherence in the entangled states. Because of their preservative entanglement features, the pairing entangled states can act as stable quantum channels. In future, the realization of quantum computers and quantum nets may receive pro ts from them. Section 2 is a brief review of coherence preserving states based on cooperative decoherence. In Section 3, we present the pairing entangled states and discuss some of their applications in quantum information. In Section 4, we give a general method for preparation of the pairing entangled states. We summarize some of their merits in the last section.
2. Decoherence and coherence preserving states Suppose that the initial state of the environment is |e0 i. For a general spin 12 qubit system (|s0 i and |s1 i are eigenvectors in two-dimensional Hilbert space) the decoherence process must be |e0 i|s0 i → |ef1 i|s0 i + |ef2 i|s1 i ; |e0 i|s1 i → |ef3 i|s0 i + |ef4 i|s1 i ;
(1)
where |ef1 i, |ef2 i, |ef3 i and |ef4 i are the states of the environment (not generally orthogonal or normalized). Owing to the interaction between quantum system and the environment, their states entangle together. So, after we take the trace over the environment, the state of the system is transformed into a mixed one from the original pure one. For a general L identical qubits system coupled with the environment
Z.W. Zhou, G.C. Guo / Physica A 275 (2000) 475–482
477
described by a collection of noninteracting linear oscillators, the whole Hamiltonian has the following form (setting ˝ = 1) [16]: HL = !0
L X
lz +
l=1
L X[ !
(!a+ !l a!l ) +
l=1
L X X l=1
!
((1) lx + (2) ly + (3) lz )g!l
×(a+ !l + a!l ) ;
(2)
where a+ !l and a!l are the creation and annihilation operators of the oscillator coupling with the l qubit (! varies from 0 to ∞). l ( = x; y; z) is the Pauli operator of the l qubit. (i) (i = 1; 2; 3) characterizes the type of dissipation. The coupling constant g!l relies on ! and l. But, when the L identical qubits decohere cooperatively g!l is independent of l, i.e., g!l = g! . Let us review two strategies of coherence preserving states in the following part of this section. We can rewrite the above Hamiltonian when collective decoherence in the qubit system takes place: X X !a+ ((1) S x + (2) S y + (3) S z )g! (a+ (3) H L = !0 S z + ! a! + ! + a! ) ;
PL
!
!
l=1 l ,
where S = ( = x; y; z). Zanardi and Rasetti have put forward a strategy by directly searching for the com0 mon eigenstates of three operators S [17]. Since [S ; S ] 6= 0 ( 6= 0 ), under usual circumstances there are no common eigenstates of S x , S y and S z . However, when even qubits decohere collectively, the subspace in which the eigenvalue of the operator 2 2 2 S 2 = S x + S y + S z is zero, is the common eigenspace of the operators S x ; S y and S z . We indicate the subspace as C. When the state of L-qubit | s i ∈ C, S | s i = 0. In this case, the whole state of the system and the environment evolves in the following way: P + (4) | (t)i = e−iHL t (| s (0)i ⊗ | env (0)i) = | s (0)i ⊗ e−i ! !a! a! t | env (0)i : Obviously, the coherence of the states in C can be preserved perfectly. Another strategy has been proposed by our group in which some eigenspace C 0 of the operator S 0 = (1) S x + (2) S y + (3) S z will act as the space of the coherence preserving states [16]. But, since [S z ; S 0 ] 6= 0, the free Hamiltonian !0 S z will urge these states in the space C 0 out of their space. Therefore, we adopt a technique called “free-Hamiltonian elimination” to surmount this obstacle. We introduce an auxiliary classical electromagnetic eld Ha , which satis es [Ha + !0 S z ; S 0 ] = 0, to act on the L-qubit system. Under the Hamiltonian Ha + HL , when | s (0)i ∈ C 0 , at time t the state of the whole system evolves into +
| (t)i = e−i(Ha +HL )t (| s (0)i ⊗ | env (0)i) = | s (0)i ⊗ e−itf(m; !; a! ; a! ) | env (0)i ; (5) 0 where f is the function about m; !; a+ ! , and a! . m is the eigenvalue of the operator S 0 in the space C . In this case, no decoherence takes place.
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3. Pairing entangled states If we combine the ideas of cooperative decoherence and independent decoherence we can nd a collection of pairing entangled states. Under some special auxiliary condition they can preserve their entanglement features for a long time. The simplest pairing entangled states are composed of four particles. Without “free Hamiltonian elimination” the Hamiltonian of the system and the environment has the following form X X X H = !0 Sqz + !q a+ (q(1) Sqx + q(2) Sqy + q(3) Sqz )g!q (a+ !q a!q + !q + a!q ) ; q
!; q
!; q
(6) where q has A and B, which refer to two di erent spatial domains without any correla tions. Sq = q1 + q2 . We suppose that particle A1 (B1 ) pairs with particle A2 (B2 ) in the domain A(B). In the same domain, the particles decohere collectively. In contrast, those between di erent domains decohere independently. So we will use the second strategy in the above section to construct the coherence preserving states of entanglement — pairing entangled states. We apply two homogeneous classical large-blue-detuned optical elds EA and EB to the domain A and B to eliminate the e ect of free Hamiltonian. Under the adiabatic approximation the additional Hamiltonian is Hdrv = −
X 2|gq |2 |Eq |2 Sqz ; q ! − ! 0 opt q
(7)
q where !opt is the frequency of the optical eld Eq . By adjusting the intensity |Eq |2 of the optical eld, we can choose the coecient in Eq. (7) to satisfy
2|gq |2 |Eq |2 = !0 : q !opt − !0
(70 )
Thus, the e ect of the free Hamiltonian is o set by the additional driving elds. Suppose that |1iqi and | − 1iqi are the eigenstates of the operator Sqi = q(1) qix + y + q(3) qiz with the eigenvalues c1q and c2q , respectively. The four-particle pairing q(2) qi entangled state has the following form: 1 | p i = √ (|1iA1 | − 1iA2 | − 1iB1 |1iB2 + | − 1iA1 |1iA2 |1iB1 | − 1iB2 ) : 2
(8)
In fact, | p i is just a four-subsystem entangled state. But only two particles decohere collectively in the domain A(B). (In Eq. (8) the coecient √12 is not necessary. For convenience of discussion, in this paper the pairing entangled states refer to those with the maximum correlation.) | p i is a special case in the model of a quantum register proposed by Zanardi [20]. In the following part, we will demonstrate that it is free from decoherence in the framework of the master equation. Following the standard Born–Markov approximation, we obtain the master equation about the reduced density matrix of the four-qubit system. ˜ ˙ = L() = i[; H 0 ] + L() ;
(9)
Z.W. Zhou, G.C. Guo / Physica A 275 (2000) 475–482
479
where the superoperator L is called Liouvillian. The action of the dissipative part is " (−) 4 X ji (−) ˜ (Si Sj + Si Sj ) L() = S S − i j ij 2 i; j=1 # (+) +
(+) ij Si Sj
ji
−
(Si Sj + Si Sj ) ;
2
(10)
where 1, 2, 3 and 4 refer to A1 , A2 , B1 and B2 , respectively. For free Hamiltonian has been o set by the classical optical elds EA and EB , so 4 X
H0 =
(+) ((−) ij + ij )Si Sj ;
(11)
i; j=1 +) (−
+) (−
where the parameters ij and ij ronment [20]. In our model +) (− 12 +) (− 12
= =
+) (− 21 +) (− 21
= =
+) (− 11 +) (− 11
= =
+) (−
;
22 +) (− 22 ;
contain all relevant information about the envi+) (− 34 +) (− 34 =
=
+) (−
43 +) (− 43 =
=
+) (−
33 +) (− 33 =
=
+) (−
44 +) (− 44 :
;
(12) +) (−
+) (−
and ij But the other parameters ij | i = | − 1iA1 |1iA2 |1iB1 | − 1iB2 then
are zero. Let | i = |1iA1 | − 1iA2 | − 1iB1 |1iB2 ,
S1 | i = c1A | i;
S2 | i = c2A | i;
S3 | i = c2B | i; S4 | i = c1B | i ;
S1 | i = c2A | i;
S2 | i = c1A | i; S3 | i = c1B | i; S4 | i = c2B | i :
(13)
By calculating one derives L(| ih |) = L(| ih |) = L(| ih |) = L(| ih |) = 0, i.e., | p i is the xed point (stationary state) of the master equation (9). It is free from decoherence. As the four-particle state with the maximum entanglement, | p i can undertake the function of the Bell states. For instance, we can make use of | p i to teleport an unknown qubit. The teleportation process is fully analogous with the scheme proposed by Bennett et al. [7]. Suppose that Alice and Bob have established the correlation of a pairing entangled state. Alice will teleport an unknown qubit a|1i + b| − 1i to Bob. The whole state is shown in the following: 1 (a|1i + b| − 1i) √ (|1A1 − 1A2 − 1B1 1B2 i + | − 1A1 1A2 1B1 − 1B2 i) 2 1 = {| − i(a|1B1 − 1B2 i − b| − 1B1 1B2 i) 2 + | + i(a|1B1 − 1B2 i + b| − 1B1 1B2 i) + |− i(a| − 1B1 1B2 i − b|1B1 − 1B2 i) + |+ i(a| − 1B1 1B2 i + b|1B1 − 1B2 i)} ;
(14)
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Z.W. Zhou, G.C. Guo / Physica A 275 (2000) 475–482
where 1 | ± i = √ (|1i| − 1A1 1A2 i ± | − 1i|1A1 − 1A2 i) ; 2 1 |± i = √ (|1i|1A1 − 1A2 i ± | − 1i| − 1A1 1A2 i) : 2
(15)
We nd if Alice projects her three particles onto one of the four states in (15) Bob’s two particles will be collapsed into a pairing coherence preserving state [16]. (Theoretically, any of the maximally entangled states for any number of particles can be identi ed by a suitable quantum network [21]. Furthermore, Pan and Zeilinger have presented practically realizable procedures for identifying two of the N -particle GHZ states based on the concept of quantum erasure [22,23].) After receiving a classical information from Alice, Bob is able to apply a quantum CNOT (Controlled NOT) operation and a unitary transformation to decode it and recover the initial unknown quantum state. For example, Alice collapses her particles into |− i. She transfers this classical information to Bob. After receiving it Bob applies a joint operation 0 () = RB2 (−)RB1 (−)CB2B1 RB2 ()RB1 () on B1 and B2 , where Ri () refers to CB2B1 the rotation operation acting: on the i particle, which maps | ± 1i to calculating bases of the CNOT operation. CB2B1 is quantum CNOT operation where the subscript B2 refers to the control bit and the subscript B1 refers to target bit. We thus have C0
()
B2B1 (a|1B2 i − b| − 1B2 i)|1B1 i : a|1B2 − 1B1 i − b| − 1B2 1B1 i →→
(16)
Afterwards, Bob performs a unitary operation U (0 )=R(−0 )z R(0 ) on the particle B2 , where R(0 ) maps | ± 1B2 i to two eigenvectors of the operator z with eigenvalue ±1, respectively. Thereby, the particle B2 is transformed into the initial unknown quantum state. An apparent good reason in applying the pairing entangled state to teleportation lies in: after one part projects his (or her) particles onto one of the GHZ states, in the other part the quantum state of the particles will collapse to a pairing coherence preserving state. In the ideal case, the coherence preserving state can be stored in an intact way within an in nite time. Only if Bob receives the classical information from his colleague can he restore the unknown quantum state at any time. Similar discussions show that pairing entangled states can be also applied to other aspects of quantum information. For instance, by employing the scheme proposed by Zeilinger et al. [24] one can use two four-particle pairing entangled states to construct a six-particle pairing entangled state related to three di erent spatial domains. The scheme for entanglement swapping [25] can be also realized by pair ingentangled states. Indeed this can be seen in a direct manner. In some local spatial domain the quantum states of the pairing particles span a two-dimensional space. If we map |1; −1i to |↑i and | − 1; 1i to |↓i the four-particle pairing entangled states will be changed into the forms of the Bell states.Therefore four-particle pairing entangled states can be applied to any elds where Bell states can work.
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4. Preparation of the pairing entangled states In the above section, we do not touch upon the preparation of the pairing entangled states. Due to exerting a classical optical eld satisfying (70 ) on the pairing particles, it is inconvenient to transfer them directly. But this point does not constitute a main obstacle to construct pairing entangled states between Alice and Bob. A general method is that Alice and Bob rst set up a Bell state; afterwards, each part pairs his (or her) particle with an auxiliary particle to prepare a pairing entangled state. Without doubt, Bell states will decohere during the process of transmission and storage. But fortunately, we can overcome this diculty by means of entanglement puri cation [15,26,27]. The fundamental idea about entanglement puri cation can be described as follows. Suppose that two parties share n pairs of impure entangled particles. By some special local operations and classical communication both numbers can abandon some of their particles. If the density matrix of quantum states abandoned is separable, i.e. their entanglement degree is zero, and local operations cause no excessive expending of entanglement, the entanglement degree of the remainder will be increased. Repeat the puri cation operation again and again. Alice and Bob are able to acquire pure Bell states in an asymptotic way. Assume that Alice and Bob have derived a Bell state | + i = √12 (|↑A1 ↓B1 i + |↓A1 ↑B1 i) by some puri cation procedures. They can use particle A1 (B1 ) as the control bit and ancillary particle A2 (B2 ) as the target bit to perform a quantum CNOT operation. (where | A2(B2) i = |↑A2(B2) i, without loss of generality, we suppose that the calculating bases of the quantum controlled NOT gate are |↑i and |↓i) After rotating all the particles and exerting the classical optical elds to satisfy (7) and (70 ), Alice and Bob will obtain the four-particle pairing entangled state. The whole process is the following: 1 √ (|↑A1 ↓B1 i + |↓A1 ↑B1 i)|↑A2 i|↑B2 i 2 RA1 ()RA2 ()RB1 ()RB2 ()CA1A2 CB1B2 −−−−−−−−−−−−−−−−−−−−−−−−−−→ 1 √ (|1A1 − 1A2 − 1B1 1B2 i + | − 1A1 1A2 1B1 − 1B2 i) : (17) 2 Of course, the two pairing particles must be set close so that the distance d between them satis es d., where is the mean e ective wave length of the environment. We can put to use the “entanglement puri cation” procedure advanced by David Deutsch et al. [27]. This scheme can e ectively get rid of the in uences from the noise and the eavesdroppers. Once Bell states are achieved, one can prepare pairing entangled states and realize the long time storage of entanglement. 5. Conclusion In this paper we put forward the new concept called pairing entangled states based on the theory of QEACs. It is a distinguished merit for the pairing entangled states to
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preserve entanglement for a long time. As four-particle pairing entangled states they have all the features of Bell states. At the same time they still have some other merits. For example, Alice teleports an unknown qubit to Bob. After she makes a projection measurement for her particles, Bob’s particles will collapse to a coherence preserving state which can e ectively avoid decoherence from the environment. So Bob is allowed to recover the original quantum state at any time only by some fundamental quantum operations. This makes the elasticity of teleportation to improve greatly. In view of the above-mentioned reasons, we can say pairing entangled states are powerful quantum channels. Through them the stable and reliable transmission of the quantum information will be realized. Acknowledgements This project was supported by the National Nature Science Foundation of China. References [1] P.W. Shor, in: Proceeding of 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, New York, 1994, pp. 124 –134. [2] L.L. Chuang, R. La amme, W.H. Zurek, Science 270 (1995) 1633. [3] E. Knill, R. La amme, W.H. Zurek, Science 279 (1998) 342. [4] C.H. Bennett, G. Brassard, in: Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, IEEE, New York, 1984, p. 175. [5] A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661. [6] C.H. Bennett, Phys. Rev. Lett. 68 (1992) 3121. [7] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [8] W.G. Unruh, Phys. Rev. A 51 (1995) 992. [9] P.W. Shor, Phys. Rev. A 52 (1995) R2493. [10] A.M. Steane, Phys. Rev. Lett. 77 (1996) 793. [11] A.R. Calderbank, P.W. Shor, Phys. Rev. A 54 (1996) 1098. [12] A.M. Steane, Proc. Roy. Soc. London Ser. A 452 (1996) 2551. [13] R. La amme, C. Miquel, J.P. Paz, W.H. Zurek, Phys. Rev. Lett. 77 (1996) 198. [14] M.B. Plenio, V. Vedral, P.L. Knight, Phys. Rev. A 55 (1997) 67. [15] C.H. Bennett, D.P. DiVincezo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. [16] L.M. Duan, G.C. Guo, Phys. Rev. Lett. 79 (1997) 1953. [17] P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79 (1997) 3306. [18] L.M. Duan, G.C. Guo, Phys. Rev. A 57 (1998) 737. [19] L.M. Duan, G.C. Guo, LANL eprint quant-ph=9809057. [20] P. Zanardi, Phys. Rev. A 57 (1998) 3276. [21] D. Bruss, A. Ekert, S.F. Huelga, J.W. Pan, A. Zeilinger, Philos. Trans. Roy. Soc. London A, to be published. [22] J.W. Pan, A. Zeilinger, A. Ekert, unpublished. [23] J.W. Pan, A. Zeilinger, Phys. Rev. A 57 (1998) 2208. [24] A. Zeilinger, M.A. Horne, H. Weinfurter, M. Zukowski, Phys. Rev. Lett. 78 (1997) 3031. [25] M. Zukowski, A. Zeilinger, M.A. Horne, A.K. Ekert, Phys. Rev. Lett. 71 (1993) 4287. [26] C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W.K. Wootters, Phys. Rev. Lett. 76 (1996) 722. [27] D. Deutsch, A.K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, Phys. Rev. Lett. 77 (1996) 2818.
www.elsevier.com/locate/physa
Pairing entangled states Zheng Wei Zhou ∗ , Guang Can Guo Department of Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei 230026, People’s Republic of China Received 6 April 1999
Abstract On the basis of coherence preserving states a new concept called pairing entangled states is proposed. Pairing entangled states can e ctively avoid the decoherence in uence from the environment and realize the long time storage of entanglement. Some of their applications in quantum information are also discussed in this paper. We nd that the pairing entangled states c 2000 Elsevier Science B.V. All rights rehave more favorable features than Bell states. served. PACS: 03.67.-a; 03.65.Bz Keywords: Entangled state; Decoherence; Teleportation
1. Introduction Of late, quantum information science has been thriving. Great progress has been made in quantum computation [1–3], quantum cryptography [4 – 6] and quantum teleportation [7], etc. An apparent, good feature of quantum information, compared with classical information, lies in its coherence. The interaction between a quantum system and the environment makes the system transform its state from a pure state to a mixed state, which is called decoherence. As a major menace to quantum information, decoherence brings about quantum parallelism loss such that quantum computation loses ecacy [8]. To overcome this fragility of quantum information, some strategies, such as quantum error-correcting codes (QECCs) [9 –15] and quantum error-avoiding codes (QEACs) [16,17], have been proposed. QECCs are quantum counterparts of classical error-correcting codes. Since the rst scheme of QECCs, proposed by Shor [9], to date, only a general theory for quantum error correction has been presented [11,12]. ∗
Corresponding author. E-mail addresses: [email protected] (Z.W. Zhou), [email protected] (G.C. Guo)
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 3 9 9 - 4
476
Z.W. Zhou, G.C. Guo / Physica A 275 (2000) 475–482
Now, perfect QECCs can correct for any error incurred by any one of the ve qubits [13]. Another strategy to conquer decoherence is QEACs which build on the frame of the cooperative decoherence. The basic idea is described as the following. In the system Hilbert space H , under some special circumstances, there exists a subspace C in which all the quantum states will perceive the same environmental noise at any time. Thus, after taking the trace over the environment, the unitary evolution of the states in the subspace C will escape from the environmental destruction. We call all the states in the subspace C the coherence preserving states [16 –18]. The codes based on the coherence preserving states are QEACs. QECCs are capable of detecting and correcting quantum errors but not avoiding them. On the contrary, the QEACs can only avoid errors. To date, only a general theory of QEACs has been developed. Furthermore, the existence of the quantum error avoiding and correcting codes in a special circumstance has been found which has the ability to avoid and at the same time correct quantum errors [19]. Entangled states have been widely applied to many aspects of quantum information. But they also cannot escape from decoherence due to the inevitable interaction with the environment. In this paper we present pairing entangled states, which depend on the idea of QEACs to overcome decoherence in the entangled states. Because of their preservative entanglement features, the pairing entangled states can act as stable quantum channels. In future, the realization of quantum computers and quantum nets may receive pro ts from them. Section 2 is a brief review of coherence preserving states based on cooperative decoherence. In Section 3, we present the pairing entangled states and discuss some of their applications in quantum information. In Section 4, we give a general method for preparation of the pairing entangled states. We summarize some of their merits in the last section.
2. Decoherence and coherence preserving states Suppose that the initial state of the environment is |e0 i. For a general spin 12 qubit system (|s0 i and |s1 i are eigenvectors in two-dimensional Hilbert space) the decoherence process must be |e0 i|s0 i → |ef1 i|s0 i + |ef2 i|s1 i ; |e0 i|s1 i → |ef3 i|s0 i + |ef4 i|s1 i ;
(1)
where |ef1 i, |ef2 i, |ef3 i and |ef4 i are the states of the environment (not generally orthogonal or normalized). Owing to the interaction between quantum system and the environment, their states entangle together. So, after we take the trace over the environment, the state of the system is transformed into a mixed one from the original pure one. For a general L identical qubits system coupled with the environment
Z.W. Zhou, G.C. Guo / Physica A 275 (2000) 475–482
477
described by a collection of noninteracting linear oscillators, the whole Hamiltonian has the following form (setting ˝ = 1) [16]: HL = !0
L X
lz +
l=1
L X[ !
(!a+ !l a!l ) +
l=1
L X X l=1
!
((1) lx + (2) ly + (3) lz )g!l
×(a+ !l + a!l ) ;
(2)
where a+ !l and a!l are the creation and annihilation operators of the oscillator coupling with the l qubit (! varies from 0 to ∞). l ( = x; y; z) is the Pauli operator of the l qubit. (i) (i = 1; 2; 3) characterizes the type of dissipation. The coupling constant g!l relies on ! and l. But, when the L identical qubits decohere cooperatively g!l is independent of l, i.e., g!l = g! . Let us review two strategies of coherence preserving states in the following part of this section. We can rewrite the above Hamiltonian when collective decoherence in the qubit system takes place: X X !a+ ((1) S x + (2) S y + (3) S z )g! (a+ (3) H L = !0 S z + ! a! + ! + a! ) ;
PL
!
!
l=1 l ,
where S = ( = x; y; z). Zanardi and Rasetti have put forward a strategy by directly searching for the com0 mon eigenstates of three operators S [17]. Since [S ; S ] 6= 0 ( 6= 0 ), under usual circumstances there are no common eigenstates of S x , S y and S z . However, when even qubits decohere collectively, the subspace in which the eigenvalue of the operator 2 2 2 S 2 = S x + S y + S z is zero, is the common eigenspace of the operators S x ; S y and S z . We indicate the subspace as C. When the state of L-qubit | s i ∈ C, S | s i = 0. In this case, the whole state of the system and the environment evolves in the following way: P + (4) | (t)i = e−iHL t (| s (0)i ⊗ | env (0)i) = | s (0)i ⊗ e−i ! !a! a! t | env (0)i : Obviously, the coherence of the states in C can be preserved perfectly. Another strategy has been proposed by our group in which some eigenspace C 0 of the operator S 0 = (1) S x + (2) S y + (3) S z will act as the space of the coherence preserving states [16]. But, since [S z ; S 0 ] 6= 0, the free Hamiltonian !0 S z will urge these states in the space C 0 out of their space. Therefore, we adopt a technique called “free-Hamiltonian elimination” to surmount this obstacle. We introduce an auxiliary classical electromagnetic eld Ha , which satis es [Ha + !0 S z ; S 0 ] = 0, to act on the L-qubit system. Under the Hamiltonian Ha + HL , when | s (0)i ∈ C 0 , at time t the state of the whole system evolves into +
| (t)i = e−i(Ha +HL )t (| s (0)i ⊗ | env (0)i) = | s (0)i ⊗ e−itf(m; !; a! ; a! ) | env (0)i ; (5) 0 where f is the function about m; !; a+ ! , and a! . m is the eigenvalue of the operator S 0 in the space C . In this case, no decoherence takes place.
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3. Pairing entangled states If we combine the ideas of cooperative decoherence and independent decoherence we can nd a collection of pairing entangled states. Under some special auxiliary condition they can preserve their entanglement features for a long time. The simplest pairing entangled states are composed of four particles. Without “free Hamiltonian elimination” the Hamiltonian of the system and the environment has the following form X X X H = !0 Sqz + !q a+ (q(1) Sqx + q(2) Sqy + q(3) Sqz )g!q (a+ !q a!q + !q + a!q ) ; q
!; q
!; q
(6) where q has A and B, which refer to two di erent spatial domains without any correla tions. Sq = q1 + q2 . We suppose that particle A1 (B1 ) pairs with particle A2 (B2 ) in the domain A(B). In the same domain, the particles decohere collectively. In contrast, those between di erent domains decohere independently. So we will use the second strategy in the above section to construct the coherence preserving states of entanglement — pairing entangled states. We apply two homogeneous classical large-blue-detuned optical elds EA and EB to the domain A and B to eliminate the e ect of free Hamiltonian. Under the adiabatic approximation the additional Hamiltonian is Hdrv = −
X 2|gq |2 |Eq |2 Sqz ; q ! − ! 0 opt q
(7)
q where !opt is the frequency of the optical eld Eq . By adjusting the intensity |Eq |2 of the optical eld, we can choose the coecient in Eq. (7) to satisfy
2|gq |2 |Eq |2 = !0 : q !opt − !0
(70 )
Thus, the e ect of the free Hamiltonian is o set by the additional driving elds. Suppose that |1iqi and | − 1iqi are the eigenstates of the operator Sqi = q(1) qix + y + q(3) qiz with the eigenvalues c1q and c2q , respectively. The four-particle pairing q(2) qi entangled state has the following form: 1 | p i = √ (|1iA1 | − 1iA2 | − 1iB1 |1iB2 + | − 1iA1 |1iA2 |1iB1 | − 1iB2 ) : 2
(8)
In fact, | p i is just a four-subsystem entangled state. But only two particles decohere collectively in the domain A(B). (In Eq. (8) the coecient √12 is not necessary. For convenience of discussion, in this paper the pairing entangled states refer to those with the maximum correlation.) | p i is a special case in the model of a quantum register proposed by Zanardi [20]. In the following part, we will demonstrate that it is free from decoherence in the framework of the master equation. Following the standard Born–Markov approximation, we obtain the master equation about the reduced density matrix of the four-qubit system. ˜ ˙ = L() = i[; H 0 ] + L() ;
(9)
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where the superoperator L is called Liouvillian. The action of the dissipative part is " (−) 4 X ji (−) ˜ (Si Sj + Si Sj ) L() = S S − i j ij 2 i; j=1 # (+) +
(+) ij Si Sj
ji
−
(Si Sj + Si Sj ) ;
2
(10)
where 1, 2, 3 and 4 refer to A1 , A2 , B1 and B2 , respectively. For free Hamiltonian has been o set by the classical optical elds EA and EB , so 4 X
H0 =
(+) ((−) ij + ij )Si Sj ;
(11)
i; j=1 +) (−
+) (−
where the parameters ij and ij ronment [20]. In our model +) (− 12 +) (− 12
= =
+) (− 21 +) (− 21
= =
+) (− 11 +) (− 11
= =
+) (−
;
22 +) (− 22 ;
contain all relevant information about the envi+) (− 34 +) (− 34 =
=
+) (−
43 +) (− 43 =
=
+) (−
33 +) (− 33 =
=
+) (−
44 +) (− 44 :
;
(12) +) (−
+) (−
and ij But the other parameters ij | i = | − 1iA1 |1iA2 |1iB1 | − 1iB2 then
are zero. Let | i = |1iA1 | − 1iA2 | − 1iB1 |1iB2 ,
S1 | i = c1A | i;
S2 | i = c2A | i;
S3 | i = c2B | i; S4 | i = c1B | i ;
S1 | i = c2A | i;
S2 | i = c1A | i; S3 | i = c1B | i; S4 | i = c2B | i :
(13)
By calculating one derives L(| ih |) = L(| ih |) = L(| ih |) = L(| ih |) = 0, i.e., | p i is the xed point (stationary state) of the master equation (9). It is free from decoherence. As the four-particle state with the maximum entanglement, | p i can undertake the function of the Bell states. For instance, we can make use of | p i to teleport an unknown qubit. The teleportation process is fully analogous with the scheme proposed by Bennett et al. [7]. Suppose that Alice and Bob have established the correlation of a pairing entangled state. Alice will teleport an unknown qubit a|1i + b| − 1i to Bob. The whole state is shown in the following: 1 (a|1i + b| − 1i) √ (|1A1 − 1A2 − 1B1 1B2 i + | − 1A1 1A2 1B1 − 1B2 i) 2 1 = {| − i(a|1B1 − 1B2 i − b| − 1B1 1B2 i) 2 + | + i(a|1B1 − 1B2 i + b| − 1B1 1B2 i) + |− i(a| − 1B1 1B2 i − b|1B1 − 1B2 i) + |+ i(a| − 1B1 1B2 i + b|1B1 − 1B2 i)} ;
(14)
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where 1 | ± i = √ (|1i| − 1A1 1A2 i ± | − 1i|1A1 − 1A2 i) ; 2 1 |± i = √ (|1i|1A1 − 1A2 i ± | − 1i| − 1A1 1A2 i) : 2
(15)
We nd if Alice projects her three particles onto one of the four states in (15) Bob’s two particles will be collapsed into a pairing coherence preserving state [16]. (Theoretically, any of the maximally entangled states for any number of particles can be identi ed by a suitable quantum network [21]. Furthermore, Pan and Zeilinger have presented practically realizable procedures for identifying two of the N -particle GHZ states based on the concept of quantum erasure [22,23].) After receiving a classical information from Alice, Bob is able to apply a quantum CNOT (Controlled NOT) operation and a unitary transformation to decode it and recover the initial unknown quantum state. For example, Alice collapses her particles into |− i. She transfers this classical information to Bob. After receiving it Bob applies a joint operation 0 () = RB2 (−)RB1 (−)CB2B1 RB2 ()RB1 () on B1 and B2 , where Ri () refers to CB2B1 the rotation operation acting: on the i particle, which maps | ± 1i to calculating bases of the CNOT operation. CB2B1 is quantum CNOT operation where the subscript B2 refers to the control bit and the subscript B1 refers to target bit. We thus have C0
()
B2B1 (a|1B2 i − b| − 1B2 i)|1B1 i : a|1B2 − 1B1 i − b| − 1B2 1B1 i →→
(16)
Afterwards, Bob performs a unitary operation U (0 )=R(−0 )z R(0 ) on the particle B2 , where R(0 ) maps | ± 1B2 i to two eigenvectors of the operator z with eigenvalue ±1, respectively. Thereby, the particle B2 is transformed into the initial unknown quantum state. An apparent good reason in applying the pairing entangled state to teleportation lies in: after one part projects his (or her) particles onto one of the GHZ states, in the other part the quantum state of the particles will collapse to a pairing coherence preserving state. In the ideal case, the coherence preserving state can be stored in an intact way within an in nite time. Only if Bob receives the classical information from his colleague can he restore the unknown quantum state at any time. Similar discussions show that pairing entangled states can be also applied to other aspects of quantum information. For instance, by employing the scheme proposed by Zeilinger et al. [24] one can use two four-particle pairing entangled states to construct a six-particle pairing entangled state related to three di erent spatial domains. The scheme for entanglement swapping [25] can be also realized by pair ingentangled states. Indeed this can be seen in a direct manner. In some local spatial domain the quantum states of the pairing particles span a two-dimensional space. If we map |1; −1i to |↑i and | − 1; 1i to |↓i the four-particle pairing entangled states will be changed into the forms of the Bell states.Therefore four-particle pairing entangled states can be applied to any elds where Bell states can work.
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4. Preparation of the pairing entangled states In the above section, we do not touch upon the preparation of the pairing entangled states. Due to exerting a classical optical eld satisfying (70 ) on the pairing particles, it is inconvenient to transfer them directly. But this point does not constitute a main obstacle to construct pairing entangled states between Alice and Bob. A general method is that Alice and Bob rst set up a Bell state; afterwards, each part pairs his (or her) particle with an auxiliary particle to prepare a pairing entangled state. Without doubt, Bell states will decohere during the process of transmission and storage. But fortunately, we can overcome this diculty by means of entanglement puri cation [15,26,27]. The fundamental idea about entanglement puri cation can be described as follows. Suppose that two parties share n pairs of impure entangled particles. By some special local operations and classical communication both numbers can abandon some of their particles. If the density matrix of quantum states abandoned is separable, i.e. their entanglement degree is zero, and local operations cause no excessive expending of entanglement, the entanglement degree of the remainder will be increased. Repeat the puri cation operation again and again. Alice and Bob are able to acquire pure Bell states in an asymptotic way. Assume that Alice and Bob have derived a Bell state | + i = √12 (|↑A1 ↓B1 i + |↓A1 ↑B1 i) by some puri cation procedures. They can use particle A1 (B1 ) as the control bit and ancillary particle A2 (B2 ) as the target bit to perform a quantum CNOT operation. (where | A2(B2) i = |↑A2(B2) i, without loss of generality, we suppose that the calculating bases of the quantum controlled NOT gate are |↑i and |↓i) After rotating all the particles and exerting the classical optical elds to satisfy (7) and (70 ), Alice and Bob will obtain the four-particle pairing entangled state. The whole process is the following: 1 √ (|↑A1 ↓B1 i + |↓A1 ↑B1 i)|↑A2 i|↑B2 i 2 RA1 ()RA2 ()RB1 ()RB2 ()CA1A2 CB1B2 −−−−−−−−−−−−−−−−−−−−−−−−−−→ 1 √ (|1A1 − 1A2 − 1B1 1B2 i + | − 1A1 1A2 1B1 − 1B2 i) : (17) 2 Of course, the two pairing particles must be set close so that the distance d between them satis es d., where is the mean e ective wave length of the environment. We can put to use the “entanglement puri cation” procedure advanced by David Deutsch et al. [27]. This scheme can e ectively get rid of the in uences from the noise and the eavesdroppers. Once Bell states are achieved, one can prepare pairing entangled states and realize the long time storage of entanglement. 5. Conclusion In this paper we put forward the new concept called pairing entangled states based on the theory of QEACs. It is a distinguished merit for the pairing entangled states to
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preserve entanglement for a long time. As four-particle pairing entangled states they have all the features of Bell states. At the same time they still have some other merits. For example, Alice teleports an unknown qubit to Bob. After she makes a projection measurement for her particles, Bob’s particles will collapse to a coherence preserving state which can e ectively avoid decoherence from the environment. So Bob is allowed to recover the original quantum state at any time only by some fundamental quantum operations. This makes the elasticity of teleportation to improve greatly. In view of the above-mentioned reasons, we can say pairing entangled states are powerful quantum channels. Through them the stable and reliable transmission of the quantum information will be realized. Acknowledgements This project was supported by the National Nature Science Foundation of China. References [1] P.W. Shor, in: Proceeding of 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, New York, 1994, pp. 124 –134. [2] L.L. Chuang, R. La amme, W.H. Zurek, Science 270 (1995) 1633. [3] E. Knill, R. La amme, W.H. Zurek, Science 279 (1998) 342. [4] C.H. Bennett, G. Brassard, in: Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, IEEE, New York, 1984, p. 175. [5] A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661. [6] C.H. Bennett, Phys. Rev. Lett. 68 (1992) 3121. [7] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [8] W.G. Unruh, Phys. Rev. A 51 (1995) 992. [9] P.W. Shor, Phys. Rev. A 52 (1995) R2493. [10] A.M. Steane, Phys. Rev. Lett. 77 (1996) 793. [11] A.R. Calderbank, P.W. Shor, Phys. Rev. A 54 (1996) 1098. [12] A.M. Steane, Proc. Roy. Soc. London Ser. A 452 (1996) 2551. [13] R. La amme, C. Miquel, J.P. Paz, W.H. Zurek, Phys. Rev. Lett. 77 (1996) 198. [14] M.B. Plenio, V. Vedral, P.L. Knight, Phys. Rev. A 55 (1997) 67. [15] C.H. Bennett, D.P. DiVincezo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. [16] L.M. Duan, G.C. Guo, Phys. Rev. Lett. 79 (1997) 1953. [17] P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79 (1997) 3306. [18] L.M. Duan, G.C. Guo, Phys. Rev. A 57 (1998) 737. [19] L.M. Duan, G.C. Guo, LANL eprint quant-ph=9809057. [20] P. Zanardi, Phys. Rev. A 57 (1998) 3276. [21] D. Bruss, A. Ekert, S.F. Huelga, J.W. Pan, A. Zeilinger, Philos. Trans. Roy. Soc. London A, to be published. [22] J.W. Pan, A. Zeilinger, A. Ekert, unpublished. [23] J.W. Pan, A. Zeilinger, Phys. Rev. A 57 (1998) 2208. [24] A. Zeilinger, M.A. Horne, H. Weinfurter, M. Zukowski, Phys. Rev. Lett. 78 (1997) 3031. [25] M. Zukowski, A. Zeilinger, M.A. Horne, A.K. Ekert, Phys. Rev. Lett. 71 (1993) 4287. [26] C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W.K. Wootters, Phys. Rev. Lett. 76 (1996) 722. [27] D. Deutsch, A.K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, Phys. Rev. Lett. 77 (1996) 2818.