- Email: [email protected]
Direct measurement of the Concurrence of spin-entangled states in a cavity–quantum dot system
Physica B 495 (2016) 50–53
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Direct measurement of the Concurrence of spin-entangled states in a cavity–quantum dot system Ping Dong a,n, Jun Liu a, Li-Hua Zhang b, Zhuo-Liang Cao a a b
School of Electronic and Information Engineering, Hefei Normal University, Hefei 230061, China School of Physics and Electrical Engineering, Anqing Normal University, Anqing 246011, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 27 October 2015 Received in revised form 28 February 2016 Accepted 7 May 2016 Available online 9 May 2016
A scheme for implementing the direct measurement of Concurrence is given in a cavity–quantum dot system. The scenario not only can directly measure the Concurrence of two-spin pure entangled state, but also suitable for the case of mixed state. More importantly, all of the operations are of geometric nature, which depend on the cavity-state-free evolution and can be robust against random operation errors. Our scheme provided an alternative method for directly measuring the degree of entanglement in solid-state system. & 2016 Elsevier B.V. All rights reserved.
Keywords: Direct measurement Concurrence Cavity–quantum dot system
1. Introduction Entanglement states are important resource in quantum information processing (QIP), such as quantum communication, quantum computation, and quantum error correction [1–5]. However, the quantification and measurement of the degree of entanglement are still the main obstacles for the deep development of quantum information science. Thus many renewed efforts have recently been given in this direction both theoretically and experimentally. Concurrence is considered as the best one of the quantified methods for calculating the degree of entanglement including entanglement of formation (EOF), partial entropy, relative entropy, negativity, and distributed entanglement, since it can be used not only in specifically bipartite system, but also in multi-body system [6]. Based on the definition of Concurrence proposed by Hill and Wooters [7], one can know that the Concurrence of a bipartite pure state can be expressed as C (|ϕ〉) = |AB 〈ϕ|θ|ϕ〉AB |, where θ|ϕ〉AB = σy ⊗ σy |ϕ〉⁎AB , |ϕ〉 is the bipartite pure state, sy is the Pauli operation and “n” denotes the complex conjugate. And the Concurrence of a bipartite mixed state can be given by C (ρ) = max {0, λ1 − λ2 − λ3 − λ4 }, where ρ is the density of the bipartite mixed state, λis are the eigenvalues of the matrix ρ (σy ⊗ σy ) ρ⁎ (σy ⊗ σy ) ρ , in decreasing order. More importantly, the Concurrence can be used to define the EOF for the bipartite mixed state and also can be generalized to investigate the n
Corresponding author. E-mail addresses: [email protected] (P. Dong), [email protected] (J. Liu).
http://dx.doi.org/10.1016/j.physb.2016.05.005 0921-4526/& 2016 Elsevier B.V. All rights reserved.
distributed entanglement and residual entanglement. Thus it has attracted much attention in this field, recently. On the other hand, due to the importance of measuring the degree of entanglement, many direction measurement schemes based on single-qubit measurement have emerged. After the first direct measurement experimental scheme of the Concurrence of two-photon entangled state was proposed by Walborn et al. [8], Romero et al. and Yang et al. have given another two schemes for directly measuring the Concurrence of two-ion entangled state [9,10]. Then Zhang et al. designed two alternate theoretical direct measurement schemes of the Concurrence of two-photon entangled state by parity-check measurement [11] and homodyne measurement [12], respectively. The direct measurement schemes of the Concurrence for two-atom pure states and mixed states have been also appeared in cavity QED systems [13]. However, as we know, the direct measurement of the Concurrence of an arbitrary spin entangled state in a quantum dot system has not been considered by now. Moreover, quantum dot system has been generally accepted to be the best promising hardware for quantum computation. Therefore, it is significant for investigating the direct measurement of the Concurrence of spin entangled states in a quantum dot system. In this paper, we will design the schemes for implementing the direct measurement of the Concurrence of the pure and mixed two-spin entangled states in a cavity–quantum dot system and discuss the effect brought by the cavity decay on the measurement of the Concurrence. In the current scheme, the quantum dots are all doped such that each quantum dot has a single conduction band electron being in the ground state orbital and a full valence band, so the spin states of the electrons are stable. The capture
P. Dong et al. / Physica B 495 (2016) 50–53
problem of the quantum dots also can be avoided, since the quantum dots are all directly embedded in a microcavity. Our scheme will provide an alternative method for directly measuring the degree of entanglement in quantum dot system.
computation basis |±〉i = 2
He =
The relevant energy levels of the quantum dot can be treated as a V-type configuration, as shown in Fig. 1(a), where the spin-up and spin-down states of the conduction-band electron are encoded as the qubit states and the valence-band state is used as an auxiliary state. The spin-up and spin-down states can be obtained by using a uniform magnetic field. Two quantum dots embedded in a microcavity excited by three classical laser fields, as shown in Fig. 1(b), the Hamiltonian of the system can be expressed as (assuming = = 1)
H = H0 + Hint
(1)
2
H0 =
∑ (ω↑ σ↑↑i + ω↓ σ↓↓i + ωv σvvi ) + ωc a†a
(| ↑〉 ± | ↓〉 ) can be reduced as i
i
i=1
⎤ i i (a†e−iΔ t + aeiΔ t )(σ↑↓ + σ↓↑ ) ⎥ ⎦ 2
(4)
where Δ and Δ are the detunings between the spin-up state and the valence-band state of the ith quantum dot driven by laser gΩ ⎛ 1 1 ⎞ fields 2 and 1, respectively, and A = 2 2 ⎜ i + i i ⎟. If we consider Δ1 +Δ ⎠ ⎝ Δ1 the single quantum dot inside a microcavity and turn off the laser field 2, the cavity field can be eliminated, the net effect of the cavity filled is an additional ac Stack shift, inversely proportional to the detuning of optical frequencies Δ1, which can be safely neglected. The effective Hamiltonian can be reduced as [14]
He′ =
i 2
Ω1Ω3 (σ↓↑ + σ↑↓ ) Δ2
(5)
The process is equivalent to a simple single-channel Raman process with two laser fields of that in Ref. [15]. It is shown that twoqubit operations can be realized by the interaction of Eq. (4) and all of the single-qubit operations can be implemented by adjusting the laser fields in terms with Eq. (5).
3. Direct measurement of the Concurrence of two-spin entangled states
2
∑ {[(Ω1e−iω1t + Ω2 e−iω2t ) σ↑iv i=1
+ (Ω3 e−iω3 t + ga) σ↓i v ] + H . c . }
(3)
where Ωj (j = 1, 2, 3) is the Rabi frequencies of the jth classical field and g is the coupling constant between each quantum dot and the cavity mode. a† and a are the annihilation and creation i operations of the cavity mode, respectively, and σmn = |m〉〈n| ( m , n = ↑ , ↓ , v ) are the operations of the ith quantum dot. In the case of adiabatically eliminating the valence-band state of each quantum dot under the condition of Δ1 , Δ2 ⪢Ωj , g , Δ1 − Δ2 ⪢Δ, (Δ1 +Δ2 ) Ω1Ω2 (Δ1 +Δ2 ) Ω2 Ω3 (2Δ1 +Δ) Ω1g (2Δ1 +Δ) Ω3 g , , 2Δ (Δ +Δ) , 2Δ (Δ +Δ) 2Δ1Δ2 2Δ1Δ2 1 1 1 1
2
(2)
i=1
Hint =
1
⎡A
∑ ⎢⎣ i 1
2. The cavity–quantum dot interaction model
51
, and
2Ω1Ω3 ⪢Δ[14], Δ2
the
effective Hamiltonian of the cavity–quantum dot system under the
To directly measure the Concurrence of two-spin entangled pure states, two identical copies of which are necessary. We first prepare four quantum dots embedded in a microcavity being in | + 〉⊗4 , then we turn on two laser fields with ω1 and ω3 on quantum dot 1 and adjust the interaction time, turn on the three laser fields with ω1, ω2 and ω3 on quantum dot 1 and one laser field with ω3 on quantum dot 2, respectively, let Δ1 = Δ2 and adjust the interaction time appropriately, we can obtain an arbitrary twospin entangled pure state of quantum dots 1 and 2, i.e.,
|ϕ〉12 = a| − − 〉12 + b| − + 〉12 + c| + − 〉12 + d|++〉12
(6)
where a, b, c and d are all real, and |a|2 + |b|2 + |c|2 + |d|2 = 1. Repeat
conduction band
dot 1 dot 2
valence band Fig. 1. (a) The relevant energy levels of a single quantum dot. | ↑ 〉, | ↓ 〉 and |v〉 denote spin up and spin down states of conduction-band electron, and valence-band state, respectively, the energies corresponding to these are =ω↑ , =ω↓ and ℏωv. ω↑↓ = ω↑ − ω↓ . ωj (j = 1, 2, 3) and ωc are the frequencies of three classical fields and the microcavity field. Δ1, Δ2 and Δ are three detunings, and Δ = ω↓ − ω v − ωc − Δ1 = ω3 + Δ2 − ωc − Δ1. (b). The schematic setup of the quantum dots embedded in a microcavity.
52
P. Dong et al. / Physica B 495 (2016) 50–53
the above process on quantum dots 3 and 4 as that on quantum dots 1 and 2, two identical quantum states can be generated. That is to say that the state of the four quantum dots becomes
(7)
|ψ 〉1234 = |ϕ〉12 ⊗ |ϕ〉34
Next, we turn on two laser fields on quantum dots 3 and 4, and control the interaction time, let
| + 〉q ⟶| − 〉q , | − 〉q ⟶ − | + 〉q
(q = 3, 4)
(8)
and then operate a single-qubit rotation on quantum dot 4 by using a classical laser field, let
1 (| − 〉4 − i| + 〉4 ), 2 1 | − 〉4 ⟶ (| + 〉4 + | − 〉4 ) 2 | + 〉4 ⟶
(9)
the total state becomes
|ψ 〉′1234 =
C (ϕ) = 2|ad − bc| = 2 2Psuc If the above two-spin entangled pure state is exposed by environment, i.e., the leakage out of the cavity mode, the collective relaxation and the dephase of the transition of the quantum dots will accompany the interaction process, the state will reduce to a mixed state. We assume that the mixed one is
ρ12 = F |φ〉1221〈φ| + (1 − F )|ψ 〉1221〈ψ |
⊗ (a|++〉 − a| + − 〉 + b|++〉 + b| + − 〉 (10)
Then we turn on three laser fields with ω1, ω2 and ω3 on quantum dots 2 and 4, respectively, the quantum dots 2 and 4 will interact under the condition of Δ2 = Δ4 = Δ, controlling the interaction , the total state of the system will become time, let t = 2πΔ 2
(14)
where 0 < F < 1, |φ〉12 = α|++〉12 + β| − − 〉12, |ψ 〉12 = | + − 〉12, and |α|2 + |β|2 = 1. Assuming that the two pure states are both reduced to the above mixed states, i.e., ρ1234 = ρ12 ⊗ ρ34 . We repeat above operations as the case of the pure states, and then detect the spinstate of the four quantum dots, the success probability of obtaining | − − − − 〉 is
Psuc ′ =
1 (a| − − 〉 + b| − + 〉 + c| + − 〉 + d|++〉)12 2 +c| − − 〉 − c| − + 〉 − d| − − 〉 − d|++〉)34
measuring the success probability Psuc, i.e.,
F2 |αβ|2 2
The Concurrence of mixed state is 2F |αβ|, thus the Concurrence of that also can be directly obtained by measuring the success probability Psuc ′ , i.e.,
C′ (ρ) = 2 2Psuc ′ If |ψ 〉12 = | − + 〉12, the Concurrence also can be measured by the above process, the result is the same as C′(ρ). The method can be generalized to measure other mixed states in quantum dot system.
A
|ψ 〉″1234
1 = (a| − − 〉 + c| + − 〉)12 ⊗ (a|++〉 − a| + − 〉 2
4. Discussions and conclusions
+ b|++〉 + b| + − 〉 + c| − − 〉 − c| − + 〉 1 −d| − − 〉 − d|++〉)34 + (b| − + 〉 2 +d|++〉)12 ⊗ ( − a|++〉 − a| + − 〉 −b|++〉 + b| + − 〉 + c| − − 〉 + c| − + 〉 − d| − − 〉 + d|++〉)34
(11)
Finally, we turn on two laser fields on quantum dot 2 and quantum dot 4, respectively, let
1 (| − 〉m − | + 〉m ), 2 1 | − 〉m ⟶ (| + 〉m + | − 〉m ) (m = 2, 4) 2 | + 〉m ⟶
(12)
The total state of the system will be
|ψ 〉‴ 1234 =
1 [(b2 − a2)| − − ++〉 − (b2 + a2)| − +++〉 2 + (bd − ac )| + − ++〉 − (bd + ac )|++++〉 + 2ab| − ++ − 〉 + (bc − ad)| + − + − 〉 + (bc + ad)|+++ − 〉 + (ac − bd)| − − − + 〉 + (ac + bd)| − + − + 〉 + (c2 − d2)| − ++ − 〉 + (c2 + d2)|++ − + 〉 + (bc − ad)| − − − − 〉 +(bc + ad)| − + − − 〉 − 2dc|++ − − 〉]1234
(13)
Then we detect the spin-state of the four quantum dots, the probability of obtaining the | − − − − 〉 can be obtained, which can be given by
Psuc =
bc − ad |bc − ad|2 = 2 2
The Concurrence of the above arbitrary bipartite state is 2|ad − bc|, thus the value of the Concurrence can be directly obtained by
Discussions on the feasibility of the current scheme are necessary for implementing QIP in experiment. The cavity plays an important role in some a cavity quantum Electrodynamics (C-QED) scheme, so the decay of the cavity state attracted much attention. In previous work, the cavity decay would affect either the fidelity of quantum operations or the success probability of these. Of course, it also can be used to play constructive role in some schemes of quantum communication [12,16–21]. In our scheme, the cavity decay can be neglected successfully, because the whole interaction is cavity-state-free evolution along with periodical evolution of a near-resonant driving. As our previous scheme [14], all of the multi-qubit operations are of geometric nature and the single-qubit operations only relied on classical laser fields. Therefore our scheme is insensitive to the thermal field and more robust against random operation errors. Decoherence is unavoidable in practical QIP, so it is necessary to research into the effect of it. To successfully achieve our scheme, we should consider the relative magnitude of the decoherent rates as compared to the gate-operation time. As Ref. [14], the time required to complete a single-qubit operation need about 1 ps, and implement a two-qubit operation need about 1–100 ps, which is based on the type of interactions. The coherent time of conduction-band electrons is about 1 μs in doped quantum well and bulk semiconductors [15], even can reach 1.2 μs based on spin-echo technology [22]. The ratio of the operation time need in our scheme to the coherent time will be less than 10 4, i.e., all of the operations can be achieved in the range of coherent time. Thus our scheme is feasible in the current experiment. In conclusion, we have proposed a scheme for directly measuring the Concurrence of the two-spin entangled states in a cavity–quantum dot system. The direct measurement method of two-spin entangled pure states can be generalized to mixed state case. The Concurrences in the two cases are both based on the success probability P, which can be expressed as C = 2 2Psuc .
P. Dong et al. / Physica B 495 (2016) 50–53
Discussions on the feasibility show that the photon-number-dependent parts in the evolution operator are canceled, i.e., the process is cavity-state-free evolution, so the current scheme is insensitive to the thermal field and robust against random operation errors. It is also noted that our scheme could be implemented within the current experimental technology.
Acknowledgments This work is supported by National Natural Science Foundation of China (NSFC) under Grant no. 61370090, Anhui Provincial Natural Science Foundation under Grant no. 1508085MA19, Provincial Academic and Technical Leaders (Candidates) Foundation of Anhui 2014H008, The key Program for Excellent Young Talents in University of Anhui Province under Grant no. gxyqZD2016206 and The “136” Talent Foundation of Hefei Normal University under Grant nos. 2014136KJB06 and 2014136KJA02.
References [1] A.K. Ekert, Phys. Rev. Lett. 679 (1991) 661. [2] C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881.
53
[3] C.H. Bennett, G. Brssard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [4] R. Laflamme, C. Miquel, J.P. Paz, W.H. Zurek, Phys. Rev. Lett. 77 (1996) 198. [5] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wooters, Phys. Rev. A 54 (1996) 3824. [6] L. Aolita, F. Mintert, Phys. Rev. Lett. 97 (2006) 050501. [7] S. Hill, W.K. Wootters, Phys. Rev. Lett. 78 (1997) 5022. [8] S.P. Walborn, P.H. Souto Ribeiro, L. Davidovich, F. Mintert, A. Buchleitner, Nature 440 (2006) 1022. [9] G. Romero, C.E. López, F. Lastra, E. Solano, J.C. Retamal, Phys. Rev. A 75 (2007) 032303. [10] R.C. Yang, X. Xin, Z.P. Huang, H.C. Li, Commun. Theor. Phys. 51 (2009) 252. [11] L.H. Zhang, M. Yang, Z.L. Cao, Phys. Lett. A 377 (2013) 1241. [12] L.H. Zhang, Q. Yang, M. Yang, W. Song, Z.L. Cao, Phys. Rev. A 88 (2013) 062342. [13] L.H. Zhang, M. Yang, Z.L. Cao, Eur. Phy. J. D 69 (2014) 109. [14] P. Dong, Z.L. Cao, Phys. Lett. A 373 (2009) 1527. [15] A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. DiVincenzo, D. Loss, M. Sherwin, A. Small, Phys. Rev. Lett. 83 (1996) 4204. [16] S. Bose, P.L. Knight, M.B. Plenio, V. Vedral, Phys. Rev. Lett. 83 (1999) 5158. [17] X.B. Zou, K. Pahlke, W. Mathis, Phys. Rev. A 68 (2003) 024302. [18] B. Yu, Z.W. Zhou, Y. Zhang, G.Y. Xiang, G.C. Guo, Phys. Rev. A 70 (2004) 014302. [19] G. Chimczak, R. Tanas, A. Miranowicz, Phys. Rev. A 71 (2005) 032316. [20] G. Chimczak, Phys. Rev. A 71 (2005) 052305. [21] Z.L. Cao, L.H. Zhang, M. Yang, Phys. Rev. A 73 (2006) 014303. [22] J.M. Taylor, J.R. Petta, A.C. Johnson, A. Yacoby, C.M. Marcus, M.D. Lukin, Phys. Rev. B 76 (2007) 035315.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Direct measurement of the Concurrence of spin-entangled states in a cavity–quantum dot system Ping Dong a,n, Jun Liu a, Li-Hua Zhang b, Zhuo-Liang Cao a a b
School of Electronic and Information Engineering, Hefei Normal University, Hefei 230061, China School of Physics and Electrical Engineering, Anqing Normal University, Anqing 246011, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 27 October 2015 Received in revised form 28 February 2016 Accepted 7 May 2016 Available online 9 May 2016
A scheme for implementing the direct measurement of Concurrence is given in a cavity–quantum dot system. The scenario not only can directly measure the Concurrence of two-spin pure entangled state, but also suitable for the case of mixed state. More importantly, all of the operations are of geometric nature, which depend on the cavity-state-free evolution and can be robust against random operation errors. Our scheme provided an alternative method for directly measuring the degree of entanglement in solid-state system. & 2016 Elsevier B.V. All rights reserved.
Keywords: Direct measurement Concurrence Cavity–quantum dot system
1. Introduction Entanglement states are important resource in quantum information processing (QIP), such as quantum communication, quantum computation, and quantum error correction [1–5]. However, the quantification and measurement of the degree of entanglement are still the main obstacles for the deep development of quantum information science. Thus many renewed efforts have recently been given in this direction both theoretically and experimentally. Concurrence is considered as the best one of the quantified methods for calculating the degree of entanglement including entanglement of formation (EOF), partial entropy, relative entropy, negativity, and distributed entanglement, since it can be used not only in specifically bipartite system, but also in multi-body system [6]. Based on the definition of Concurrence proposed by Hill and Wooters [7], one can know that the Concurrence of a bipartite pure state can be expressed as C (|ϕ〉) = |AB 〈ϕ|θ|ϕ〉AB |, where θ|ϕ〉AB = σy ⊗ σy |ϕ〉⁎AB , |ϕ〉 is the bipartite pure state, sy is the Pauli operation and “n” denotes the complex conjugate. And the Concurrence of a bipartite mixed state can be given by C (ρ) = max {0, λ1 − λ2 − λ3 − λ4 }, where ρ is the density of the bipartite mixed state, λis are the eigenvalues of the matrix ρ (σy ⊗ σy ) ρ⁎ (σy ⊗ σy ) ρ , in decreasing order. More importantly, the Concurrence can be used to define the EOF for the bipartite mixed state and also can be generalized to investigate the n
Corresponding author. E-mail addresses: [email protected] (P. Dong), [email protected] (J. Liu).
http://dx.doi.org/10.1016/j.physb.2016.05.005 0921-4526/& 2016 Elsevier B.V. All rights reserved.
distributed entanglement and residual entanglement. Thus it has attracted much attention in this field, recently. On the other hand, due to the importance of measuring the degree of entanglement, many direction measurement schemes based on single-qubit measurement have emerged. After the first direct measurement experimental scheme of the Concurrence of two-photon entangled state was proposed by Walborn et al. [8], Romero et al. and Yang et al. have given another two schemes for directly measuring the Concurrence of two-ion entangled state [9,10]. Then Zhang et al. designed two alternate theoretical direct measurement schemes of the Concurrence of two-photon entangled state by parity-check measurement [11] and homodyne measurement [12], respectively. The direct measurement schemes of the Concurrence for two-atom pure states and mixed states have been also appeared in cavity QED systems [13]. However, as we know, the direct measurement of the Concurrence of an arbitrary spin entangled state in a quantum dot system has not been considered by now. Moreover, quantum dot system has been generally accepted to be the best promising hardware for quantum computation. Therefore, it is significant for investigating the direct measurement of the Concurrence of spin entangled states in a quantum dot system. In this paper, we will design the schemes for implementing the direct measurement of the Concurrence of the pure and mixed two-spin entangled states in a cavity–quantum dot system and discuss the effect brought by the cavity decay on the measurement of the Concurrence. In the current scheme, the quantum dots are all doped such that each quantum dot has a single conduction band electron being in the ground state orbital and a full valence band, so the spin states of the electrons are stable. The capture
P. Dong et al. / Physica B 495 (2016) 50–53
problem of the quantum dots also can be avoided, since the quantum dots are all directly embedded in a microcavity. Our scheme will provide an alternative method for directly measuring the degree of entanglement in quantum dot system.
computation basis |±〉i = 2
He =
The relevant energy levels of the quantum dot can be treated as a V-type configuration, as shown in Fig. 1(a), where the spin-up and spin-down states of the conduction-band electron are encoded as the qubit states and the valence-band state is used as an auxiliary state. The spin-up and spin-down states can be obtained by using a uniform magnetic field. Two quantum dots embedded in a microcavity excited by three classical laser fields, as shown in Fig. 1(b), the Hamiltonian of the system can be expressed as (assuming = = 1)
H = H0 + Hint
(1)
2
H0 =
∑ (ω↑ σ↑↑i + ω↓ σ↓↓i + ωv σvvi ) + ωc a†a
(| ↑〉 ± | ↓〉 ) can be reduced as i
i
i=1
⎤ i i (a†e−iΔ t + aeiΔ t )(σ↑↓ + σ↓↑ ) ⎥ ⎦ 2
(4)
where Δ and Δ are the detunings between the spin-up state and the valence-band state of the ith quantum dot driven by laser gΩ ⎛ 1 1 ⎞ fields 2 and 1, respectively, and A = 2 2 ⎜ i + i i ⎟. If we consider Δ1 +Δ ⎠ ⎝ Δ1 the single quantum dot inside a microcavity and turn off the laser field 2, the cavity field can be eliminated, the net effect of the cavity filled is an additional ac Stack shift, inversely proportional to the detuning of optical frequencies Δ1, which can be safely neglected. The effective Hamiltonian can be reduced as [14]
He′ =
i 2
Ω1Ω3 (σ↓↑ + σ↑↓ ) Δ2
(5)
The process is equivalent to a simple single-channel Raman process with two laser fields of that in Ref. [15]. It is shown that twoqubit operations can be realized by the interaction of Eq. (4) and all of the single-qubit operations can be implemented by adjusting the laser fields in terms with Eq. (5).
3. Direct measurement of the Concurrence of two-spin entangled states
2
∑ {[(Ω1e−iω1t + Ω2 e−iω2t ) σ↑iv i=1
+ (Ω3 e−iω3 t + ga) σ↓i v ] + H . c . }
(3)
where Ωj (j = 1, 2, 3) is the Rabi frequencies of the jth classical field and g is the coupling constant between each quantum dot and the cavity mode. a† and a are the annihilation and creation i operations of the cavity mode, respectively, and σmn = |m〉〈n| ( m , n = ↑ , ↓ , v ) are the operations of the ith quantum dot. In the case of adiabatically eliminating the valence-band state of each quantum dot under the condition of Δ1 , Δ2 ⪢Ωj , g , Δ1 − Δ2 ⪢Δ, (Δ1 +Δ2 ) Ω1Ω2 (Δ1 +Δ2 ) Ω2 Ω3 (2Δ1 +Δ) Ω1g (2Δ1 +Δ) Ω3 g , , 2Δ (Δ +Δ) , 2Δ (Δ +Δ) 2Δ1Δ2 2Δ1Δ2 1 1 1 1
2
(2)
i=1
Hint =
1
⎡A
∑ ⎢⎣ i 1
2. The cavity–quantum dot interaction model
51
, and
2Ω1Ω3 ⪢Δ[14], Δ2
the
effective Hamiltonian of the cavity–quantum dot system under the
To directly measure the Concurrence of two-spin entangled pure states, two identical copies of which are necessary. We first prepare four quantum dots embedded in a microcavity being in | + 〉⊗4 , then we turn on two laser fields with ω1 and ω3 on quantum dot 1 and adjust the interaction time, turn on the three laser fields with ω1, ω2 and ω3 on quantum dot 1 and one laser field with ω3 on quantum dot 2, respectively, let Δ1 = Δ2 and adjust the interaction time appropriately, we can obtain an arbitrary twospin entangled pure state of quantum dots 1 and 2, i.e.,
|ϕ〉12 = a| − − 〉12 + b| − + 〉12 + c| + − 〉12 + d|++〉12
(6)
where a, b, c and d are all real, and |a|2 + |b|2 + |c|2 + |d|2 = 1. Repeat
conduction band
dot 1 dot 2
valence band Fig. 1. (a) The relevant energy levels of a single quantum dot. | ↑ 〉, | ↓ 〉 and |v〉 denote spin up and spin down states of conduction-band electron, and valence-band state, respectively, the energies corresponding to these are =ω↑ , =ω↓ and ℏωv. ω↑↓ = ω↑ − ω↓ . ωj (j = 1, 2, 3) and ωc are the frequencies of three classical fields and the microcavity field. Δ1, Δ2 and Δ are three detunings, and Δ = ω↓ − ω v − ωc − Δ1 = ω3 + Δ2 − ωc − Δ1. (b). The schematic setup of the quantum dots embedded in a microcavity.
52
P. Dong et al. / Physica B 495 (2016) 50–53
the above process on quantum dots 3 and 4 as that on quantum dots 1 and 2, two identical quantum states can be generated. That is to say that the state of the four quantum dots becomes
(7)
|ψ 〉1234 = |ϕ〉12 ⊗ |ϕ〉34
Next, we turn on two laser fields on quantum dots 3 and 4, and control the interaction time, let
| + 〉q ⟶| − 〉q , | − 〉q ⟶ − | + 〉q
(q = 3, 4)
(8)
and then operate a single-qubit rotation on quantum dot 4 by using a classical laser field, let
1 (| − 〉4 − i| + 〉4 ), 2 1 | − 〉4 ⟶ (| + 〉4 + | − 〉4 ) 2 | + 〉4 ⟶
(9)
the total state becomes
|ψ 〉′1234 =
C (ϕ) = 2|ad − bc| = 2 2Psuc If the above two-spin entangled pure state is exposed by environment, i.e., the leakage out of the cavity mode, the collective relaxation and the dephase of the transition of the quantum dots will accompany the interaction process, the state will reduce to a mixed state. We assume that the mixed one is
ρ12 = F |φ〉1221〈φ| + (1 − F )|ψ 〉1221〈ψ |
⊗ (a|++〉 − a| + − 〉 + b|++〉 + b| + − 〉 (10)
Then we turn on three laser fields with ω1, ω2 and ω3 on quantum dots 2 and 4, respectively, the quantum dots 2 and 4 will interact under the condition of Δ2 = Δ4 = Δ, controlling the interaction , the total state of the system will become time, let t = 2πΔ 2
(14)
where 0 < F < 1, |φ〉12 = α|++〉12 + β| − − 〉12, |ψ 〉12 = | + − 〉12, and |α|2 + |β|2 = 1. Assuming that the two pure states are both reduced to the above mixed states, i.e., ρ1234 = ρ12 ⊗ ρ34 . We repeat above operations as the case of the pure states, and then detect the spinstate of the four quantum dots, the success probability of obtaining | − − − − 〉 is
Psuc ′ =
1 (a| − − 〉 + b| − + 〉 + c| + − 〉 + d|++〉)12 2 +c| − − 〉 − c| − + 〉 − d| − − 〉 − d|++〉)34
measuring the success probability Psuc, i.e.,
F2 |αβ|2 2
The Concurrence of mixed state is 2F |αβ|, thus the Concurrence of that also can be directly obtained by measuring the success probability Psuc ′ , i.e.,
C′ (ρ) = 2 2Psuc ′ If |ψ 〉12 = | − + 〉12, the Concurrence also can be measured by the above process, the result is the same as C′(ρ). The method can be generalized to measure other mixed states in quantum dot system.
A
|ψ 〉″1234
1 = (a| − − 〉 + c| + − 〉)12 ⊗ (a|++〉 − a| + − 〉 2
4. Discussions and conclusions
+ b|++〉 + b| + − 〉 + c| − − 〉 − c| − + 〉 1 −d| − − 〉 − d|++〉)34 + (b| − + 〉 2 +d|++〉)12 ⊗ ( − a|++〉 − a| + − 〉 −b|++〉 + b| + − 〉 + c| − − 〉 + c| − + 〉 − d| − − 〉 + d|++〉)34
(11)
Finally, we turn on two laser fields on quantum dot 2 and quantum dot 4, respectively, let
1 (| − 〉m − | + 〉m ), 2 1 | − 〉m ⟶ (| + 〉m + | − 〉m ) (m = 2, 4) 2 | + 〉m ⟶
(12)
The total state of the system will be
|ψ 〉‴ 1234 =
1 [(b2 − a2)| − − ++〉 − (b2 + a2)| − +++〉 2 + (bd − ac )| + − ++〉 − (bd + ac )|++++〉 + 2ab| − ++ − 〉 + (bc − ad)| + − + − 〉 + (bc + ad)|+++ − 〉 + (ac − bd)| − − − + 〉 + (ac + bd)| − + − + 〉 + (c2 − d2)| − ++ − 〉 + (c2 + d2)|++ − + 〉 + (bc − ad)| − − − − 〉 +(bc + ad)| − + − − 〉 − 2dc|++ − − 〉]1234
(13)
Then we detect the spin-state of the four quantum dots, the probability of obtaining the | − − − − 〉 can be obtained, which can be given by
Psuc =
bc − ad |bc − ad|2 = 2 2
The Concurrence of the above arbitrary bipartite state is 2|ad − bc|, thus the value of the Concurrence can be directly obtained by
Discussions on the feasibility of the current scheme are necessary for implementing QIP in experiment. The cavity plays an important role in some a cavity quantum Electrodynamics (C-QED) scheme, so the decay of the cavity state attracted much attention. In previous work, the cavity decay would affect either the fidelity of quantum operations or the success probability of these. Of course, it also can be used to play constructive role in some schemes of quantum communication [12,16–21]. In our scheme, the cavity decay can be neglected successfully, because the whole interaction is cavity-state-free evolution along with periodical evolution of a near-resonant driving. As our previous scheme [14], all of the multi-qubit operations are of geometric nature and the single-qubit operations only relied on classical laser fields. Therefore our scheme is insensitive to the thermal field and more robust against random operation errors. Decoherence is unavoidable in practical QIP, so it is necessary to research into the effect of it. To successfully achieve our scheme, we should consider the relative magnitude of the decoherent rates as compared to the gate-operation time. As Ref. [14], the time required to complete a single-qubit operation need about 1 ps, and implement a two-qubit operation need about 1–100 ps, which is based on the type of interactions. The coherent time of conduction-band electrons is about 1 μs in doped quantum well and bulk semiconductors [15], even can reach 1.2 μs based on spin-echo technology [22]. The ratio of the operation time need in our scheme to the coherent time will be less than 10 4, i.e., all of the operations can be achieved in the range of coherent time. Thus our scheme is feasible in the current experiment. In conclusion, we have proposed a scheme for directly measuring the Concurrence of the two-spin entangled states in a cavity–quantum dot system. The direct measurement method of two-spin entangled pure states can be generalized to mixed state case. The Concurrences in the two cases are both based on the success probability P, which can be expressed as C = 2 2Psuc .
P. Dong et al. / Physica B 495 (2016) 50–53
Discussions on the feasibility show that the photon-number-dependent parts in the evolution operator are canceled, i.e., the process is cavity-state-free evolution, so the current scheme is insensitive to the thermal field and robust against random operation errors. It is also noted that our scheme could be implemented within the current experimental technology.
Acknowledgments This work is supported by National Natural Science Foundation of China (NSFC) under Grant no. 61370090, Anhui Provincial Natural Science Foundation under Grant no. 1508085MA19, Provincial Academic and Technical Leaders (Candidates) Foundation of Anhui 2014H008, The key Program for Excellent Young Talents in University of Anhui Province under Grant no. gxyqZD2016206 and The “136” Talent Foundation of Hefei Normal University under Grant nos. 2014136KJB06 and 2014136KJA02.
References [1] A.K. Ekert, Phys. Rev. Lett. 679 (1991) 661. [2] C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881.
53
[3] C.H. Bennett, G. Brssard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [4] R. Laflamme, C. Miquel, J.P. Paz, W.H. Zurek, Phys. Rev. Lett. 77 (1996) 198. [5] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wooters, Phys. Rev. A 54 (1996) 3824. [6] L. Aolita, F. Mintert, Phys. Rev. Lett. 97 (2006) 050501. [7] S. Hill, W.K. Wootters, Phys. Rev. Lett. 78 (1997) 5022. [8] S.P. Walborn, P.H. Souto Ribeiro, L. Davidovich, F. Mintert, A. Buchleitner, Nature 440 (2006) 1022. [9] G. Romero, C.E. López, F. Lastra, E. Solano, J.C. Retamal, Phys. Rev. A 75 (2007) 032303. [10] R.C. Yang, X. Xin, Z.P. Huang, H.C. Li, Commun. Theor. Phys. 51 (2009) 252. [11] L.H. Zhang, M. Yang, Z.L. Cao, Phys. Lett. A 377 (2013) 1241. [12] L.H. Zhang, Q. Yang, M. Yang, W. Song, Z.L. Cao, Phys. Rev. A 88 (2013) 062342. [13] L.H. Zhang, M. Yang, Z.L. Cao, Eur. Phy. J. D 69 (2014) 109. [14] P. Dong, Z.L. Cao, Phys. Lett. A 373 (2009) 1527. [15] A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. DiVincenzo, D. Loss, M. Sherwin, A. Small, Phys. Rev. Lett. 83 (1996) 4204. [16] S. Bose, P.L. Knight, M.B. Plenio, V. Vedral, Phys. Rev. Lett. 83 (1999) 5158. [17] X.B. Zou, K. Pahlke, W. Mathis, Phys. Rev. A 68 (2003) 024302. [18] B. Yu, Z.W. Zhou, Y. Zhang, G.Y. Xiang, G.C. Guo, Phys. Rev. A 70 (2004) 014302. [19] G. Chimczak, R. Tanas, A. Miranowicz, Phys. Rev. A 71 (2005) 032316. [20] G. Chimczak, Phys. Rev. A 71 (2005) 052305. [21] Z.L. Cao, L.H. Zhang, M. Yang, Phys. Rev. A 73 (2006) 014303. [22] J.M. Taylor, J.R. Petta, A.C. Johnson, A. Yacoby, C.M. Marcus, M.D. Lukin, Phys. Rev. B 76 (2007) 035315.