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Experimentally simulating the violation of Bell-type inequalities for generalized GHZ states
Physics Letters A 373 (2009) 4222–4226
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Experimentally simulating the violation of Bell-type inequalities for generalized GHZ states Changliang Ren a , Dawei Lu b , Mingjun Shi a , Xinhua Peng a,b , Jiangfeng Du a,b,∗ a b
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Hefei National Laboratory for Physical Sciences at Microscale, Hefei, Anhui 230026, China
a r t i c l e
i n f o
Article history: Received 8 October 2008 Received in revised form 11 September 2009 Accepted 17 September 2009 Available online 22 September 2009 Communicated by P.R. Holland
a b s t r a c t Using NMR techniques, we implemented the simulation of the violations of two Bell-type inequalities: Mermin–Ardehali–Belinskii–Klyshko (MABK) inequality and Chen’s inequality, for the 3-qubit generalized GHZ states. The experimental results are in good agreement with the quantum predictions and show that Chen’s inequality is more efficient than MABK inequality in the case of the generalized GHZ entangled states. © 2009 Elsevier B.V. All rights reserved.
PACS: 03.65.Ud 76.60.-k Keywords: NMR system Bell-type inequality 3-qubit generalized GHZ states Nonmaximal entangled states
1. Introduction In 1964, Bell showed that in all local realistic theories, correlations between the outcomes of measurements in different parts of a physical system satisfy a certain class of inequalities [1]. However, it is easy to find that entangled states violate these inequalities in quantum mechanics, which shows the crucial conflict between classical theory and quantum mechanics. Hence, Bell’s work was described as “the most profound discovery of science” [2] or “one of the greatest discoveries of modern science” [3]. Later, more important generalizations, including the Clauser–Horne–Shimony– Holt (CHSH) [4] and Mermin–Ardehali–Belinskii–Klyshko (MABK) inequalities [5] were developed. More recently, Werner and Wolf ˙ and Zukowski and Brukner (WWZB) derived a set of multipartite Bell inequalities, by using two dichotomic observables per site [6]. There has been increasing interest in the subject of Bell’s inequalities, because of not only fundamental problems of quantum mechanics, but also their relation to quantum communication [7–9] and quantum cryptography [10,11]. For example, the
*
Corresponding author at: Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China. Tel.: +86 551 3600039; fax: +86 551 3601164. E-mail address: [email protected] (J. Du). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.09.052
security of some quantum communication protocols are based on the loophole-free violation of Bell inequalities [11,12]. Furthermore, Bells inequalities can be a useful tool to detect entanglement which is a powerful computational resource in quantum computation [13]. Various experiments to test Bell inequality have been performed in various systems including photons [14–16], atoms systems [17], atomic ensembles [18,19], and trapped ions [20]. Recently, an experiment to simulate the violation of CHSH inequality was carried out on NMR [21]. All the experiments before were mainly implemented on the maximal entangled states, such as Bell state and the standard GHZ state, but rarely on nonmaximal entangled states, such as the generalized GHZ states. We will focus on the violation of two different Bell-type inequality for generalized GHZ states in experiment. For the three-qubit generalized GHZ states
|Ψ = cos θ|000 + sin θ|111,
(1)
which θ ∈ (0, π2 ), Scarani and Gisin [22] firstly found that there existed a region of them satisfying the MABK inequality. It was shown that for θ π /12 or θ 5π /12 the states (1) do not vi˙ olate the three-qubit MABK inequality. Later on, Zukowski et al. proved that [23] (i) for N = even, the generalized GHZ states violate the WWZB inequality, rather than MABK inequalities; (ii) for
C. Ren et al. / Physics Letters A 373 (2009) 4222–4226
4223
√
N = odd and sin(2θ) 1/ 2 N −1 , the generalized GHZ states satisfy both MABK and WWZB inequalities. Soon, Chen and Wu et al. developed several Bell inequalities for three qubits, which can be numerically violated by arbitrary generalized GHZ state [24,25]. Recently, more significant progress was achieved by Chen et al. [26]. They presented a family of Bell inequalities involving only two measurement settings of each observer for N > 2 qubits, which is violated by any N-qubit generalized GHZ state, and moreover the amount of maximal violation grows exponentially as 2( N −2)/2 . However, there is theoretical conclusion, no experiments aim to display them so far. In this letter, we implemented the simulation of the violation of two different Bell-type inequalities, i.e., MABK inequality [5] and Chen’s inequality [26], for the generalized GHZ states in an NMR system. The experimental results clearly show that the high efficiency of Chen’s inequality and the limitation of MABK inequality for any generalized GHZ entangled state and predict the behaviors of quantum mechanics.
For NMR experimental implementation, there are still two problems to be solved. Firstly, the thermal equilibrium state of a NMR system at room temperature is highly mixed. We can use pseudopure state (PPS) [27] technique to overcome this. Instead of a pure state |000, we prepared a PPS:
2. Simulating violation of MABK inequality for GHZ state
ρpps =
Let us consider such a scenario: there are three observers Alice ( A), Bob (B), and Charlie (C ), each having one qubit. The formulation of the MABK inequality is based on the assumption that every observer is allowed to choose one observable between two dichotomic observables. Denote the outcome of observer X ’s measurement by X i , X = A , B , C , with i = 1, 2. Under the assumption of local realism, each outcome can either take value +1 or −1. In a specific run of the experiment, the correlations between the measurement outcomes of all three observers can be represented by the product A i B j C k , where i , j , k = 1, 2. In a local realistic theory, the correlation function of the measurements performed by all three observers is the average of A i B j C k over many runs of the experiment,
It is a mixture of the totally mixed state I 2n unchanged when applying with unitary transformations and a pure state |000 with the polarization ε ≈ 10−5 . So ignoring I 2n which does not affect NMR experiments and using the entanglement (strictly, pseudoentanglement) of the pure part, we can simulate violation of the Bell-type inequalities we mentioned in this letter. The second problem is only the spin projection values under a computational basis can be directly measured. The solution is to rotate the state or density matrix instead of changing the projective direction,
E ( A i , B j , C k ) = A i B j C k avg .
(2)
The MABK inequality reads as [5]
E ( A1, B 2, C2) + E ( A2, B 1, C2)
+ E ( A 2 , B 2 , C 1 ) − E ( A 1 , B 1 , C 1 ) 2.
(3)
We denote the left-hand side of the MABK inequality by |BMABK | where −2 BMABK 2. In any local hidden variable (LHV) theory, the absolute value of a particular combination of correlations is bounded by 2. However, if one turns to quantum mechanics, this inequality can be violated. For MABK inequality, the maximal violation allowed by quantum mechanics is 4 [22], by the standard GHZ state, i.e., the state with θ = π4 in Eq. (1). As an example, in this section, we implemented the simulation of MABK inequality for the standard GHZ state in NMR system. To prepare standard GHZ state from |000, we used the network as shown in Fig. 1, by selecting the rotation angle θ = π /4. After that, we will measure the spin projection σ · n, where σ = (σx , σ y , σz ) is the vector form of Pauli matrices and the two measurement directions for every qubit we chose here are n1 = (1, 0, 0) and n2 = (cos α , sin α , 0). In other words, the two dichotomic observables allowed to be chosen for A, B, C are σ n1 and σ n2 . For this special spin projection measurement, the theoretical result of BMABK is (for convenience we just ignore the absolute value sign)
BMABK = 3 cos2 α − sin2 α − 1,
(4)
demonstrating that for α = 0.3041π ∼ 0.6959π , |BMABK | > 2 to violate MABK inequality and reach the maximal violation value 4 when α = π /2.
Fig. 1. Quantum network for creating a generalized GHZ state. The input state is |000. R 2yθ denotes a rotation of an angle 2θ along y axis, following by two controlled-not (CNOT) gates. The output state is a generalized GHZ state cos θ|000 + sin θ|111.
(1 − ε ) 2n
I 2n + ε |000000|.
M = Tr(ρ · M 1 ) = Tr
(5)
ρ · U † M 2 U = Tr U ρ U † · M 2 ,
(6)
where M 1 and M 2 are the desired and experimental measurements, respectively. U is one unitary operation satisfying M 1 = U † M 2 U . In NMR experiments we can apply U to the density matrix and then perform measurement of M 2 , which is equivalent to measuring M 1 . All experiments were performed at room temperature on a Bruker Avance 400 MHz NMR spectrometer. We used the spins of three 13 C Nuclei in alanine dissolved in D 2 O . The system Hamiltonian can be written as
H sys = 2π
3 i =1
ωi I iz + 2π
3
j
J i j I iz I z ,
(7)
i< j
with the resonance frequencies ωi and J -coupling constants J i j . The chemical shifts of the three carbon nuclei are ω1 = 5128.2 Hz, ω2 = 17740 Hz, and ω3 = 1676.7 Hz; the J -coupling strengths are J 12 = 53.98 Hz, J 23 = −1.18 Hz, and J 13 = 34.88 Hz. The whole experiment was divided into three steps. Firstly, to prepare ρpps from the thermal equilibrium state by using the spatial average technique [28]. Secondly, to prepare a standard GHZ state by using the network in Fig. 1 with θ = π /4. Finally, to rotate the required qubits and execute the projective measurements. In the experiment A, B and C are replaced by three spins, while the subscripts 1 and 2 means the spin projection are σ · n1 and σ · n2 . E ( A 1 , B 2 , C 2 ) represents the value of the projective operator σx ⊗ (cos ασx + sin ασ y ) ⊗ (cos ασx + sin ασ y ) measured in the final state ρ
Tr
ρ · σx ⊗ (cos ασx + sin ασ y ) ⊗ (cos ασx + sin ασ y )
and so on with the other three. If we can measure the values of E ( A 1 , B 2 , C 2 ), E ( A 2 , B 1 , C 2 ), E ( A 2 , B 2 , C 1 ) and E ( A 1 , B 1 , C 1 ), the value of the left of MABK inequality would be obtained. E ( A 1 , B 1 , C 1 ) can be easily got through a single measurement.
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√
Fig. 2. Theoretical (a) and experimental (b) density matrices of the standard GHZ state (|000 + |111)/ 2. The numbers 1–8 on the x and y axes represent the product states |000 to |111.
For alanine on an NMR system, we just need apply a rotation 1, 3 R y (π /2) and calculate the integrations of the spectrum of spin 2. However, for the other three, such as E ( A 1 , B 2 , C 2 ), the projective operator is σx ⊗ (cos ασx + sin ασ y ) ⊗ (cos ασx + sin ασ y ), whose general measuring method for NMR is to obtain the values of σx ⊗ σx ⊗ σx , σx ⊗ σ y ⊗ σx , σx ⊗ σx ⊗ σ y , σx ⊗ σ y ⊗ σ y respectively and implement their linear combinations. It is not suitable for this experiment as we require only one measurement for each E. The solution is using (6) where M 1 = σx ⊗ (cos ασx + 2, 3 sin ασ y ) ⊗ (cos ασx + sin ασ y ), M 2 = σx ⊗ σx ⊗ σx and U = R z (α ). 2, 3
That is to say, if we apply R z (α ) on the final state ρ , the value of σx ⊗ σx ⊗ σx measured in the new final state is equivalent to σx ⊗ (cos ασx + sin ασ y ) ⊗ (cos ασx + sin ασ y ) measured in ρ . Thus we obtain E ( A 1 , B 2 , C 2 ) through one measurement σx ⊗ σx ⊗ σx . In order to improve the accuracy of radio frequency (RF) pulses, we used strongly modulating pulse (SMP) techniques [29]. We also maximized the effective gate fidelity by averaging over a weighted distribution of RF field strengths to overcome the inhomogeneity of the RF fields over the sample. The gate fidelity we calculated for every pulse is higher than 0.995 considering the RF field inhomogeneity. The range of the pulse lengths are about from 200 to 700 μs. Fig. 2(b) shows a full state tomography of the standard GHZ state prepared in experiment. The overall fidelity is
F=
Tr(ρth ρexp ) 2 2 (Tr(ρth ) Tr(ρexp ))
= 0.98.
(8)
We took the observers mentioned above (σ n1 , σ n2 ) to do the corresponding measurement on the standard GHZ state. The experimental result is shown in Fig. 3, where the blue squares stand for the experiment results, and the red thick line stands for the theoretical result. Clearly, the experimental results are in excellent agreement with the theoretical expectation of quantum mechanics. 3. Simulating violation of MABK inequality for generalized GHZ states So far, almost all previous Bell experiments were performed on maximal entangled states, such as Bell state and the standard GHZ state. Recently, much work about nonmaximal entangled states have been done [30,31]. In this section, we implemented the simulation the violation of MABK inequality for the generalized GHZ state. In this experiment, we choose the directions of the two measurements for every particle is n1 = (1, 0, 0) and n2 = (0, 1, 0). For these special spin projection measurements, the theoretical result of BMABK for the generalized GHZ states satisfies such a function,
|BMABK | = −4 sin(2θ).
(9)
Fig. 3. Experimental test of the MABK inequality for a standard GHZ state. The red thick line stands for the theoretical expectation, and the blue square stands for the experimental data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
From Eq. (9), one can see that the maximal violation is obtained when θ = π4 just the standard GHZ state. Obviously, the MABK in-
π , 5π ]; in other equality is efficient only in the region of θ ∈ [ 12 12 words, only in such a region the inequality can be violated. We measured a set of generalized GHZ states with particular angles θ . Fig. 4(a) shows the experimental date along with the theoretical expectation.
4. Simulating violation of Chen’s inequality for generalized GHZ states For a three-qubit system, Chen’s inequality can be written as
BChen =
1 2
E ( A1, B 1, C1) + E ( A1, B 2, C1) + E ( A2, B 1, C1)
− E ( A2, B 2, C1) + E ( A1, B 1, C2) + E ( A1, B 2, C2) + E ( A2, B 1, C2) − E ( A2, B 2, C2)
+ E (C 1 ) − E (C 2 ),
(10)
with |BChen | 2 in the LHV model. In experiment, we took the directions of two measurement about A and B as n1 = (1, 0, 0) and n2 = (0, 1, 0). For C , the directions of two measurement were chosen as n1 = (sin α cos(− π4 ), sin α sin(− π4 ), cos α ) and n2 = (sin(π − α ) cos(− π4 ), sin(π − α ) × sin(− π4 ), cos(π − α )), where
C. Ren et al. / Physics Letters A 373 (2009) 4222–4226
(a)
4225
(b)
Fig. 4. For the generalized GHZ states, (a) the values of BMABK as a function of θ , (b) the values of BChen as a function of θ . The red thick line stands for the theoretical expectation, and the blue square stands for the experiment data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
α = tan−1 α = tan−1
√
2 tan(2θ) ,
√
0θ
2 tan(2θ) + π ,
π 4
π 4
,
θ
π 2
.
(11)
Then, we obtain BChen as
1/2 BChen = 2 2 sin2 (2θ) + cos2 (2θ) ,
(12)
which tells us that BChen always larger than 2 no matter whatever θ is. It illustrates that, the whole region of the generalized GHZ states can violate the inequality by a set of suitable observation angles. Obviously, Chen’s inequality is more efficient than MABK inequality for generalized GHZ states. The experimental result is shown in Fig. 4(b), which perfectly simulates the violation of Chen’s inequality for the generalize GHZ states. 5. Conclusions In summary, we have investigated the simulation of the violation of Bell-type inequalities, including MABK inequality and Chen’s inequality for the generalized GHZ states in an NMR system. In the range of the generalized GHZ states, Chen’s inequality is more efficient than MABK inequality. The experimental results are well in agreement with the expectation of quantum mechanics. It is necessary to emphasize that, in strict, because NMR qubits are many nuclear spins of atoms bounded together in a single molecule, separated by a few angstroms, the NMR experiment is inherently local. Furthermore, the initial state in NMR is prepared from the thermal equilibrium into a highly mixed state called pseudo-pure state (PPS) [27], where ε ≈ 10−5 is the polarization at room temperature. Braunstein et al. [32] pointed out that any state of the form (5) is not entangled whenever ε (1+212n−1 ) . Hence,
our work neither provides an experimental procedure to prove or disprove non-local effects nor reveals entanglement in NMR experiments at room temperature. Whereas, the meaning is that, when
we experimentally simulate the violation of different Bell-type inequalities for arbitrary generalized three-qubit GHZ states in NMR, the results are excellently in accord with the quantum predictions. It tells us, despite of many existed disputes, NMR may contribute more on some fundamentals of quantum mechanics. As a refined tool and technique for experimentally realizing quantum computation in the last decade, NMR is still contributing to numerous fundamental problems of quantum mechanics. Acknowledgements The authors are grateful to Ya Wang, Ping Zou, Jing Zhu, and Zhaokai Li for their help and interesting comments and discussions. Financial support comes from National Natural Science Foundation of China. References [1] [2] [3] [4] [5]
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[16] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, A. Zeilinger, Phys. Rev. Lett. 81 (1998) 5039. [17] D.L. Moehring, M.J. Madsen, B.B. Blinov, C. Monroe, Phys. Rev. Lett. 93 (2004) 090410. [18] D.N. Matsukevich, et al., Phys. Rev. Lett. 96 (2006) 030405. [19] C.-W. Chou, et al., Science 316 (2007) 1316. [20] M.A. Rowe, et al., Nature (London) 409 (2001) 791. [21] A.M. Souza, et al., New J. Phys. 10 (2008) 033020. [22] V. Scarani, N. Gisin, J. Phys. A 34 (2001) 6043. ˇ Brukner, W. Laskowski, M. Wiesniak, Phys. Rev. Lett. 88 (2002) ˙ [23] M. Zukowski, C. 210402.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Experimentally simulating the violation of Bell-type inequalities for generalized GHZ states Changliang Ren a , Dawei Lu b , Mingjun Shi a , Xinhua Peng a,b , Jiangfeng Du a,b,∗ a b
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Hefei National Laboratory for Physical Sciences at Microscale, Hefei, Anhui 230026, China
a r t i c l e
i n f o
Article history: Received 8 October 2008 Received in revised form 11 September 2009 Accepted 17 September 2009 Available online 22 September 2009 Communicated by P.R. Holland
a b s t r a c t Using NMR techniques, we implemented the simulation of the violations of two Bell-type inequalities: Mermin–Ardehali–Belinskii–Klyshko (MABK) inequality and Chen’s inequality, for the 3-qubit generalized GHZ states. The experimental results are in good agreement with the quantum predictions and show that Chen’s inequality is more efficient than MABK inequality in the case of the generalized GHZ entangled states. © 2009 Elsevier B.V. All rights reserved.
PACS: 03.65.Ud 76.60.-k Keywords: NMR system Bell-type inequality 3-qubit generalized GHZ states Nonmaximal entangled states
1. Introduction In 1964, Bell showed that in all local realistic theories, correlations between the outcomes of measurements in different parts of a physical system satisfy a certain class of inequalities [1]. However, it is easy to find that entangled states violate these inequalities in quantum mechanics, which shows the crucial conflict between classical theory and quantum mechanics. Hence, Bell’s work was described as “the most profound discovery of science” [2] or “one of the greatest discoveries of modern science” [3]. Later, more important generalizations, including the Clauser–Horne–Shimony– Holt (CHSH) [4] and Mermin–Ardehali–Belinskii–Klyshko (MABK) inequalities [5] were developed. More recently, Werner and Wolf ˙ and Zukowski and Brukner (WWZB) derived a set of multipartite Bell inequalities, by using two dichotomic observables per site [6]. There has been increasing interest in the subject of Bell’s inequalities, because of not only fundamental problems of quantum mechanics, but also their relation to quantum communication [7–9] and quantum cryptography [10,11]. For example, the
*
Corresponding author at: Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China. Tel.: +86 551 3600039; fax: +86 551 3601164. E-mail address: [email protected] (J. Du). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.09.052
security of some quantum communication protocols are based on the loophole-free violation of Bell inequalities [11,12]. Furthermore, Bells inequalities can be a useful tool to detect entanglement which is a powerful computational resource in quantum computation [13]. Various experiments to test Bell inequality have been performed in various systems including photons [14–16], atoms systems [17], atomic ensembles [18,19], and trapped ions [20]. Recently, an experiment to simulate the violation of CHSH inequality was carried out on NMR [21]. All the experiments before were mainly implemented on the maximal entangled states, such as Bell state and the standard GHZ state, but rarely on nonmaximal entangled states, such as the generalized GHZ states. We will focus on the violation of two different Bell-type inequality for generalized GHZ states in experiment. For the three-qubit generalized GHZ states
|Ψ = cos θ|000 + sin θ|111,
(1)
which θ ∈ (0, π2 ), Scarani and Gisin [22] firstly found that there existed a region of them satisfying the MABK inequality. It was shown that for θ π /12 or θ 5π /12 the states (1) do not vi˙ olate the three-qubit MABK inequality. Later on, Zukowski et al. proved that [23] (i) for N = even, the generalized GHZ states violate the WWZB inequality, rather than MABK inequalities; (ii) for
C. Ren et al. / Physics Letters A 373 (2009) 4222–4226
4223
√
N = odd and sin(2θ) 1/ 2 N −1 , the generalized GHZ states satisfy both MABK and WWZB inequalities. Soon, Chen and Wu et al. developed several Bell inequalities for three qubits, which can be numerically violated by arbitrary generalized GHZ state [24,25]. Recently, more significant progress was achieved by Chen et al. [26]. They presented a family of Bell inequalities involving only two measurement settings of each observer for N > 2 qubits, which is violated by any N-qubit generalized GHZ state, and moreover the amount of maximal violation grows exponentially as 2( N −2)/2 . However, there is theoretical conclusion, no experiments aim to display them so far. In this letter, we implemented the simulation of the violation of two different Bell-type inequalities, i.e., MABK inequality [5] and Chen’s inequality [26], for the generalized GHZ states in an NMR system. The experimental results clearly show that the high efficiency of Chen’s inequality and the limitation of MABK inequality for any generalized GHZ entangled state and predict the behaviors of quantum mechanics.
For NMR experimental implementation, there are still two problems to be solved. Firstly, the thermal equilibrium state of a NMR system at room temperature is highly mixed. We can use pseudopure state (PPS) [27] technique to overcome this. Instead of a pure state |000, we prepared a PPS:
2. Simulating violation of MABK inequality for GHZ state
ρpps =
Let us consider such a scenario: there are three observers Alice ( A), Bob (B), and Charlie (C ), each having one qubit. The formulation of the MABK inequality is based on the assumption that every observer is allowed to choose one observable between two dichotomic observables. Denote the outcome of observer X ’s measurement by X i , X = A , B , C , with i = 1, 2. Under the assumption of local realism, each outcome can either take value +1 or −1. In a specific run of the experiment, the correlations between the measurement outcomes of all three observers can be represented by the product A i B j C k , where i , j , k = 1, 2. In a local realistic theory, the correlation function of the measurements performed by all three observers is the average of A i B j C k over many runs of the experiment,
It is a mixture of the totally mixed state I 2n unchanged when applying with unitary transformations and a pure state |000 with the polarization ε ≈ 10−5 . So ignoring I 2n which does not affect NMR experiments and using the entanglement (strictly, pseudoentanglement) of the pure part, we can simulate violation of the Bell-type inequalities we mentioned in this letter. The second problem is only the spin projection values under a computational basis can be directly measured. The solution is to rotate the state or density matrix instead of changing the projective direction,
E ( A i , B j , C k ) = A i B j C k avg .
(2)
The MABK inequality reads as [5]
E ( A1, B 2, C2) + E ( A2, B 1, C2)
+ E ( A 2 , B 2 , C 1 ) − E ( A 1 , B 1 , C 1 ) 2.
(3)
We denote the left-hand side of the MABK inequality by |BMABK | where −2 BMABK 2. In any local hidden variable (LHV) theory, the absolute value of a particular combination of correlations is bounded by 2. However, if one turns to quantum mechanics, this inequality can be violated. For MABK inequality, the maximal violation allowed by quantum mechanics is 4 [22], by the standard GHZ state, i.e., the state with θ = π4 in Eq. (1). As an example, in this section, we implemented the simulation of MABK inequality for the standard GHZ state in NMR system. To prepare standard GHZ state from |000, we used the network as shown in Fig. 1, by selecting the rotation angle θ = π /4. After that, we will measure the spin projection σ · n, where σ = (σx , σ y , σz ) is the vector form of Pauli matrices and the two measurement directions for every qubit we chose here are n1 = (1, 0, 0) and n2 = (cos α , sin α , 0). In other words, the two dichotomic observables allowed to be chosen for A, B, C are σ n1 and σ n2 . For this special spin projection measurement, the theoretical result of BMABK is (for convenience we just ignore the absolute value sign)
BMABK = 3 cos2 α − sin2 α − 1,
(4)
demonstrating that for α = 0.3041π ∼ 0.6959π , |BMABK | > 2 to violate MABK inequality and reach the maximal violation value 4 when α = π /2.
Fig. 1. Quantum network for creating a generalized GHZ state. The input state is |000. R 2yθ denotes a rotation of an angle 2θ along y axis, following by two controlled-not (CNOT) gates. The output state is a generalized GHZ state cos θ|000 + sin θ|111.
(1 − ε ) 2n
I 2n + ε |000000|.
M = Tr(ρ · M 1 ) = Tr
(5)
ρ · U † M 2 U = Tr U ρ U † · M 2 ,
(6)
where M 1 and M 2 are the desired and experimental measurements, respectively. U is one unitary operation satisfying M 1 = U † M 2 U . In NMR experiments we can apply U to the density matrix and then perform measurement of M 2 , which is equivalent to measuring M 1 . All experiments were performed at room temperature on a Bruker Avance 400 MHz NMR spectrometer. We used the spins of three 13 C Nuclei in alanine dissolved in D 2 O . The system Hamiltonian can be written as
H sys = 2π
3 i =1
ωi I iz + 2π
3
j
J i j I iz I z ,
(7)
i< j
with the resonance frequencies ωi and J -coupling constants J i j . The chemical shifts of the three carbon nuclei are ω1 = 5128.2 Hz, ω2 = 17740 Hz, and ω3 = 1676.7 Hz; the J -coupling strengths are J 12 = 53.98 Hz, J 23 = −1.18 Hz, and J 13 = 34.88 Hz. The whole experiment was divided into three steps. Firstly, to prepare ρpps from the thermal equilibrium state by using the spatial average technique [28]. Secondly, to prepare a standard GHZ state by using the network in Fig. 1 with θ = π /4. Finally, to rotate the required qubits and execute the projective measurements. In the experiment A, B and C are replaced by three spins, while the subscripts 1 and 2 means the spin projection are σ · n1 and σ · n2 . E ( A 1 , B 2 , C 2 ) represents the value of the projective operator σx ⊗ (cos ασx + sin ασ y ) ⊗ (cos ασx + sin ασ y ) measured in the final state ρ
Tr
ρ · σx ⊗ (cos ασx + sin ασ y ) ⊗ (cos ασx + sin ασ y )
and so on with the other three. If we can measure the values of E ( A 1 , B 2 , C 2 ), E ( A 2 , B 1 , C 2 ), E ( A 2 , B 2 , C 1 ) and E ( A 1 , B 1 , C 1 ), the value of the left of MABK inequality would be obtained. E ( A 1 , B 1 , C 1 ) can be easily got through a single measurement.
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√
Fig. 2. Theoretical (a) and experimental (b) density matrices of the standard GHZ state (|000 + |111)/ 2. The numbers 1–8 on the x and y axes represent the product states |000 to |111.
For alanine on an NMR system, we just need apply a rotation 1, 3 R y (π /2) and calculate the integrations of the spectrum of spin 2. However, for the other three, such as E ( A 1 , B 2 , C 2 ), the projective operator is σx ⊗ (cos ασx + sin ασ y ) ⊗ (cos ασx + sin ασ y ), whose general measuring method for NMR is to obtain the values of σx ⊗ σx ⊗ σx , σx ⊗ σ y ⊗ σx , σx ⊗ σx ⊗ σ y , σx ⊗ σ y ⊗ σ y respectively and implement their linear combinations. It is not suitable for this experiment as we require only one measurement for each E. The solution is using (6) where M 1 = σx ⊗ (cos ασx + 2, 3 sin ασ y ) ⊗ (cos ασx + sin ασ y ), M 2 = σx ⊗ σx ⊗ σx and U = R z (α ). 2, 3
That is to say, if we apply R z (α ) on the final state ρ , the value of σx ⊗ σx ⊗ σx measured in the new final state is equivalent to σx ⊗ (cos ασx + sin ασ y ) ⊗ (cos ασx + sin ασ y ) measured in ρ . Thus we obtain E ( A 1 , B 2 , C 2 ) through one measurement σx ⊗ σx ⊗ σx . In order to improve the accuracy of radio frequency (RF) pulses, we used strongly modulating pulse (SMP) techniques [29]. We also maximized the effective gate fidelity by averaging over a weighted distribution of RF field strengths to overcome the inhomogeneity of the RF fields over the sample. The gate fidelity we calculated for every pulse is higher than 0.995 considering the RF field inhomogeneity. The range of the pulse lengths are about from 200 to 700 μs. Fig. 2(b) shows a full state tomography of the standard GHZ state prepared in experiment. The overall fidelity is
F=
Tr(ρth ρexp ) 2 2 (Tr(ρth ) Tr(ρexp ))
= 0.98.
(8)
We took the observers mentioned above (σ n1 , σ n2 ) to do the corresponding measurement on the standard GHZ state. The experimental result is shown in Fig. 3, where the blue squares stand for the experiment results, and the red thick line stands for the theoretical result. Clearly, the experimental results are in excellent agreement with the theoretical expectation of quantum mechanics. 3. Simulating violation of MABK inequality for generalized GHZ states So far, almost all previous Bell experiments were performed on maximal entangled states, such as Bell state and the standard GHZ state. Recently, much work about nonmaximal entangled states have been done [30,31]. In this section, we implemented the simulation the violation of MABK inequality for the generalized GHZ state. In this experiment, we choose the directions of the two measurements for every particle is n1 = (1, 0, 0) and n2 = (0, 1, 0). For these special spin projection measurements, the theoretical result of BMABK for the generalized GHZ states satisfies such a function,
|BMABK | = −4 sin(2θ).
(9)
Fig. 3. Experimental test of the MABK inequality for a standard GHZ state. The red thick line stands for the theoretical expectation, and the blue square stands for the experimental data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
From Eq. (9), one can see that the maximal violation is obtained when θ = π4 just the standard GHZ state. Obviously, the MABK in-
π , 5π ]; in other equality is efficient only in the region of θ ∈ [ 12 12 words, only in such a region the inequality can be violated. We measured a set of generalized GHZ states with particular angles θ . Fig. 4(a) shows the experimental date along with the theoretical expectation.
4. Simulating violation of Chen’s inequality for generalized GHZ states For a three-qubit system, Chen’s inequality can be written as
BChen =
1 2
E ( A1, B 1, C1) + E ( A1, B 2, C1) + E ( A2, B 1, C1)
− E ( A2, B 2, C1) + E ( A1, B 1, C2) + E ( A1, B 2, C2) + E ( A2, B 1, C2) − E ( A2, B 2, C2)
+ E (C 1 ) − E (C 2 ),
(10)
with |BChen | 2 in the LHV model. In experiment, we took the directions of two measurement about A and B as n1 = (1, 0, 0) and n2 = (0, 1, 0). For C , the directions of two measurement were chosen as n1 = (sin α cos(− π4 ), sin α sin(− π4 ), cos α ) and n2 = (sin(π − α ) cos(− π4 ), sin(π − α ) × sin(− π4 ), cos(π − α )), where
C. Ren et al. / Physics Letters A 373 (2009) 4222–4226
(a)
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(b)
Fig. 4. For the generalized GHZ states, (a) the values of BMABK as a function of θ , (b) the values of BChen as a function of θ . The red thick line stands for the theoretical expectation, and the blue square stands for the experiment data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)
α = tan−1 α = tan−1
√
2 tan(2θ) ,
√
0θ
2 tan(2θ) + π ,
π 4
π 4
,
θ
π 2
.
(11)
Then, we obtain BChen as
1/2 BChen = 2 2 sin2 (2θ) + cos2 (2θ) ,
(12)
which tells us that BChen always larger than 2 no matter whatever θ is. It illustrates that, the whole region of the generalized GHZ states can violate the inequality by a set of suitable observation angles. Obviously, Chen’s inequality is more efficient than MABK inequality for generalized GHZ states. The experimental result is shown in Fig. 4(b), which perfectly simulates the violation of Chen’s inequality for the generalize GHZ states. 5. Conclusions In summary, we have investigated the simulation of the violation of Bell-type inequalities, including MABK inequality and Chen’s inequality for the generalized GHZ states in an NMR system. In the range of the generalized GHZ states, Chen’s inequality is more efficient than MABK inequality. The experimental results are well in agreement with the expectation of quantum mechanics. It is necessary to emphasize that, in strict, because NMR qubits are many nuclear spins of atoms bounded together in a single molecule, separated by a few angstroms, the NMR experiment is inherently local. Furthermore, the initial state in NMR is prepared from the thermal equilibrium into a highly mixed state called pseudo-pure state (PPS) [27], where ε ≈ 10−5 is the polarization at room temperature. Braunstein et al. [32] pointed out that any state of the form (5) is not entangled whenever ε (1+212n−1 ) . Hence,
our work neither provides an experimental procedure to prove or disprove non-local effects nor reveals entanglement in NMR experiments at room temperature. Whereas, the meaning is that, when
we experimentally simulate the violation of different Bell-type inequalities for arbitrary generalized three-qubit GHZ states in NMR, the results are excellently in accord with the quantum predictions. It tells us, despite of many existed disputes, NMR may contribute more on some fundamentals of quantum mechanics. As a refined tool and technique for experimentally realizing quantum computation in the last decade, NMR is still contributing to numerous fundamental problems of quantum mechanics. Acknowledgements The authors are grateful to Ya Wang, Ping Zou, Jing Zhu, and Zhaokai Li for their help and interesting comments and discussions. Financial support comes from National Natural Science Foundation of China. References [1] [2] [3] [4] [5]
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