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Bell’s inequalities for three-qubit entangled states with white noise
Chaos, Solitons and Fractals 41 (2009) 1201–1207
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Bell’s inequalities for three-qubit entangled states with white noise Jinho Chang a, Younghun Kwon a,b,* a b
Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea School of Electrical Engineering, University of Central Florida, Orlando, FL 32816-2450, USA
a r t i c l e
i n f o
Article history: Accepted 29 April 2008 Communicated by Prof. Ji-Huan He
a b s t r a c t We consider three-qubit entangled states classified by Acin et al. and evaluate Bell’s inequalities for them when the white noise exists, which may be a real situation for the experiment of the Bell inequality to three-qubit entangled states. We obtain the maximum violation for the Bell inequality in each case and find the condition for exceeding the classical limit. And we observe that even when there would exist quite amount of white noise, some of three-qubit entangled states(for example 2b, 3a, 3b-I, 3b-II and 3b-III types) might show the violation of the Bell inequality. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction In last 20 years there has been considerable progress in understanding quantum information. Shors factoring algorithm [1] and Grovers search algorithm [2] can be thought as major achievements in investigation of quantum information. Actually these discoveries tell us that quantum information may be an essential ingredient in understanding a future computer [3]. Entanglement is one of the most important building blocks in quantum information processing. There exist various ways to understand it. Among them Bell inequality is thought as the best way to understand the nonlocality of it. Recently there have been many experiments which would perform the Bells inequality. One of the important issue in laboratory experiment for Bells inequality is to consider noise [4]. So in this study we consider the Bells inequality of three-qubit entangled state and provide the condition for violation of Bells inequality when white noise exists. In Section 2, we first review the Bell’s inequality for a three-qubit state. In Section 3, we consider three-qubit states classified by Acin et al. and evaluate the Bell’s inequality of them when the white noise exists. And we obtain the maximum value of the Bell’s inequality and the range of the noise for the violation of the Bell’s inequality. In Section 4, we discuss the result and make a conclusion. 2. Three-qubit entangled state with noise and Bell inequality A three-qubit entangled state w with white noise can be written as follows:
q ¼ pjwihwj þ
1p I 8
ð1Þ
Here p is the probability that the state is unaffected by white noise and I the identity matrix. The Bell’s inequality for a three-qubit state was given by Mermin
jbj 2
* Corresponding author. Address: Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea. E-mail address: [email protected] (Y. Kwon). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.054
ð2Þ
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J. Chang, Y. Kwon / Chaos, Solitons and Fractals 41 (2009) 1201–1207
where b ¼ hA1 B2 B3 i þ hB1 A2 B3 i þ hB1 B2 A3 i hA1 A2 A3 i, which implies that the maximum value for the inequality in classical sense is 2 [5,6] (here Ai or Bj denotes the local observable). 3. Bell inequality for the three-qubit entangled state classified by Acin et al. 3.1. The three-qubit entangled state classified by Acin et al. Recently Acin et al. showed that the three-qubit entangled states can be classified by five different ways. Table 1 shows the detailed description of the classification [7]. 3.2. Bell Inequality for the three-qubit entangled states with white noise When we evaluate Bell Inequality for the three-qubit entangled states with white noise, the maximum violation of it can be obtained in 2b state j 000iþ j 111i (known as GHZ state). For the type 2b state with noise, when the detectors are located in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X and B3 ¼ cos /3 Y þ sin /3 X where X, Y and Z denote the Pauli’s spin matrices, the value of Bell inequality is given by bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ. And we can see that the maximum value of b of 2b state is 4, when h1 þ h2 þ h3 ¼ p2 or p2 ; h3 þ /1 þ /2 ¼ p2 or p2 ; h2 þ /1 þ /3 ¼ p2 or p2 p p and h1 þ /2 þ /3 ¼ 2 or 2 . Actually this result agrees with [4]. Fig. 1 shows the value of j b j of 2b state when p ¼ 1; h2 ¼ 0:26242; h3 ¼ 1:79517; /2 ¼ 1:30837 and /3 ¼ 0:22438. For the 2b state, the range of p which can be beyond the classical limit is between 1 and 0.5. Here p denotes the degree of noiseless state(p = 1 means noiseless state). The state with the second maximum violation turns out to be 3a state j 000iþ j 101iþ j 110i. For the state with white noise, we find bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ 13 pðcos h2 cos /1 cos /3 þ cos h3 ð3 cos /1 sin /2 2 sin h1 sin h2 Þ 2 cos h2 sin h1 sin h3 þ cos h1 ð3 cos h2 cos h3 þ 3 cos /2 cos /3 þ 2 sin h2 sin /3 Þ þ 2 cos h3 sin h2 sin /1 þ 2 cos /2 sin h3 sin /1 þ 2 cos /3 sin h1 sin with the locations of detectors such as /2 2 sinðh3 /1 Þ sin /2 þ 2ð sinðh2 /1 Þ þ sinðh1 /2 ÞÞ sin /3 Þ, A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X and B3 ¼ (or A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /3 Y þ sin /3 X cos /2 Z þ sin /2 X and B3 ¼ cos /3 Z þ sin /3 X). The maximum value of b in type 3a state is 3.04596. Fig. 2 shows the value of j b j of the 3a state when p ¼ 1; h2 ¼ 0:62776; h3 ¼ 1:98783; /2 ¼ 1:15377 and /3 ¼ 0:62776. For the 3a state, the range of p which can exceed the classical limit is between 1 and 0.656607. The state with the third maximum violation turns out to be 3b-I or 3b-II or 3b-III state; j 000iþ j 110iþ j 111i or j 000iþ j 100iþ j 111i or j 000iþ j 101iþ j 111i. When each state exists with white noise, we find the same value such as bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ 23 pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ, with the locations of detectors such as A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X and B3 ¼ cos /3 Y þ sin /3 X. The maximum value of b in type 3b-I or 3b-II or 3b-III state is 2.66667. Fig. 3 shows the value of j b j of 3b state when p ¼ 1; h2 ¼ 0:29120; h3 ¼ 1:81241; /2 ¼ 1:27960 and /3 ¼ 0:24161. For the 3b states, the range of p which can be beyond the classical limit is between 1 and 0.749999. For all three-qubit entangled states with white noise, values of b can be found in Appendix. Table 2 shows the summary of them.
Table 1 Equivalence classes of three-qubit (pure) states Types
States
1
(product states)
2a-I 2a-II 2b
j 000iþ j 100iþ j 101i j 000iþ j 100iþ j 110i j 000iþ j 111i
3a 3b-I 3b-II 3b-III
j 000iþ j 101iþ j 110i j 000iþ j 110iþ j 111i j 000iþ j 100iþ j 111i j 000iþ j 101iþ j 111i
4a 4b-I 4b-II 4c
j 000iþ j 100iþ j 101iþ j 110i j 000iþ j 100iþ j 110iþ j 111i j 000iþ j 100iþ j 101iþ j 111i j 000iþ j 101iþ j 110iþ j 111i
5
j 000iþ j 100iþ j 101iþ j 110iþ j 111i
J. Chang, Y. Kwon / Chaos, Solitons and Fractals 41 (2009) 1201–1207
Fig. 1. j b j of 2b state when p ¼ 1; h2 ¼ 0:26242; h3 ¼ 1:79517; /2 ¼ 1:30837 and /3 ¼ 0:22438.
Fig. 2. j b j of the 3a state when p ¼ 1; h2 ¼ 0:62776; h3 ¼ 1:98783; /2 ¼ 1:15377 and /3 ¼ 0:62776.
Fig. 3. j b j of 3b state when p ¼ 1; h2 ¼ 0:29120; h3 ¼ 1:81241; /2 ¼ 1:27960 and /3 ¼ 0:24161.
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Table 2 Maximum Bell violation of three-qubit entangled states and range of p for exceeding the classical limit Types
Maximum violation
Range of p exceeding the classical limit
2a-I 2a-II 2b
2.4037 2.4037 4
1 6 p 6 0:832051 1 6 p 6 0:832051 1 6 p 6 0:5
3a 3b-I 3b-II 3b-III
3.04596 2.66667 2.66667 2.66667
1 6 p 6 0:656607 1 6 p 6 0:749999 1 6 p 6 0:749999 1 6 p 6 0:749999
4a 4b-I 4b-II 4c
2.62568 2.41421 2.41421 2.55313
1 6 p 6 0:761707 1 6 p 6 0:828428 1 6 p 6 0:828428 1 6 p 6 0:783352
5
2.2432
1 6 p 6 0:891583
4. Concluding remark In this paper, We considered the three-qubit entangled state classified by Acin et al. and evaluated Bell’s inequalities when these states with white noise would be used, which would be a real situation for the experiment of the Bell inequality to three-qubit entangled states. And we obtained the maximum violation for the Bell inequality in each case and found the condition for exceeding the classical limit. We found that even when there would exist quite amount of white noise, some of three-qubit entangled states(specially 2b, 3a, 3b-I, 3b-II and 3b-III) might show the violation of the Bell inequality. Meanwhile it was reported that the Bell inequalities in E-infinity theory [8] may be violated [9]. Ahmed argued that the nonlocal network structure of E-infinity theory could guarantee the violation of the Bell inequalities. The E-infinity approach to quantum mechanics seems to be intriguing and is worth investigating further. Acknowledgements Y. Kwon was supported by the research fund of Hanyang University (HY-2006-I). Y. Kwon thanks Prof. Qu at UCF for his hospitality. Appendix A. Bell Inequalities for the type 2 states with white noise A.1. Bell Inequality for the type 2a-I state with white noise In this subsection, we will evaluate Bell’s inequality when we use the type 2a-I state with white noise. When the local observables are located in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pðcos h2 cos /1 cos /3 þ 2 cos /2 cos /3 sin h1 2 cos /1 cos /2 sin h3 2 cos h2 sin h1 sin h3 3 þ 2 cos h2 cos /3 sin /1 þ 2 cos /2 sin h3 sin /1 þ cos h3 ð2 cos h2 sin h1 þ cos /2 ðcos /1 þ 2 sin /1 ÞÞ þ cos h1 ð cos h2 ðcos h3 2 sin h3 Þ þ cos /2 ðcos /3 2 sin /3 ÞÞ 2 cos h2 cos /1 sin /3 þ 2 cos /2 sin h1 sin /3 þ 2 cos h2 sin /1 sin /3 Þ
The maximum value of the above is 2.4037. The maximum value of p exceeding the classical limit is 0.83205. A.2. Bell Inequality for the type 2a-II state with white noise In this subsection, we evaluate the Bell inequality when we use the type 2a-II state with white noise. When the measurement may be done in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pð cos /3 ð3 cosðh2 /1 Þ þ cosðh2 þ /1 Þ 3 cosðh1 /2 Þ þ cosðh1 þ /2 Þ þ cosðh2 /1 Þ 6 4 sinðh1 /2 Þ þ cos h3 ð3 cosðh1 h2 Þ þ cosðh1 þ h2 Þ 4 sinðh1 h2 Þ þ 2 cos /1 ðcos /2 2 sin /2 Þ þ 4 sin /1 ðcos /2 þ sin /2 ÞÞÞÞ
The maximum value of the above is 2.4037. The maximum value of p exceeding the classical limit may be 0.83205.
J. Chang, Y. Kwon / Chaos, Solitons and Fractals 41 (2009) 1201–1207
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A.3. Bell Inequality for the type 2b state with white noise In this subsection, we evaluate the Bell inequality when we use the type 2b state with white noise. When the local observables are located in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X, and B3 ¼ cos /3 Y þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ The maximum value of the above is 4 and the maximum value of p exceeding the classical limitation can be 0.5. Appendix B. Bell Inequalities for the type 3 states with white noise B.1. Bell Inequality for the type 3a state with white noise In this subsection, we evaluate the Bell inequality when we use the type 3a state with white noise. Having the detectors located in A1 ¼ cos h1 Y þ sinh1 X; A2 ¼ cos h2 Y þ sinh2 X; A3 ¼ cos h3 Y þ sinh3 X; B1 ¼ cos /1 Y þ sin /1 X;B2 ¼ cos /2 Y þ sin/2 X, and B3 ¼ cos /3 Y þ sin/3 X or A1 ¼ cos h1 Z þ sinh1 X;A2 ¼ cos h2 Z þ sinh2 X;A3 ¼ cos h3 Z þ sinh3 X; B1 ¼ cos /1 Z þ sin /1 X;B2 ¼ cos /2 Zþ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pðcos h2 cos /1 cos /3 þ cos h3 ð3 cos /1 sin /2 2 sin h1 sin h2 Þ 2 cos h2 sin h1 sin h3 3 þ cos h1 ð3 cos h2 cos h3 þ 3 cos /2 cos /3 þ 2 sin h2 sin /3 Þ þ 2 cos h3 sin h2 sin /1 þ 2 cos /2 sin h3 sin /1 þ 2 cos /3 sin h1 sin /2 2 sinðh3 /1 Þ sin /2 þ 2ð sinðh2 /1 Þ þ sinðh1 /2 ÞÞ sin /3 Þ
The maximum value of the above is 3.04596. The maximum value of p exceeding the classical limit can be 0.656607. B.2. Bell Inequality for the type 3b-I state with white noise In this subsection, we will evaluate the Bell inequality when we use the type 3b-I state with white noise. When the measurement may be done in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X, and B3 ¼ cos /3 Y þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
2 pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ 3
The maximum value of the above is 2.66667 and the maximum value of p exceeding the classical limit can be 0.749999. B.3. Bell Inequality for the type 3b-II state with white noise In this subsection, we evaluate the Bell inequality when we use the type 3b-II state with white noise. Having the measurement be done in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X, and B3 ¼ cos /3 Y þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
2 pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ 3
The maximum value of the above is 2.66667. The maximum value of p exceeding the classical limit can be 0.749999. B.4. Bell Inequality for the type 3b-III state with white noise In this subsection, we evaluate the Bell inequality when we use the type 3b-III state with white noise. When the measurement can be done in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X, and B3 ¼ cos /3 Y þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
2 pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ 3
The maximum value of the above is 2.66667. The maximum value of p exceeding the classical limit will be 0.749999. Appendix C. Bell Inequalities for the type 4 states with white noise C.1. Bell Inequality for the type 4a state with white noise
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J. Chang, Y. Kwon / Chaos, Solitons and Fractals 41 (2009) 1201–1207
In this subsection, we evaluate the Bell inequality when we use the type 4a state with white noise. Having the detectors located in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
1 bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ pðcosðh1 h2 h3 Þ cosðh3 /1 þ /2 Þ þ cos /1 cos /3 sin h2 cos /3 sin h2 sin /1 2 þ cos h3 cos /1 sin /2 cos /3 sin h1 sin /2 cos h2 ðcosð/1 /3 Þ sinðh1 h3 Þ þ sinð/1 /3 ÞÞ þ cos /1 sin h2 sin h3 þ cos /2 ðsinðh3 /1 Þ sin h1 ðcos /3 þ sin /3 ÞÞ cos h1 ðcosð/2 þ /3 Þ þ cos h3 sin h2 sinð/2 þ /3 ÞÞÞ The maximum value of the above is 2.62568. The maximum value of p exceeding the classical limit can be 0.761707. C.2. Bell Inequality for the type 4b-I state with white noise In this subsection, we evaluate the Bell inequality when we use the type 4b-I state with white noise. When the measurement may be done in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pð cosðh1 h2 h3 Þ þ cosðh1 þ h2 h3 Þ cosðh3 /1 /2 Þ þ cosðh3 /1 þ /2 Þ 4 cosðh2 þ /1 /3 Þ þ cosðh1 /2 /3 Þ cosðh1 þ /2 /3 Þ þ cosðh2 /1 þ /3 Þ 2 sinðh1 h2 þ h3 Þ þ 2 sinðh3 þ /1 /2 Þ 2 sinðh2 /1 /3 Þ þ 2 sinðh1 /2 þ /3 ÞÞ
The maximum value of the above is 2.41421 and the maximum value of p exceeding the classical limit can be 0.828428. C.3. Bell Inequality for the type 4b-II state with white noise In this subsection, we evaluate the Bell inequality when we use the type 4b-II state with white noise. Having the measurement be done in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pð cosðh1 h2 h3 Þ þ cosðh1 h2 þ h3 Þ cosðh3 þ /1 /2 Þ þ cosðh3 /1 þ /2 Þ 4 2ðsinðh1 þ h2 h3 Þ þ sinðh3 /1 /2 Þ cos /3 ðsinðh2 þ /1 Þ þ sinðh1 þ /2 ÞÞÞ 2ðcosðh2 þ /1 Þ þ cosðh1 þ /2 Þ þ sinðh2 /1 Þ cosðh1 /2 ÞÞ sin /3 Þ
The maximum value of the above is 2.41421. The maximum value of p exceeding the classical limit can be 0.828428. C.4. Bell Inequality for the type 4c state with white noise In this subsection, we evaluate the Bell inequality when we use the type 4c state with white noise. When the measurement can be done in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pð cosðh1 h2 h3 Þ þ sinðh3 ð cosðh1 h2 Þ þ cosð/1 /2 Þ þ sinð/1 /2 ÞÞÞ 2 þ cosð/3 ðcos h1 cos /2 þ cos /1 ðcos h2 þ sin h2 Þ þ sin h2 sin /1 þ ðcos h1 þ sin h1 Þ sin /2 Þ þ cos h3 ðcos /1 cos /2 cos h1 sin h2 þ ðcos /1 þ sin /Þ sin /2 Þ þ ðcosðh2 /1 Þ þ cosðh1 /2 Þ sinðh2 /1 Þ þ sinðh1 /2 ÞÞ sin /3 ÞÞ
The maximum value of the above is 2.55313. The maximum value of p exceeding the classical limit can be 0.783352. Appendix D. Bell Inequality for the type 5 state with white noise In this subsection, we evaluate the Bell inequality when we use the type 5 state with white noise. When the detectors are located in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
J. Chang, Y. Kwon / Chaos, Solitons and Fractals 41 (2009) 1201–1207
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1207
1 pðcos h2 cos /1 cos /3 þ 2 cos /2 cos /3 sin h1 2 cos h2 sin h1 sin h3 2 cos h1 sin h2 sin h3 5 þ 2 cos h2 cos /3 sin /1 þ 2 cos /3 sin h2 sin /1 þ 2 cos /2 sin h3 sin /1 þ 2 cos /3 sin h1 sin /2 4 cos /1 sin h3 sin /2 2 sin h1 ðcos h2 þ sin h2 Þ þ 2 sin /1 ðcos /2 þ sin /2 Þ þ 2ð2 cos /1 sin h2 þ ðcos h2 þ sin h2Þ sin /1 þ sin h1 ðcos /2 þ sin /2 ÞÞ sin /3 þ cos h1 ð cos h2 cos h3 þ cos /2 cos /3 þ 4 sin h2 sin h3 4 sin /2 sin /3 ÞÞ
The maximum value of the above is 2.2432. The maximum value of p exceeding the classical limit can be 0.891583. References [1] Shor P. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Sci Stat Comput Rev 1999;41:3031484. [2] Grover LK. Quantum mechanics helps in searching for a needle in a haystack. Phys Rev Lett 1997;79:325. [3] Kwon Y. Quantum mechanism of biological search. Chaos, Solitons & Fractals 2007;34:1037. [4] Cabello A, Feito A, Lamas-Linares A. Bells inequalities with realistic noise for polarization-entangled photons. Phys Rev A 2005;72:052112. [5] Mermin ND. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys Rev Lett 1990;65:1838. [6] Klyshko DN. The Bell and GHZ theorems: a possible three-photon interference experiment and the question of nonlocality. Phys Lett A 1993;172:399. [7] Acin A, Andrianov A, Costa L, Jane E, Latorre JI, Tarrach R. Generalized Schmidt decomposition and classification of three-quantum-bit states. Phys Rev Lett 2000;85:1560. [8] El Naschie MS. The concepts of E-infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22:495. [9] Ahmed N. Cantorian small world, Machs principle, and the universal mass network. Chaos, Solitons & Fractals 2004;21:773.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Bell’s inequalities for three-qubit entangled states with white noise Jinho Chang a, Younghun Kwon a,b,* a b
Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea School of Electrical Engineering, University of Central Florida, Orlando, FL 32816-2450, USA
a r t i c l e
i n f o
Article history: Accepted 29 April 2008 Communicated by Prof. Ji-Huan He
a b s t r a c t We consider three-qubit entangled states classified by Acin et al. and evaluate Bell’s inequalities for them when the white noise exists, which may be a real situation for the experiment of the Bell inequality to three-qubit entangled states. We obtain the maximum violation for the Bell inequality in each case and find the condition for exceeding the classical limit. And we observe that even when there would exist quite amount of white noise, some of three-qubit entangled states(for example 2b, 3a, 3b-I, 3b-II and 3b-III types) might show the violation of the Bell inequality. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction In last 20 years there has been considerable progress in understanding quantum information. Shors factoring algorithm [1] and Grovers search algorithm [2] can be thought as major achievements in investigation of quantum information. Actually these discoveries tell us that quantum information may be an essential ingredient in understanding a future computer [3]. Entanglement is one of the most important building blocks in quantum information processing. There exist various ways to understand it. Among them Bell inequality is thought as the best way to understand the nonlocality of it. Recently there have been many experiments which would perform the Bells inequality. One of the important issue in laboratory experiment for Bells inequality is to consider noise [4]. So in this study we consider the Bells inequality of three-qubit entangled state and provide the condition for violation of Bells inequality when white noise exists. In Section 2, we first review the Bell’s inequality for a three-qubit state. In Section 3, we consider three-qubit states classified by Acin et al. and evaluate the Bell’s inequality of them when the white noise exists. And we obtain the maximum value of the Bell’s inequality and the range of the noise for the violation of the Bell’s inequality. In Section 4, we discuss the result and make a conclusion. 2. Three-qubit entangled state with noise and Bell inequality A three-qubit entangled state w with white noise can be written as follows:
q ¼ pjwihwj þ
1p I 8
ð1Þ
Here p is the probability that the state is unaffected by white noise and I the identity matrix. The Bell’s inequality for a three-qubit state was given by Mermin
jbj 2
* Corresponding author. Address: Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea. E-mail address: [email protected] (Y. Kwon). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.054
ð2Þ
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where b ¼ hA1 B2 B3 i þ hB1 A2 B3 i þ hB1 B2 A3 i hA1 A2 A3 i, which implies that the maximum value for the inequality in classical sense is 2 [5,6] (here Ai or Bj denotes the local observable). 3. Bell inequality for the three-qubit entangled state classified by Acin et al. 3.1. The three-qubit entangled state classified by Acin et al. Recently Acin et al. showed that the three-qubit entangled states can be classified by five different ways. Table 1 shows the detailed description of the classification [7]. 3.2. Bell Inequality for the three-qubit entangled states with white noise When we evaluate Bell Inequality for the three-qubit entangled states with white noise, the maximum violation of it can be obtained in 2b state j 000iþ j 111i (known as GHZ state). For the type 2b state with noise, when the detectors are located in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X and B3 ¼ cos /3 Y þ sin /3 X where X, Y and Z denote the Pauli’s spin matrices, the value of Bell inequality is given by bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ. And we can see that the maximum value of b of 2b state is 4, when h1 þ h2 þ h3 ¼ p2 or p2 ; h3 þ /1 þ /2 ¼ p2 or p2 ; h2 þ /1 þ /3 ¼ p2 or p2 p p and h1 þ /2 þ /3 ¼ 2 or 2 . Actually this result agrees with [4]. Fig. 1 shows the value of j b j of 2b state when p ¼ 1; h2 ¼ 0:26242; h3 ¼ 1:79517; /2 ¼ 1:30837 and /3 ¼ 0:22438. For the 2b state, the range of p which can be beyond the classical limit is between 1 and 0.5. Here p denotes the degree of noiseless state(p = 1 means noiseless state). The state with the second maximum violation turns out to be 3a state j 000iþ j 101iþ j 110i. For the state with white noise, we find bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ 13 pðcos h2 cos /1 cos /3 þ cos h3 ð3 cos /1 sin /2 2 sin h1 sin h2 Þ 2 cos h2 sin h1 sin h3 þ cos h1 ð3 cos h2 cos h3 þ 3 cos /2 cos /3 þ 2 sin h2 sin /3 Þ þ 2 cos h3 sin h2 sin /1 þ 2 cos /2 sin h3 sin /1 þ 2 cos /3 sin h1 sin with the locations of detectors such as /2 2 sinðh3 /1 Þ sin /2 þ 2ð sinðh2 /1 Þ þ sinðh1 /2 ÞÞ sin /3 Þ, A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X and B3 ¼ (or A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /3 Y þ sin /3 X cos /2 Z þ sin /2 X and B3 ¼ cos /3 Z þ sin /3 X). The maximum value of b in type 3a state is 3.04596. Fig. 2 shows the value of j b j of the 3a state when p ¼ 1; h2 ¼ 0:62776; h3 ¼ 1:98783; /2 ¼ 1:15377 and /3 ¼ 0:62776. For the 3a state, the range of p which can exceed the classical limit is between 1 and 0.656607. The state with the third maximum violation turns out to be 3b-I or 3b-II or 3b-III state; j 000iþ j 110iþ j 111i or j 000iþ j 100iþ j 111i or j 000iþ j 101iþ j 111i. When each state exists with white noise, we find the same value such as bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ 23 pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ, with the locations of detectors such as A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X and B3 ¼ cos /3 Y þ sin /3 X. The maximum value of b in type 3b-I or 3b-II or 3b-III state is 2.66667. Fig. 3 shows the value of j b j of 3b state when p ¼ 1; h2 ¼ 0:29120; h3 ¼ 1:81241; /2 ¼ 1:27960 and /3 ¼ 0:24161. For the 3b states, the range of p which can be beyond the classical limit is between 1 and 0.749999. For all three-qubit entangled states with white noise, values of b can be found in Appendix. Table 2 shows the summary of them.
Table 1 Equivalence classes of three-qubit (pure) states Types
States
1
(product states)
2a-I 2a-II 2b
j 000iþ j 100iþ j 101i j 000iþ j 100iþ j 110i j 000iþ j 111i
3a 3b-I 3b-II 3b-III
j 000iþ j 101iþ j 110i j 000iþ j 110iþ j 111i j 000iþ j 100iþ j 111i j 000iþ j 101iþ j 111i
4a 4b-I 4b-II 4c
j 000iþ j 100iþ j 101iþ j 110i j 000iþ j 100iþ j 110iþ j 111i j 000iþ j 100iþ j 101iþ j 111i j 000iþ j 101iþ j 110iþ j 111i
5
j 000iþ j 100iþ j 101iþ j 110iþ j 111i
J. Chang, Y. Kwon / Chaos, Solitons and Fractals 41 (2009) 1201–1207
Fig. 1. j b j of 2b state when p ¼ 1; h2 ¼ 0:26242; h3 ¼ 1:79517; /2 ¼ 1:30837 and /3 ¼ 0:22438.
Fig. 2. j b j of the 3a state when p ¼ 1; h2 ¼ 0:62776; h3 ¼ 1:98783; /2 ¼ 1:15377 and /3 ¼ 0:62776.
Fig. 3. j b j of 3b state when p ¼ 1; h2 ¼ 0:29120; h3 ¼ 1:81241; /2 ¼ 1:27960 and /3 ¼ 0:24161.
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Table 2 Maximum Bell violation of three-qubit entangled states and range of p for exceeding the classical limit Types
Maximum violation
Range of p exceeding the classical limit
2a-I 2a-II 2b
2.4037 2.4037 4
1 6 p 6 0:832051 1 6 p 6 0:832051 1 6 p 6 0:5
3a 3b-I 3b-II 3b-III
3.04596 2.66667 2.66667 2.66667
1 6 p 6 0:656607 1 6 p 6 0:749999 1 6 p 6 0:749999 1 6 p 6 0:749999
4a 4b-I 4b-II 4c
2.62568 2.41421 2.41421 2.55313
1 6 p 6 0:761707 1 6 p 6 0:828428 1 6 p 6 0:828428 1 6 p 6 0:783352
5
2.2432
1 6 p 6 0:891583
4. Concluding remark In this paper, We considered the three-qubit entangled state classified by Acin et al. and evaluated Bell’s inequalities when these states with white noise would be used, which would be a real situation for the experiment of the Bell inequality to three-qubit entangled states. And we obtained the maximum violation for the Bell inequality in each case and found the condition for exceeding the classical limit. We found that even when there would exist quite amount of white noise, some of three-qubit entangled states(specially 2b, 3a, 3b-I, 3b-II and 3b-III) might show the violation of the Bell inequality. Meanwhile it was reported that the Bell inequalities in E-infinity theory [8] may be violated [9]. Ahmed argued that the nonlocal network structure of E-infinity theory could guarantee the violation of the Bell inequalities. The E-infinity approach to quantum mechanics seems to be intriguing and is worth investigating further. Acknowledgements Y. Kwon was supported by the research fund of Hanyang University (HY-2006-I). Y. Kwon thanks Prof. Qu at UCF for his hospitality. Appendix A. Bell Inequalities for the type 2 states with white noise A.1. Bell Inequality for the type 2a-I state with white noise In this subsection, we will evaluate Bell’s inequality when we use the type 2a-I state with white noise. When the local observables are located in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pðcos h2 cos /1 cos /3 þ 2 cos /2 cos /3 sin h1 2 cos /1 cos /2 sin h3 2 cos h2 sin h1 sin h3 3 þ 2 cos h2 cos /3 sin /1 þ 2 cos /2 sin h3 sin /1 þ cos h3 ð2 cos h2 sin h1 þ cos /2 ðcos /1 þ 2 sin /1 ÞÞ þ cos h1 ð cos h2 ðcos h3 2 sin h3 Þ þ cos /2 ðcos /3 2 sin /3 ÞÞ 2 cos h2 cos /1 sin /3 þ 2 cos /2 sin h1 sin /3 þ 2 cos h2 sin /1 sin /3 Þ
The maximum value of the above is 2.4037. The maximum value of p exceeding the classical limit is 0.83205. A.2. Bell Inequality for the type 2a-II state with white noise In this subsection, we evaluate the Bell inequality when we use the type 2a-II state with white noise. When the measurement may be done in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pð cos /3 ð3 cosðh2 /1 Þ þ cosðh2 þ /1 Þ 3 cosðh1 /2 Þ þ cosðh1 þ /2 Þ þ cosðh2 /1 Þ 6 4 sinðh1 /2 Þ þ cos h3 ð3 cosðh1 h2 Þ þ cosðh1 þ h2 Þ 4 sinðh1 h2 Þ þ 2 cos /1 ðcos /2 2 sin /2 Þ þ 4 sin /1 ðcos /2 þ sin /2 ÞÞÞÞ
The maximum value of the above is 2.4037. The maximum value of p exceeding the classical limit may be 0.83205.
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A.3. Bell Inequality for the type 2b state with white noise In this subsection, we evaluate the Bell inequality when we use the type 2b state with white noise. When the local observables are located in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X, and B3 ¼ cos /3 Y þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ The maximum value of the above is 4 and the maximum value of p exceeding the classical limitation can be 0.5. Appendix B. Bell Inequalities for the type 3 states with white noise B.1. Bell Inequality for the type 3a state with white noise In this subsection, we evaluate the Bell inequality when we use the type 3a state with white noise. Having the detectors located in A1 ¼ cos h1 Y þ sinh1 X; A2 ¼ cos h2 Y þ sinh2 X; A3 ¼ cos h3 Y þ sinh3 X; B1 ¼ cos /1 Y þ sin /1 X;B2 ¼ cos /2 Y þ sin/2 X, and B3 ¼ cos /3 Y þ sin/3 X or A1 ¼ cos h1 Z þ sinh1 X;A2 ¼ cos h2 Z þ sinh2 X;A3 ¼ cos h3 Z þ sinh3 X; B1 ¼ cos /1 Z þ sin /1 X;B2 ¼ cos /2 Zþ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pðcos h2 cos /1 cos /3 þ cos h3 ð3 cos /1 sin /2 2 sin h1 sin h2 Þ 2 cos h2 sin h1 sin h3 3 þ cos h1 ð3 cos h2 cos h3 þ 3 cos /2 cos /3 þ 2 sin h2 sin /3 Þ þ 2 cos h3 sin h2 sin /1 þ 2 cos /2 sin h3 sin /1 þ 2 cos /3 sin h1 sin /2 2 sinðh3 /1 Þ sin /2 þ 2ð sinðh2 /1 Þ þ sinðh1 /2 ÞÞ sin /3 Þ
The maximum value of the above is 3.04596. The maximum value of p exceeding the classical limit can be 0.656607. B.2. Bell Inequality for the type 3b-I state with white noise In this subsection, we will evaluate the Bell inequality when we use the type 3b-I state with white noise. When the measurement may be done in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X, and B3 ¼ cos /3 Y þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
2 pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ 3
The maximum value of the above is 2.66667 and the maximum value of p exceeding the classical limit can be 0.749999. B.3. Bell Inequality for the type 3b-II state with white noise In this subsection, we evaluate the Bell inequality when we use the type 3b-II state with white noise. Having the measurement be done in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X, and B3 ¼ cos /3 Y þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
2 pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ 3
The maximum value of the above is 2.66667. The maximum value of p exceeding the classical limit can be 0.749999. B.4. Bell Inequality for the type 3b-III state with white noise In this subsection, we evaluate the Bell inequality when we use the type 3b-III state with white noise. When the measurement can be done in A1 ¼ cos h1 Y þ sin h1 X; A2 ¼ cos h2 Y þ sin h2 X; A3 ¼ cos h3 Y þ sin h3 X; B1 ¼ cos /1 Y þ sin /1 X; B2 ¼ cos /2 Y þ sin /2 X, and B3 ¼ cos /3 Y þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
2 pðsinðh1 þ h2 þ h3 Þ sinðh3 þ /1 þ /2 Þ sinðh2 þ /1 þ /3 Þ sinðh1 þ /2 þ /3 ÞÞ 3
The maximum value of the above is 2.66667. The maximum value of p exceeding the classical limit will be 0.749999. Appendix C. Bell Inequalities for the type 4 states with white noise C.1. Bell Inequality for the type 4a state with white noise
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In this subsection, we evaluate the Bell inequality when we use the type 4a state with white noise. Having the detectors located in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
1 bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼ pðcosðh1 h2 h3 Þ cosðh3 /1 þ /2 Þ þ cos /1 cos /3 sin h2 cos /3 sin h2 sin /1 2 þ cos h3 cos /1 sin /2 cos /3 sin h1 sin /2 cos h2 ðcosð/1 /3 Þ sinðh1 h3 Þ þ sinð/1 /3 ÞÞ þ cos /1 sin h2 sin h3 þ cos /2 ðsinðh3 /1 Þ sin h1 ðcos /3 þ sin /3 ÞÞ cos h1 ðcosð/2 þ /3 Þ þ cos h3 sin h2 sinð/2 þ /3 ÞÞÞ The maximum value of the above is 2.62568. The maximum value of p exceeding the classical limit can be 0.761707. C.2. Bell Inequality for the type 4b-I state with white noise In this subsection, we evaluate the Bell inequality when we use the type 4b-I state with white noise. When the measurement may be done in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pð cosðh1 h2 h3 Þ þ cosðh1 þ h2 h3 Þ cosðh3 /1 /2 Þ þ cosðh3 /1 þ /2 Þ 4 cosðh2 þ /1 /3 Þ þ cosðh1 /2 /3 Þ cosðh1 þ /2 /3 Þ þ cosðh2 /1 þ /3 Þ 2 sinðh1 h2 þ h3 Þ þ 2 sinðh3 þ /1 /2 Þ 2 sinðh2 /1 /3 Þ þ 2 sinðh1 /2 þ /3 ÞÞ
The maximum value of the above is 2.41421 and the maximum value of p exceeding the classical limit can be 0.828428. C.3. Bell Inequality for the type 4b-II state with white noise In this subsection, we evaluate the Bell inequality when we use the type 4b-II state with white noise. Having the measurement be done in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pð cosðh1 h2 h3 Þ þ cosðh1 h2 þ h3 Þ cosðh3 þ /1 /2 Þ þ cosðh3 /1 þ /2 Þ 4 2ðsinðh1 þ h2 h3 Þ þ sinðh3 /1 /2 Þ cos /3 ðsinðh2 þ /1 Þ þ sinðh1 þ /2 ÞÞÞ 2ðcosðh2 þ /1 Þ þ cosðh1 þ /2 Þ þ sinðh2 /1 Þ cosðh1 /2 ÞÞ sin /3 Þ
The maximum value of the above is 2.41421. The maximum value of p exceeding the classical limit can be 0.828428. C.4. Bell Inequality for the type 4c state with white noise In this subsection, we evaluate the Bell inequality when we use the type 4c state with white noise. When the measurement can be done in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1 pð cosðh1 h2 h3 Þ þ sinðh3 ð cosðh1 h2 Þ þ cosð/1 /2 Þ þ sinð/1 /2 ÞÞÞ 2 þ cosð/3 ðcos h1 cos /2 þ cos /1 ðcos h2 þ sin h2 Þ þ sin h2 sin /1 þ ðcos h1 þ sin h1 Þ sin /2 Þ þ cos h3 ðcos /1 cos /2 cos h1 sin h2 þ ðcos /1 þ sin /Þ sin /2 Þ þ ðcosðh2 /1 Þ þ cosðh1 /2 Þ sinðh2 /1 Þ þ sinðh1 /2 ÞÞ sin /3 ÞÞ
The maximum value of the above is 2.55313. The maximum value of p exceeding the classical limit can be 0.783352. Appendix D. Bell Inequality for the type 5 state with white noise In this subsection, we evaluate the Bell inequality when we use the type 5 state with white noise. When the detectors are located in A1 ¼ cos h1 Z þ sin h1 X; A2 ¼ cos h2 Z þ sin h2 X; A3 ¼ cos h3 Z þ sin h3 X; B1 ¼ cos /1 Z þ sin /1 X; B2 ¼ cos /2 Z þ sin /2 X, and B3 ¼ cos /3 Z þ sin /3 X, we can obtain the following result for Bell inequality:
J. Chang, Y. Kwon / Chaos, Solitons and Fractals 41 (2009) 1201–1207
bðp; h1 ; h2 ; h3 ; /1 ; /2 ; /3 Þ ¼
1207
1 pðcos h2 cos /1 cos /3 þ 2 cos /2 cos /3 sin h1 2 cos h2 sin h1 sin h3 2 cos h1 sin h2 sin h3 5 þ 2 cos h2 cos /3 sin /1 þ 2 cos /3 sin h2 sin /1 þ 2 cos /2 sin h3 sin /1 þ 2 cos /3 sin h1 sin /2 4 cos /1 sin h3 sin /2 2 sin h1 ðcos h2 þ sin h2 Þ þ 2 sin /1 ðcos /2 þ sin /2 Þ þ 2ð2 cos /1 sin h2 þ ðcos h2 þ sin h2Þ sin /1 þ sin h1 ðcos /2 þ sin /2 ÞÞ sin /3 þ cos h1 ð cos h2 cos h3 þ cos /2 cos /3 þ 4 sin h2 sin h3 4 sin /2 sin /3 ÞÞ
The maximum value of the above is 2.2432. The maximum value of p exceeding the classical limit can be 0.891583. References [1] Shor P. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Sci Stat Comput Rev 1999;41:3031484. [2] Grover LK. Quantum mechanics helps in searching for a needle in a haystack. Phys Rev Lett 1997;79:325. [3] Kwon Y. Quantum mechanism of biological search. Chaos, Solitons & Fractals 2007;34:1037. [4] Cabello A, Feito A, Lamas-Linares A. Bells inequalities with realistic noise for polarization-entangled photons. Phys Rev A 2005;72:052112. [5] Mermin ND. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys Rev Lett 1990;65:1838. [6] Klyshko DN. The Bell and GHZ theorems: a possible three-photon interference experiment and the question of nonlocality. Phys Lett A 1993;172:399. [7] Acin A, Andrianov A, Costa L, Jane E, Latorre JI, Tarrach R. Generalized Schmidt decomposition and classification of three-quantum-bit states. Phys Rev Lett 2000;85:1560. [8] El Naschie MS. The concepts of E-infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22:495. [9] Ahmed N. Cantorian small world, Machs principle, and the universal mass network. Chaos, Solitons & Fractals 2004;21:773.