Article Description

(1)

International Journal of Engineering Inventions

ISSN: 2278-7461, www.ijeijournal.com

Volume 1, Issue 6 (October2012) PP: 55-59

Stronger Violation of Local Realism for Three Qubit Systems with

a New Bell-Type Inequality

Fatemeh Shirdel

and Hossein Movahhedian

Department of Physics, Shahrood University of Technology, Shahrood, Iran

Abstract––In this paper we show the new Bell-type inequality for three qubit systems (i.e a tripartite system with two measurements in each side and two outputs for each measurement). This inequality is violated by quantum theory with a factor violation of 3.5 tolerating 0.5 fraction of white noise. Our inequality includes only 19 different joint probabilities whereas in other works it is much more than this. We will also show violation amount of this inequality is 2.5. Note that the maximum violation factor and amount of violation in available inequalities are both 2, also t h e w h i t e n o i s e t o l e r a n c e o f o u r i n e q u a l i t y i s i n a g r e e m e n t w i t h m a x i m u m v a l u e c a l c u l a t e d s o f a r .

K e y w o r d s – – n o n l o c a l i t y ; B e l l i n e q u a l i t y ; a m o u n t o f v i o l a t i o n ; v i o l a t i o n f a c t o r ; w h i t e n o i s e t o l e r a n c e .

I. INTRODUCTION

Quantum mechanics cannot be described by local hidden variable theories. In quantum theory, the tests of local realism are based on Bell-type inequalities. Original Bell inequality d i d n o t h a v e a n y c a p a b i l i t i e s t o b e s t u d i e d e mp i r i c a l l y i n t h e l a b o r a t o r i e s [1]. Since then, many attempts have been made to obtain Bell-type inequalities which are violated by a higher factor so that it would be experimentally easy to test the non-locality feature of quantum theory. A s t h e n o n - l o c a l i t y f e a t u r e o f q u a n t u m t h e o r y i s i n t e n s i v e l y u s e d i n q u a n t u m i n f o r ma t i o n , B e l l t yp e i n e q u a l i t i e s h a v e r e c e i v e d mo r e a t t e n t i o n i n r e c e n t ye a r s [ 2 ] .

A variant of Bell-type inequality, being more general and more useful for an experimental test, was derived by Clauser, Horne, Shimony and Holt (CHSH) [3]. In 1982, Aspect, et al. performed a verification experiment for a possible violation of Bell inequalities in CHSH form [4]. CHSH inequality is derived from two particles. N-particle generalizations of the CHSH inequality were proposed by Mermin, and later developed by Ardehali, Bleinski and Khyshko and others. See [5-8]. Quantum predictions can violate such inequalities by an amount increasing exponentially with the particle number [9]. for N=3, maximal violation factor (i.e. the ratio of the value of Bell-type inequalities in quantum theory to their value in local theory) and maximum amount of violation (i.e. the difference between the value of Bell expression according to quantum theory and its extermum value according to local theory) are both 2. In these systems, white noise tolerance is 0.5 [10]. In this paper, we consider a three qubit system (i.e a three particles system with two dichotomic measurements on each particle, and two outcomes for each measurement). Then for local theory, we introduce a new Bell type expression for this system based on numerical calculations. We will show that the violation factor and the amount of violation of this inequality exceed those of available inequalities [5-9], while its white noise tolerance agrees with the previous results.

II. THREE QUBIT SYSTEMS

To verify the non-locality in quantum theory We would introduce a three qubit system consisting of particles

A

₁

,

A

₂

and

A

₃ in which two dichotomic measurements

B

₁i,

B

₂j and

B

k₃ can be performed on particles

A

₁

,

A

₂ and

A

₃

respectively, where

i

,

j

,

k



{

1 ,

2 }

. The outcomes of these measurements are denoted

b

₁i

,

b

₂j and

b

k₃ which can take the values 0, 1.

T h e l o c a l r e a l i s m a s s u me s t h e e x i s t e n c e o f p o s i t i v e t r i p l e j o i n t p r o b a b i l i t i e s i n v o l v i n g a l l p o s s i b l e o b s e r v a t i o n s f r o m wh i c h i t s h o u l d b e p o s s i b l e t o o b t a i n a l l t h e q u a n t u m p r e d i c t i o n s a s

ma r g i n a l s . L e t ' s denote these triple joint probabilities by

2 3 1 3 2 2 1 2 2 1 1 1

b , b , b , b , b , b

B , B , B , B , B , B

q

, where

b

1₁

and

b

₁2 represent the

outcome values for measurements

B

1₁

and

B

₁2 on particle

A

₁,

b

1₂

and

b

2₂ represent the outcome values for

measurements

B

1₂

and

B

2₂ on particle

A

₂ and

b

1₃

and

b

2₃ represent the outcome values for measurements

2 3 1

3

and

B

(2)

1 q

2 3 1 3 2 2 1 2 2 1 1 1 2 3 1 3 2 2 1 2 2 1 1 1 2 3 1 3 2 2 1 2 2 1 1 1 b b b b b b b b b b b b B B B B B B





, , , , , , , , , , , , , , , (1)

The total number of

2 3 1 3 2 2 1 2 2 1 1 1 2 3 1 3 2 2 1 2 2 1 1 1 b b b b b b B B B B B B

q

, , , , ,

’s, denoted as

N

_q, is 64.

Also assume k 3 j 2 i 1 k 3 j 2 i 1 b b b B B B

P

, ,

r e p r e s e n t s t h e j o i n t p r o b a b i l i t y t h a t i n a p a r t i c u l a r s e t t i n g , (i.e. the probability

of obtaining the values

b

₁i,

b

₂j and

b

k₃ in a simultaneous measurement of observables i j k

3 2

1

B

and

B

,

on the

particles

A

₁

,

A

₂

and

A

₃ respectively).

The joint probabilities take the form:



        



k 3 j 2 i 1 k 3 k 3 j 2 j 2 i 1 i 1 k 3 k 3 j 2 j 2 i 1 i 1 k 3 j 2 i 1 k 3 j 2 i 1 b , b , b b , b , b , b , b , b B , B , B , B , B , B b , b , b B , B , B

q

P

(2)

where

i



,

j



,

k





{

1 ,

2 }

and

i





i

,

j





j

and

k





k

. From Eqs. (1) and (2) we have





k 3 j 2 i 1 k 3 j 2 i 1 k 3 j 2 i 1 b b b b b b B B B

1 P

, , , , , ,

. The total number of

k 3 j 2 i 1 k 3 j 2 i 1 b b b B B B

P

, ,

’s is

N

_P



64

.

As it is clear, for a local theory a Bell type expression,

ß

, is a linear combination of joint probabilities which can be written as:







n m l K J I n m l K J I n m l K J I

P

ß

, , , , , , , , , , , ,

, ( 3 )

w h e r e

I



{

B

1₁

,

B

₁2

}

,

J



{

B

1₂

,

B

2₂

}

,

K





B

1₃

,

B

2₃



,

l



{b

1₁

,b

₁2

}

,

m



{

b

1₂

,

b

2₂

}

and

n



{

b

1₃

,

b

₃2

}

. Using equation (2) the Bell inequality in terms of

q

'

s

would become:



















_

_

_



2 1 2 1 2 1 2 3 1 3 2 2 1 2 2 1 1 1 2 3 1 3 2 2 1 2 2 1 1 1 2 1 2 1 2 1 2 c , 1 c , 2 b , 1 b , 2 a , 1 a c , c , b , b , a , a b , b , b , b , b , b B , B , B , B , B , B c , c , b , b , a , a

q

ß

( 4 )

It is clear that:

d

e



ß





( 5 ) w h e r e d ( e ) i s t h e g r e a t e s t o f p o s i t i v e r e a l n u mb e r s



'

 



'

s

i n e q u a t i o n ( 5 ) .

O n e o f t h e we l l – k n o wn B e l l t yp e i n e q u a l i t i e s f o r t h r e e p a r t i t e s ys t e ms i s M e r mi n i n e q u a l i t y , w h i c h i s e x p r e s s e d a s [ 6 ] :

)

B

,

B

,

B

(

E

)

B

,

B

,

B

(

E

)

B

,

B

,

B

(

E

)

B

,

B

,

B

(

E

1 3 1 2 1 1 1 3 2 2 1 2 3 1 2 2 1 2 3 2 1 1

M 2 2

ß







( 6 )

W h e r e

E

(

B

₁i

,

B

₂j

,

B

k₃

)

i s [ 5 ] :







i j k

k 3 j 2 i 1 k 3 j 2 i 1

a b c k

b , b , b B , B , B u k 3 j 2 i 1 k 3 j 2 i

1

,

B

,

B

)

B

,

B

,

B

(

1 )

P

B

(

E

( 7 )

In the above equation "u" is the number of zero’s resulted in each particular setting. It is shown in [6] that Mermin inequality for local theories is:

2

2 

ß

_M





(8)

Mermin inequality satisfies in Eq. (5) for e=d=2, however, according to quantum theory, the upper bound of Mermin inequality is 4. Here, the violation factor and amount of violation are both 2 and the maximum white noise tolerance calculated is 0.5 [10, 11].

III. A NEW TREE QUBIT BELL INEQUALITY

(3)

inequality, for e=1 and d=0 in Eq. (5), are 1.414 and 0.414 respectively. In [12] several Bell type inequalities, with different values of e and d, are introduced. The numerical results in [12, 13] show that amount of violation and the violation factor of inequalities, as defined in section 2, are related to e and d, and maximum violation increases when the upper/lower bound in Eq. (5) increases/decreases.

One of these Bell expressions, for a tow qubit system, is as follows [12]:

2 P

P

2 P

P

1

0 , 1

B , B 0

, 0

B , B 1

, 1

B , B 1

, 0

B , B 1

, 1

B , B

0 , 1

B , B 0 , 0

B , B 0

, 1

B , B 1

, 0

B , B 0 , 0

B , B

2 2 2 1 2

2 2 1 1 2 2 1 1

2 2 1 2 2 1 1

2 2 1 1 2 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1































(9)

where

B

₁i

and

B

j₂ are two possible measurements performed on particles

2

A

and

A

1 respectively and

 

1 ,

2 j

,

i



. With this notation, outcomes of these measurements are i₁ j

2

b

and

b

which

b

i₁

,

b

₂j



 

0 ,

1

. So

j 2 i 1

j 2 i 1 b , b

B , B

P

denotes the probability that in a particular experiment, me a s u r e me n t

B

i₁ o n p a r t i c l e A1 r e s u l t s

i 1

b

a n d

me a s u r e me n t

B

i₂ o n p a r t i c l e A2 r e s u l t s

j 2

b

. The calculated violation factor and amount of violation, for inequality (9), are 1.621 and 0.621 respectively which are more than the previous results in the literature.

Inspired by the result obtained in [12, 13], we have looked for three qubit Bell- type inequality in local theories which violates quantum theory with stronger violation.

Writing Eq. (2) in matrix form, i.e.

P



Uq

, wh e r e

P

i s a n

N

_P



1

ma t r i x ,

N

_P is the total number of j o i n t p r o b a b i l i t i e s ,

q

is an

N

_q



1

matrix, the total number of triple j o i n t p r o b a b i l i t i e s is

N

_q, and U is the conversion matrix with dimension

N

_P



N

_q. Using numerical method mentioned in [12, 13] we have solved equation

P



Uq

to

find all possible Bell expressions that satisfy eq. (5). Using the three qubit GHZ state, we obtain the values of derived Bell expressions in quantum theory. The violation factor and the amount of violation for the expressions that are violated by quantum theory have been calculated and the inequality that is violated more strongly than those other expressions in quantum theory has been obtained. However we don't discuss these here, because the result that we are going to use in following can be tested directly and easily.

Let's consider

B

1_



x

and

B

_2



y

where





1 ,

2 ,

3

, we find the following Bell expression with the help of numerical calculations mentioned above:

01 1 xx x 10 1

xx x 00 1 xx x 11 1 xx x 10 0 xx x 11 1

yy x 00 1 yy x 01 0 yy x 10 0

yy x

00 1 yx y 10 0

yx y 01 1

xy y 00 0 xy y 10 0 xy y 11 1

xy y 10 0

yy y 01 1

yy y 11 0

yy y 00 1 yy y new

P

4 P

P

5 P

P

4 P

5 P

P

4 P

P

ß















(10) I n a p p e n d i x A , i t i s s h o wn t h a t :

1

n ew

ß



( 1 1 ) Now we show that inequality (11) is violated by quantum theory. Consider a three-qubit Greenberger-Horne-Zeilinger state [14] which is:

)

(|

2

1

GHZ











( 1 2 )

w h e r e



a n d

↓

a r e s p i n p o l a r i z a t i o n a l o n g z a x i s . S o i n q u a n t u m t h e o r y,

ß

_{n ew}w o u l d b e c o me :

2

7

0

4

1

4

1

4

5

0

4

1

4

1

0

8

4

8

4

8

1

8

1

new

ß

















( 1 3 )

It can be seen that the violation factor and the amount of violation of the inequality (11) are 3.5 and 2.5 respectively, whereas the maximum violation factor and maximum amount of violation of the available inequalities so far, are both 2.

To calculate the white noise tolerance of

n ew

ß

in three qubit systems, we consider the following density matrix [11]:

I

8 )

1 (

GHZ GHZ Noise

White















( 1 4 )

w h e r e



i s t h e t o l e r a n c e o f t h e B e l l e x p r e s s i o n , i.e. the maximum fraction of white noise admixture for which a Bell expression stops being violated.

(4)

8 P

)

1 (

k 3 j 2 i 1

k 3 j 2 i 1 k

3 j 2 i 1

k 3 j 2 i 1

b , b , b

B , B , B b

, b , b

B , B , B











P

( 1 5 )

k 3 j 2 i 1

b , b , b

B , B , B

P

i s the joint probability i n t h e p r e s e n c e o f wh i t e n o i s e . F r o m e q u a t i o n s ( 1 1 ) a n d ( 1 5 ) , w e

h a v e :

8 g

f

QM new

L new QM new

ß







( 1 6 )

w h e r e QM

n ew

ß

i s t h e v a l u e o f o u r B e l l e x p r e s s i o n ,

n ew

ß

, a c c o r d i n g t o q u a n t u m t h e o r y a n d L

n ew

ß

i s

t h e u p p e r b o u n d o f

n ew

ß

a c c o r d i n g t o l o c a l t h e o r y, a n d f ( g ) i s t h e n u mb e r o f p o s i t i v e ( n e g a t i v e ) t e r ms i n B e l l e x p r e s s i o n .

So the white n o i s e tolerance of our equality, i.e. Eq. (11), is:

5 .

0

8

32

20

2

7

1

2

7 







(17)

w h i c h a g r e e s wi t h ma x i mu m v a l u e c a l c u l a t e d u p t o n o w.

IV. CONCLUSION

In this paper, it is shown that a new bell-type inequality exists for the three-qubit system, i.e. a three particles system with two measurements for each side and two outputs for each measurement, in local theories and shown that this inequality is violated by quantum theory with a violation factor of 3.5 and violation amount of 2.5. We have also shown that white noise tolerance in this new Bell type expression is 0.5. Two typical bipartite qubit inequalities are CHSH and CH inequalities in which the violation factor is 0.414 and the tolerance of white noise is 0.292. The bipartite qutrits inequalities have been studied widely in recent years. Their maximum value of the amount of violation and the tolerance are predicted to be 0.87293 and 0.30385 respectively for maximally entangled state. Note that for inequalities in three qubit systems, the violation factor and amount of violation predicted are both 2 which are less than those of our inequality. Also t h e ma x i mu m w h i t e n o i s e t o l e r a n c e o f a v a i l a b l e t h r e e q u b i t i n e q u a l i t i e s i s 0 . 5 , w h i c h a g r e e s w i t h w h i t e n o i s e t o l e r a n c e o f o u r i n e q u a l i t y. As no experiment is error- free, there was an endeavor to gain a kind of Bell type inequality that could be violated as much as possible, and would be experimentally easy to test non-locality feature of quantum theory, so in our inequality, this increment of violation factor and the amount of violation increase the accuracy of experiments in which the errors are inevitable. Also one of the advantages of our inequality is that it includes only 19 different terms of joint probabilities whereas in other works it is much more than this ( for example in Mermin and Svetlichny inequalities, it is 32 and 64 respectively). So, our inequality requires less measurement which in turn, reduces the errors due to experiment. See [13].

A p p e n d i x A

In this appendix we derive equation (11) From the definition (2) and denoting

2 3 b , 1 3 b , 2 2 b , 1

2 b , 2 1 b , 1 1 b 2 3 1 3 2 2 1

2 2 1 1 1

2 3 1 3 2 2 1 2 2 1 1 1

q

b,b ,b ,b ,b ,b

B , B , B , B , B ,

B



, for simplicity, we have:

111110 111100

110110 110100

011110 011100

010110

010100

_q

q

P

110

yyy





101011 101001

100011 100001

001011 001001

000011

000001

_q

q

P

_{y y y}0 0 1





111010 111000

110010 110000

011010 011000

010010

010000

_q

q

P

y y y1 0 0





101111 101101

100111 100101

001111 001101

000111

000101

_q

q

P

0 1 1

y y y





011010 011000

010010 010000

001010 001000

000010

000000

_q

q

p

0 0 0

x y y





011111 011101

010111 010101

001111 001101

000111 000101

xy y

q

p

011





111111 111110

110111 110110

011111 011110

010111

010110

_q

q

P

1 1 1

x y y





111010 111000

110010 110000

101010 101000

100010

100000

_q

q

p

100

xyy





101011 101010

100011 100010

001011 001010

000011

000010

_q

q

P

0 0 1

y y x





(5)

111001 111000

110001 110000

011001 011000

010001

010000

_q

q

P

y y x1 0 0





111111 111110

110111 110110

011111 011110

010111

010110

_q

q

P

y y x1 1 1





111111 111110

111011 111010

101111 101110

101011

101010

_q

q

P

XXX1,1,1





110101 110100

110001 110000

100101 100100

100001

100000

_q

q

P

1,0,0

XXX





010111 010110

010011 010010

000111 000110

000011

000010

_q

q

P

0,0,1

XXX





110111 110110

110011 110010

100111 100110

100011

100010

_q

q

P

_XXX1,0,1





011111 011110

011011 011010

001111 001110

001011

001010

_q

q

P

0,1,1

XXX





110110 110100

110010 110000

010110 010100

010010

010000

_q

q

P

1 0 0

y x y





100111 100101

100011 100001

000111 000101

000011

000001

_q

q

P

0 0 1

y x y





After simplifying we obtain:

000111 0011010

001110 011001

011010 011100

110001 110010

110100 111000

000011 000101

000110 001001

001010 001100

010001 010010

010100 011000

100001 100010

100100 101000

110000 000001

000010 000100

001000 010000

100000 000000

101001 010110

010011 010101

101010 100110

100101 100011

001011 101100

111100 111010

111001 110110

110101 110011

101110 101101

101011 100111

011110 011101

011011 010111

001111 111110

111101 111011

110111 101111

011111 111111

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

q

4 q

4

new

ß















































































Please note that all q’s are positive here, so according to equation 12,

n ew

ß

is less than or equal to 1.

REF ER EN CE S

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