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arXiv:1106.5126

An Improved Tripartite Bell-type Inequality

Fatemeh Shirdel, Hossein Movahhedian

Published Jun 25, 2011• 0 citations
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Abstract

So far, various Bell type inequalities have been introduced to test the local realism in tripartite systems. In this article we consider a tripartite system with two measurements in each side and two outputs for each measurement. Then we will introduce a new bell type inequality for this system and we show that this inequality is violated by quantum theory with a violation factor and amount of violation of 3.5 and 2.5 respectively, which exceed those of available inequalities in both cases. Also we will show that the white noise tolerance of this new Bell type expression is 0.5, which agrees with the maximum amount of white noise tolerance of available inequalities up to now.

DOI
10.48550/arxiv.1106.5126

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So far, various Bell type inequalities have been introduced to test the local realism in tripartite systems. In this article we consider a tripartite system with two measurements in each side and two outputs for each measurement. Then we will introduce a new bell type inequality for this system and we show that this inequality is violated by quantum theory with a violation factor and amount of violation of 3.5 and 2.5 respectively, which exceed those of available inequalities in both cases. Also we will show that the white noise tolerance of this new Bell type expression is 0.5, which agrees with the maximum amount of white noise tolerance of available inequalities up to now.
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 ax i m um 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 .
We have recently shown that for the special case of a bipartite system with binary inputs and outputs there exist equalities in local theories which are violated by quantum theory. The amount of white noise tolerated by these equalities are twice that of inequalities. In this paper we will first introduce an inequality in bipartite qutrits systems which, if non-maximally entangled state is used instead of maximally entangled state, is violated more strongly by quantum theory. Hence reproducing the results obtained in the literature. We will then prove that our equalities in this case are violated by quantum theory too, and they tolerate much more white noise than inequalities.
We investigate the violation factor of the original Bell-Mermin inequality. Until now, we have used an assumption that the results of measurement are . In this case, the maximum violation factor is as follows: and . The quantum predictions by n-partite Greenberger-Horne-Zeilinger state violate the Bell-Mermin inequality by an amount that grows exponentially with n. Recently, a new measurement theory is proposed [K. Nagata and T. Nakamura, International Journal of Theoretical Physics, 49, 162 (2010)]. The values of measurement outcome are . Here we use the new measurement theory. We consider a multipartite GHZ state. We use the original Bell-Mermin inequality. It turns out that the original Bell-Mermin inequality is satisfied irrespective of the number of particles. In this case, the maximum violation factor is as follows: and . Thus the original Bell-Mermin inequality is satisfied by the new measurement theory. We propose the following conjecture: All the two-orthogonal-settings experimental correlation functions admit local realistic theories irrespective of a state if we use the new measurement theory.
With Bell's inequalities one has a formal expression to show how essentially all local theories of natural phenomena that are formulated within the framework of realism may be tested using a simple experimental arrangement. For the case of entangled pairs of spin-1/2 particles we propose an alternative measurement setup which is consistent to the necessary assumptions corresponding to the derivation of the Bell inequalities. We find that the Bell inequalities are never violated with respect to our suggested measurement process.