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arXiv:0706.0245 · Quantum Information and Computation

Violation of equalities in bipartite qutrits systems

Hossein Movahhedian

Published Jan 1, 2009• 1 citation
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Abstract

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.

DOI
10.26421/qic9.1-2-6

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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.
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 .
Bell-type local realist inequalities are developed for probing quantum nonlocality of bipartite entangled qutrit states by generalizing Wigner's argument that was originally formulated for the bipartite qubit singlet state. This treatment is based upon assuming the existence of the overall joint probability distributions for the measurement outcomes pertaining to the relevant trichotomic observables, satisfying the locality condition, and yielding the measurable marginal probabilities. We first show that such generalized Wigner inequalities (GWI) are violated by quantum mechanics (QM) for both the bipartite qutrit isotropic and singlet states, thereby demonstrating their nonlocality. The efficacy of GWIs are also probed by comparing their QM violation with that obtained for other types of local realist inequalities that have been used for probing nonlocality of entangled bipartite qutrits. This is done in two different ways, (i) by employing unsharp measurements, and (ii) by incorporating white noise in the isotropic and singlet qutrit states. We find that in both these cases, GWIs outperform the two other types of local realist inequalities considered here in terms of their QM violation, by being (i) more robust against the usharpness of measurement, and (ii) more tolerant to the white noise incorporated in the states considered.
We show a new Bell-Clauser-Horne inequality for two entangled three-dimensional quantum systems (so-called qutrits). This inequality is not violated by a maximally entangled state of two qutrits observed through a symmetric three-input- and three-output-port beam splitter only if the amount of noise in the system is greater than $(11\ensuremath{-}6\sqrt{3})/2\ensuremath{\approx}0.308.$ This result is in a perfect agreement with the previous numerical calculations presented in Kaszlikowski et al. [Phys. Rev. Lett. 85, 4418 (2000)]. Moreover, we prove that for noiseless case, the necessary and sufficient condition for the threshold quantum efficiency of detectors below which there is no violation of local realism for the optimal choice of observables is equal to $6(15\ensuremath{-}4\sqrt{3})/59\ensuremath{\approx}0.821.$ This efficiency result again agrees with the numerical predictions.