A stronger form of Bell’s inequality

D. Gutkowski, G. Masotto

Christopher C. Gerry, Jihane Mimih, Adil Benmoussa

Recently, Wildfeuer et al. [Phys. Rev. A 76, 052101 (2007)] studied possible experiments demonstrating nonlocal correlation effects through the violation of various Bell-type inequalities by maximally path-entangled number states of the form $(|N⟩|0⟩+{e}^{i\ensuremath{\varphi}}|0⟩|N⟩)/\sqrt{2}$, the so-called $N00N$ states, and some strong violations were found. In this paper, we re-examine the same Bell-type inequalities with respect to maximally path-entangled coherent states of the form $\mathcal{N}({|\ensuremath{\alpha}⟩}_{a}{|0⟩}_{b}+{e}^{i\ensuremath{\Phi}}{|0⟩}_{a}{|\ensuremath{\alpha}\text{ }{e}^{i\ensuremath{\theta}}⟩}_{b})$. We find in many cases even stronger violations of the Bell-type inequalities than appear to be possible with the $N00N$ states.

Ivanović, I. D.

Mu Yang, Hui-Xian Meng, Jie Zhou +7 more

Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.

Pierre A. Millette

Tsuyoshi Ito, Hiroshi Imai, David Avis

We show that some two-party Bell inequalities with two-valued observables are stronger than the CHSH inequality for $3\ensuremath{\bigotimes}3$ isotropic states in the sense that they are violated by some isotropic states in the $3\ensuremath{\bigotimes}3$ system that do not violate the CHSH inequality. These Bell inequalities are obtained by applying triangular elimination to the list of known facet inequalities of the cut polytope on nine points. This gives a partial solution to an open problem posed by Collins and Gisin. The results of numerical optimization suggest that they are candidates for being stronger than the ${\mathrm{I}}_{3322}$ Bell inequality for $3\ensuremath{\bigotimes}3$ isotropic states. On the other hand, we found no Bell inequalities stronger than the CHSH inequality for $2\ensuremath{\bigotimes}2$ isotropic states. In addition, we illustrate an inclusion relation among some Bell inequalities derived by triangular elimination.