Information-theoretic bell inequalities and the relative measure of probability

Emilio Santos

Andrei Yu. Khrennikov

We analyze the proof of Bell’s inequality and demonstrate that this inequality is related to one particular model of probability theory, namely Kolmogorov measure-theoretical axiomatics from 1933. We found a (numerical) statistical correction to Bell’s inequality. Such an additional term εφ on the right-hand side of Bell’s inequality can be considered as a probability invariant of a quantum state φ. This is a measure of nonreproducibility of hidden variables in different runs of experiments. Experiments to verify Bell’s inequality can be considered as just experiments to estimate the constant εφ. It seems that Bell’s inequality could not be used as a crucial reason to deny local realism. We consider deterministic as well as stochastic hidden variables models.

Marek Wajs, Paweł Kurzyński, Dagomir Kaszlikowski

We numerically investigate entropic Bell inequalities for a pair of entangled qutrits using information-theoretic distances. We show that for this class of inequalities the Tsallis entropy is more suitable than the Shannon entropy, as it reveals nonclassicality for a larger set of quantum states. Finally, we find that, like probability-based inequalities, entropic ones are maximally violated by the non--maximally entangled qutrit state.

James Espinosa

Loran V. Akopyan, Vladimir I. Man’ko

Wei Li, Shengmei Zhao

Bell inequality testing, a well-established method to demonstrate quantum non-locality between remote two-partite entangled systems, is playing an important role in the field of quantum information. The extension to high-dimensional entangled systems, using the so-called Bell-CGLMP inequality, points the way in measuring joint probabilities, the kernel block to construct high dimensional Bell inequalities. Here we show that in theory the joint probability of a two-partite system entangled in a Hilbert space can be measured by choosing a set of basis vectors in its dual space that are related by a Fourier transformation. We next propose an experimental scheme to generate a high-dimensional position-entangled two-photon state aided by a combination of a multiple-slit and a 4 f system, and describe a method to test Bell's inequality using correlated diffraction. Finally, we discuss in detail consequences of such Bell-test violations and experimental requirements.