Elena R. Loubenets

Quantum violation of Bell inequalities is now used in many quantum information applications and it is important to analyze it both quantitatively and conceptually. In the present paper, we analyze violation of multipartite Bell inequalities via the local probability model—the LqHV model, incorporating the LHV model only as a particular case and correctly reproducing the probabilistic description of every quantum correlation scenario, more generally, every nonsignaling scenario. The LqHV probability framework allows us to construct nonsignaling analogs of Bell inequalities and to specify parameters quantifying violation of Bell inequalities—Bell’s nonlocality—in a general nonsignaling case. For quantum correlation scenarios on an N-qudit state, we evaluate these nonlocality parameters analytically in terms of dilation characteristics of an N-qudit state and also, numerically—in d and N. In view of our rigorous mathematical description of Bell’s nonlocality in a general nonsignaling case via the local probability model, we argue that violation of Bell inequalities in a quantum case is not due to violation of the Einstein–Podolsky–Rosen locality conjectured by Bell but due to the improper HV modelling of “quantum realism”.

Based on the new general framework for the probabilistic description of experiments, introduced in [E.R. Loubenets, Research Report No 8, MaPhySto, University of Aarhus, Denmark (2003); Proceedings Conference “Quantum Theory, Reconsideration of Foundations”, Ser. Math. Modeling, Vol. 10 (University Press, Vaxjo, 2004), pp. 365–385], we analyze in mathematical terms the link between the validity of Bell-type inequalities under joint experiments upon a system of any type and the physical concept of “local realism”. We prove that the violation of Bell-type inequalities in the quantum case has no connection with the violation of “local realism”. In a general setting, we formulate in mathematical terms the condition on “local realism” under a joint experiment and consider examples of quantum “locally realistic” joint experiments. We, in particular, show that quantum joint experiments of the Alice/Bob type are “locally realistic”. For an arbitrary bipartite quantum state, we derive quantum analogs of the original Bell inequality. In view of our results, we argue that the violation of Bell-type inequalities in the quantum case cannot be a valid argument in the discussion on locality or non-locality of quantum interactions

We show that the so-called quantum probabilistic rule, usually introduced in the physical literature as an argument of the essential distinction between the probability relations under quantum and classical measurements, is not, as it is commonly accepted, in contrast to the rule for the addition of probabilities of mutually exclusive events. The latter is valid under all experimental situations upon classical and quantum systems. We discuss also the quantum measurement situation that is similar to the classical one, described by the Bayes formula. We show the compatibility of the description of this quantum measurement situation in the frame of purely classical and experimentally justified straightforward frequency arguments and in the frame of the quantum stochastic approach to the description of generalized quantum measurements. In view of derived results, we argue that even under experiments upon classical systems the classical Bayes formula describes particular experimental situations that are specific for context-independent measurements. The similarity of the forms of the relation between the transformation of probabilities, which we derive in the frame of the quantum stochastic approach and in the frame of the approach, based on straightforward frequency arguments, underlines once more that the projective (von Neumann) measurements represent only a special type of measurement situations in quantum physics

For the probabilistic description of all the joint von Neumann measurements on a D-dimensional quantum system, we present the specific example of a context-invariant quasi hidden variable model, proved in Loubenets to exist for each Hilbert space. In this model, a quantum observable X is represented by a variety of random variables satisfying the functional condition required in quantum foundations but, in contrast to a contextual model, each of these random variables equivalently models X under all joint von Neumann measurements, regardless of their contexts. This, in particular, implies the specific local qHV model for an N-qudit state and allows us to derive the new exact upper bound on the maximal violation of 2\2-setting Bell-type inequalities of any type under N-partite joint von Neumann measurements on an N-qudit state. For d = 2, this new upper bound coincides with the maximal violation by an N-qubit state of the Mermin–Klyshko inequality. Based on our results, we discuss the conceptual and mathematical advantages of context-invariant and local qHV modelling