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Open Systems & Information Dynamics

Kolmogorov Probability Spaces Describing Accardi Models of Quantum Correlations

Andrei Yu. Khrennikov, Oleg Georgievich Smolyanov, Aubrey Truman

Published Nov 11, 2005• 12 citations
Publisher

Abstract

In this paper we consider several Kolmogorov type models describing some quantum correlations. The main ideas beyond most of these models comes back to pioneering works of Accardi (see [1, 10, 15] and references therein); because of that we call some of them the Accardi models. Namely we discuss, in our frame, the Kolmogorov type Accardi model of the Aspect version [2] of the Einstein-Podolsky-Rosen (EPR) experiment [3].

DOI
10.1007/s11080-005-4487-y

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In this paper we consider several Kolmogorov type models describing some quantum correlations. The main ideas beyond most of these models comes back to pioneering works of Accardi (see [1, 10, 15] and references therein); because of that we call some of them the Accardi models. Namely we discuss, in our frame, the Kolmogorov type Accardi model of the Aspect version [2] of the Einstein-Podolsky-Rosen (EPR) experiment [3].
A classical deterministic, reversible dynamical systems, reproducing the Einstein--Podolsky--Rosen (EPR) correlations in full respect of causality and locality and without the introduction of any ad hoc selection procedure, was constructed in the paper [AIR02]. In the present paper we prove that the above mentioned model is unique (see Theorem (2) in the sense that any local causal probability measure which reproduces the EPR correlations must coincide, under natural and generic assumptions, with the one constructed in [AIR02]. [AIR02] L. Accardi, K. Imafuku and M. Regoli, "ON THE EPR--CHAMELEON EXPERIMENT", Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 5 (2002) 1--20.
Models of a two-photon interference experiment proposed by Franson to study the Einstein-Podolsky-Rosen (EPR) effect are presented. The experiment considers a pair of photons correlated in time and in frequency, each of which is passed through an interferometer. When the path difference through the two interferometers is set so there is no first-order interference, Franson showed that second-order interference between the two photons could occur with 100% visibility. In this paper both quantum and classical models are considered and the importance of the coincidence time window and the effect of the frequency correlation are shown. Calculations are presented for all settings of the interferometers. The EPR effect arises from entangled states; a brief discussion is given of the nature of the entangled state for this type of experiment.
Einstein-Podolsky-Rosen (EPR) correlations Explaining the correlations predicted by quantum mechanics in the case of joint quantum systems of the EPR type has proved to be a major challenge for the Common Cause Principle. The EPR correlations have been confirmed by a large number of sophisticated experiments carried out since the early 1980s, and, since quantum theory does not give us any hints as to where to look for possible common causes that could explain the correlations in question, the suspicion arose that common causes of EPR correlations might not exist at all. If this were indeed the case, then the EPR correlations would constitute strong empirical evidence against Reichenbach's Common Cause Principle. But how could this be possible in view of the results presented in the previous chapters? As we have seen (Chapter 3), every common cause incomplete probability space is common cause extendible (even strongly common cause extendible with respect to any finite set of correlations; Proposition 3.9); therefore every correlation can, in principle, be explained by a (possibly hidden) common cause, including the (finite number of) EPR correlations. Could the EPR correlations (not) be explained by referring to such hidden common causes? To answer this question, one has to realize first that it is not quite obvious how to link the Common Cause Principle to quantum mechanics because Reichenbach's notion of common cause was defined in terms of classical probability theory (Chapter 2), not in terms of quantum probability theory.
Nowadays, no one should doubt that quantum entanglement is (at least, one of) the \ncharacteristic traits of quantum mechanics. Furthermore, quantum information science \nis strongly influenced by the advantages provided by quantum entanglement, as the \nlatter can be regarded as a genuine resource for an efficient (better-than-classical) \nprocessing of information in certain computational tasks. The very basic notion of \ncorrelation is fundamental for the conceptualization of the physical communicational \nproblem and, as a consequence, for the notion of information itself. Thus, the question \nabout the nature of quantum information is close (and unavoidably linked) to the \nquestion about the nature of quantum correlations.