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arXiv:0706.1813

Uniqueness of the EPR--chemeleon model

Luigi Accardi, Satoshi Uchiyama

Published Jun 13, 2007• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.0706.1813

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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.
A classical deterministic, reversible dynamical system, 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 Ref. 4. In this paper we prove that the above-mentioned model is unique (see Theorem 3.1) 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 Ref. 4.
Starting from the late 60’s many experiments have been performed to verify the violation Bell’s inequality by Einstein-Podolsky-Rosen (EPR) type correlations. The idea of these experiments being that: (i) Bell’s inequality is a consequence of locality, hence its experimental violation is an indication of non locality; (ii) this violation is a typical quantum phenomenon because any classical system making local choices (either deterministic or random) will produce correlations satisfying this inequality. Both statements (i) and (ii) have been criticized by quantum probability on theoretical grounds (not discussed in the present paper) and the experiment discussed below has been devised to support these theoretical arguments. We emphasize that the goal of our experiment is not to reproduce classically the EPR correlations but to prove that there exist perfectly local classical dynamical systems violating Bell’s inequality. The conclusions of the present experiment are: no contradiction between quantum theory and locality can be deduced from the violation of Bell’s inequality. The Copenhagen interpretation of quantum theory becomes quite reasonable and not metaphyisic if interpreted at the light of the chameleon effect. One can realize quantum entanglement by classical computers. In section (7) we prove that our experiment also provides a classical analogue of the type of logical (i.e. independent of statistics) incompatibilities pointed out by Greenberger, Home and Zeilinger. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We prove that the structure of the classical dynamical system, constructed in [AcImRe01] to reproduce the EPR correlations, is a natural consequence of the combination of the general theory of classical dynamical systems with the extension of von Neumann measurement theory, proposed in [Ac93] to include in it the requirements of locality and causality.We then prove some a priori estimates which guarantee the convergence of the simulation procedure.The program to run the experiment is available from the WEB-page: http://volterra.mat.uniroma2.it.
On September 27 (2001), as a side activity to the "Japan-Italy Joint workshop on: Quantum open systems and quantum measurement", the first; public demonstration of the dynamical EPR-chameleon experiment was performed at Waseda University in order to give an experimental answer to a long standing question in the foundations of quantum theory: do there exist classical macroscopic systems which, by local independent choices, produce sequences of data which reproduce the singlet correlations, hence violating Bell's inequality? The EPR-chameleon experiment gives an affirmative answer to this question by concretely producing an example of such systems in the form of three personal computers which realize a local deterministic dynamical evolution whose mathematical structure is very simple and transparent. In the experiment performed on September 27 the local dynamics used was not a reversible one because the interaction with the degrees of freedom of the apparatus was integrated out giving rise to an effective Markovian dynamics which, although mapping probability measures into probability measures, did not preserve the +/- 1-values of the spin (or polarization) observables. This feature was criticized by some of the partecipants and the following two questions arose: i) is it possible to prove that the Markovian evolution, used in the experiment, is indeed the reduced evolution of a bona fide reversible evolution? ii) if the answer to question (i) is affirmative, is it possible to reproduce the EPR correlations by simply considering empirical averages of +/- 1-values, as one does in usual EPR type experiments? An affirmative answer to these questions was given in the paper [AcImRe01] and it is briefly reviewed in what follows.
We describe an experiment in which two non communicating computers, starting from a common input in the form of sequences of pseudo--random numbers in the interval [0,2π][0,2π][0,2π], and computing deterministic {±1}\{\pm 1\}{±1}--valued functions, chosen at random and independently, produce sequences of numbers whose correlations coincide with the EPR correlations and therefore violate Bell's inequality. The experiment is the practical implementation of a mathematical model of a classical, deterministic system whose initial state is chosen at random from its state space, with a known initial probability distribution, and whose dynamics exhibits the chameleon effect described below. Such a system satisfies the constraints of pre--determination, locality, causality, local independent choices, singlet law and reproduces the EPR correlations.
Starting from the late 60's many experiments have been performed to verify the violation Bell's inequality by Einstein-Podolsky-Rosen (EPR) type correlations. The idea of these experiments being that: (i) Bell's inequality is a consequence of locality, hence its experimental violation is an indication of non locality; (ii) this violation is a typical quantum phenomenon because any classical system making local choices (either deterministic or random) will produce correlations satisfying this inequality. Both statements (i) and (ii) have been criticized by quantum probability on theoretical grounds (not discussed in the present paper) and the experiment discussed below has been devised to support these theoretical arguments. We emphasize that the goal of our experiment is not to reproduce classically the EPR correlations but to prove that there exist perfectly local classical dynamical systems violating Bell's inequality.