Entanglement and Bell Inequalities

Abstract

The entangled quantum states play a key role in quantum information. The association of the quantum state vector with each individual physical system in an attributive way is a source of many paradoxes and inconsistencies. The paradoxes are avoided if the purely statistical interpretation of the quantum-state vector is adopted. According to the statistical interpretation, the quantum theory does not provide any deterministic prediction for any individual experimental result obtained for a free physical system, for a trapped ion, or for a quantum dot. In this article, it is shown that, if the statistical interpretation is used, then contrary to the general belief, the quantum theory does not predict for the ideal spin singlet state perfect anti-correlation of the coincidence counts for the distant detectors. Subsequently the various proofs of the Bell's theorem are reanalyzed and, in particular, the importance and implications of using the unique probability space in these proofs are elucidated. The use of the unique probability space is shown to be equivalent to the use of joint probability distributions for noncommuting observables. The experimental violation of the Bell's inequalities proves that the naive realistic particle, like the spatio-temporal description of various quantum mechanical experiments, is impossible. Of course, it does not give any argument for the action at a distance and it does not provide the proof of the completeness of quantum mechanics. The fact that the quantum-state vector is not an attribute of a single quantum system and that the quantum observables are contextual has to be taken properly into account in any implementation of the quantum computing device.