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On the problem of completeness of QM: von Neumann against Einstein, Podolsky, and

Andrei Yu. Khrennikov

Published Apr 12, 2008• 3 citations
OpenAlex

Abstract

We performed a comparative analysis of the arguments of Einstein, Podolsky and Rosen – EPR, 1935: [1] (against the completeness of QM) and the theoretical formalism of QM (due to von Neumann, 1932: [2]). We found that the EPR considerations do not match at all with the von Neumann’s theory. Thus EPR did not criticize the real theoretical model of QM. The root of EPR’s paradoxical conclusion on incompleteness of QM is the misuse of von Neumann’s projection postulate. EPR applied this postulate to observables with degenerate spectra (which is totally forbidden by the axiomatics of QM).

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We performed a comparative analysis of the arguments of Einstein, Podolsky and Rosen – EPR, 1935: [1] (against the completeness of QM) and the theoretical formalism of QM (due to von Neumann, 1932: [2]). We found that the EPR considerations do not match at all with the von Neumann’s theory. Thus EPR did not criticize the real theoretical model of QM. The root of EPR’s paradoxical conclusion on incompleteness of QM is the misuse of von Neumann’s projection postulate. EPR applied this postulate to observables with degenerate spectra (which is totally forbidden by the axiomatics of QM).
We performed a comparative analysis of the arguments of Einstein, Podolsky and Rosen -- EPR, 1935 (against the completeness of QM) and the theoretical formalism of QM (due to von Neumann, 1932). We found that the EPR considerations do not match at all with the von Neumann's theory. Thus EPR did not criticize the real theoretical model of QM. The root of EPR's paradoxical conclusion on incompleteness of QM is the misuse of von Neumann's projection postulate. EPR applied this postulate to observables with degenerate spectra (which is totally forbidden by the axiomatics of QM).
This paper is devoted to analysis of the role of projection postulate in Einstein, Podolsky, and Rosen-EPR [1] argument against completeness of QM. It will be shown that the EPR considerations do not match with von Neumann’s quantum measurement theory (QMT). The root of EPR’s conclusion on incompleteness of QM is the misuse of von Neumann’s projection postulate. EPR applied this postulate to observables with degenerate spectra (which is totally forbidden by von Neumann’s axiomatics of QM). Consequences for Bell’s inequality and quantum information theory are analyzed. This paper might stimulate discussion on the role of projection postulate in quantum information theory, especially quantum cryptography and teleportation.
This chapter temporarily departs from conventional "orthodoxy" in that the authors set aside the completeness hypothesis and its generalizations. In 1935 A. Einstein and his two younger collaborators, B. Podolsky and N. Rosen (EPR) issued a paper in which they showed that, assuming the validity of but very general and highly plausible assumptions, the incompleteness of quantum mechanics could be demonstrated. The opinion that quantum mechanics is a complete description of the atomic and subatomic phenomena was so much engrained in the mind of the physicists that this EPR result was soon referred to as a "paradox." The EPR argument was based on assumptions, or premises, that, prima facie look most plausible. On the theoretical side it has been shown that the D. Greenberger, M. Home, and A. Zeilinger argument can be developed without involving any spin, essentially by replacing the spin-component measurements by presence measurements performed by means of counters.
Modern development of quantum technologies based on quantum information theory stimulated analysis of proposed computational, cryptographic and teleportational schemes from the viewpoint of quantum foundations. It is evident that not all mathematical calculations performed in complex Hilbert space can be directly realized in physical space. Recently by analyzing the original EPR paper we found that they argument was based on the misuse of the von Neumann's projection postulate. Opposite to von Neumann, Einstein, Podolsky and Rosen (EPR) applied this postulate to observables represented by operators with degenerate spectra. It was completely forbidden by von Neumann's axiomatics of QM. It is impossible to repeat the EPR considerations in the von Neumann's framework. In this note we analyze quantum teleportation by taking into account von Neumann's projection postulate. Our analysis shows that so called quantum teleportation is impossible in von Neumann's framework. On the other hand, our analysis implies that the main quantum algorithms are totally consistent with von Neumann's projection postulate.
Notwithstanding it is well known that von Neumann's projection postulate is inapplicable to most realistic measurement procedures, it keeps haunting the foundations of quantum mechanics. In particular its applicability to EPR experiments is often assumed. In the present contribution this problem is considered from the point of view of a quantum mechanical theory of measurement, allowing a treatment of projection in EPR experiments as a special case of conditional preparation. The conditions are spelled out under which the postulate may be applicable.
The ‘EPR paradox’ is directly related to the 1935 paper of Einstein, Podolsky, and Rosen, even though Einstein did not include the word ‘paradox’ in it. All Einstein and his co authors did was to construct an argument that concluded that quantum mechanics was ‘not complete.’ This EPR conclusion is examined in detail here and found to be based on the erroneous assumption of an ‘all values’ spread in x that leads to an unphysical exact measurement of the momentum p . The word paradox, in relation the second part of the EPR paper, was introduced by Schrödinger in 1936 and popularized by Bohm in the 1950s. Hence, we should not attach responsibility to EPR for the ‘paradox’ situation. It was Schrödinger and Bohm who attached and propagated the term paradox in relation to the EPR paper. And the literature made it famous. This is particularly lamentable since the physics of quantum entanglement is utterly free of paradoxes .