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arXiv:0804.2006

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

Andrei Yu. Khrennikov

Published Apr 12, 2008• 0 citations
arXivHTMLPDF

Abstract

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).

DOI
10.48550/arxiv.0804.2006

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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).
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).
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.
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.
From a phenomenological and experimental viewpoint, quantum mechanics is remarkably successful in explaining effects and processes in the microscopic world. Recently, however, a growing body of literature has revisited the theory's own formal foundations, specifically addressing issues of self-consistency and completeness. As a major historical example of these foundational challenges, the Einstein-Podolsky-Rosen (EPR) argument famously claimed that the theory is incomplete. While many valid criticisms and refutations of the EPR logic exist, they generally rely on a combination of technical results and conceptual objections to EPR's interpretive assumptions, thus transcending the plain quantum framework itself. Motivated by these trends of examining the theory's structural limits, here we revisit the EPR argument. Focusing on elements essential to the EPR reasoning, we first analyze general quantum correlations for EPR states: (i) the fact that their observables are always associated with non-commuting operators, and (ii) the nature of the correlated information obtained through measurement. From these two points alone, and relying strictly on the core rules of quantum mechanics, we demonstrate how to overturn the alleged EPR incompleteness without invoking any extraneous propositions. Consequently, we show that the standard formalism alone suffices to resolve such type of skepticism regarding the theory's physical reach. This offers, at least in a paradigmatic instance, a powerful indication of a structurally well-founded, self-consistent quantum theory.
We show that the projection postulate plays a crucial role in the discussion on the so-called quantum nonlocality, in particular, in the Einstein, Podolsky, and Rosen argument. We stress that the original von Neumann projection postulate was crucially modified by extending it to observables with degenerate spectra (the Lüders postulate) and we show that this modification is highly questionable from a physical point of view and is the real source of quantum nonlocality. The use of the original von Neumann postulate eliminates this problem: instead of an action at a distance nonlocality we obtain a classical measurement nonlocality, which is related to the synchronization of two measurements (on the two parts of a composite system).
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.