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International Journal of Quantum Information

EPR "PARADOX", PROJECTION POSTULATE, TIME SYNCHRONIZATION "NONLOCALITY"

Andrei Yu. Khrennikov

Published Jan 1, 2009• 11 citations
Publisher

Abstract

We show that the projection postulate plays a crucial role in the discussion on the so called "quantum nonlocality," in particular in the EPR-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 it is the real source of "quantum nonlocality." The use of the original von Neumann postulate eliminates this problem: instead of "action at the distance"-nonlocality, we obtain a classical measurement nonlocality. The latter is related to the synchronization of two measurements (on the two parts of a composite system). It seems that EPR did mistake in their 1935-paper: if one uses correctly von Neumann projection postulate, no "elements of reality" can be assigned to entangled systems. Our analysis of the EPR and projection postulate makes clearer Bohr's considerations in his reply to Einstein.

DOI
10.1142/s0219749909004724

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We show that the projection postulate plays a crucial role in the discussion on the so called "quantum nonlocality," in particular in the EPR-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 it is the real source of "quantum nonlocality." The use of the original von Neumann postulate eliminates this problem: instead of "action at the distance"-nonlocality, we obtain a classical measurement nonlocality. The latter is related to the synchronization of two measurements (on the two parts of a composite system). It seems that EPR did mistake in their 1935-paper: if one uses correctly von Neumann projection postulate, no "elements of reality" can be assigned to entangled systems. Our analysis of the EPR and projection postulate makes clearer Bohr's considerations in his reply to Einstein.
We show that the projection postulate plays a crucial role in the discussion on the so called "quantum nonlocality", in particular in the EPR-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 it is the real source of "quantum nonlocality". The use of the original von Neumann postulate eliminates this problem: instead of "action at the distance"-nonlocality, we obtain a classical measurement nonlocality. It seems that EPR did mistake in their 1935-paper: if one uses correctly von Neumann projection postulate, no ``elements of reality'' can be assigned to entangled systems. Our analysis of the EPR and projection postulate makes clearer Bohr's considerations in his reply to Einstein.
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).
Nowadays it is practically forgotten that for observables with degenerate spectra the original von Neumann projection postulate differs crucially from the version of the projection postulate which was later formalized by Lüders. The latter (and not that due to von Neumann) plays the crucial role in the basic constructions of quantum information theory. We start this paper with the presentation of the notions related to the projection postulate. Then we remind that the argument of Einstein‐Podolsky‐Rosen against completeness of QM was based on the version of the projection postulate which is nowadays called Lüders postulate. Then we recall that all basic measurements on composite systems are represented by observables with degenerate spectra. This implies that the difference in the formulation of the projection postulate (due to von Neumann and Lüders) should be taken into account seriously in the analysis of the basic constructions of quantum information theory. This paper is a review devoted to such an analysis.
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.
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).
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.