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arXiv:0905.4398 · International Journal of Quantum Information

VON NEUMANN AND LUDERS POSTULATES AND QUANTUM INFORMATION THEORY

Andrei Yu. Khrennikov

Published Oct 1, 2009• 11 citations
arXivHTMLPDF

Abstract

This note is devoted to some foundational aspects of quantum mechanics (QM) related to quantum information (QI), especially quantum teleportation and "one way quantum computing." We emphasize the role of the projection postulate (determining post-measurement states) in QI and the difference between its Lüders and von Neumann versions. As is well-known, these postulates differ in the case of observables with degenerate spectra. Such observables are important in operations with entangled states: any measurement on one subsystem is represented by an observable with degenerate spectrum in the Hilbert space of a composite system. Some QI schemes (e.g. quantum teleportation and "one way quantum computing") are based on the use of Lüders postulate. The formal application of von Neumann postulate can block these QI schemes. In this note, we present a list of natural conditions under which von Neumann's description of measurements via refinement implies Lüders projection postulate.

DOI
10.1142/s0219749909005845

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This note is devoted to some foundational aspects of quantum mechanics (QM) related to quantum information (QI), especially quantum teleportation and "one way quantum computing." We emphasize the role of the projection postulate (determining post-measurement states) in QI and the difference between its Lüders and von Neumann versions. As is well-known, these postulates differ in the case of observables with degenerate spectra. Such observables are important in operations with entangled states: any measurement on one subsystem is represented by an observable with degenerate spectrum in the Hilbert space of a composite system. Some QI schemes (e.g. quantum teleportation and "one way quantum computing") are based on the use of Lüders postulate. The formal application of von Neumann postulate can block these QI schemes. In this note, we present a list of natural conditions under which von Neumann's description of measurements via refinement implies Lüders projection postulate.
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 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).
The projection postulate is a description of the effect on a quantum system, assumed to be in a pure state, of a measurement of an observable with a discrete spectrum, in nonrelativistic quantum mechanics. It is often called "von Neumann's projection postulate" or "the Lüders rule". This paper is an examination of the versions of this postulate due to Dirac, von Neumann and Lüders. It is shown that Dirac, in 1930, proposed what is now generally known as the projection postulate. Von Neumann, in 1932, gave a different theory which only applies in special and rather unusual cases. Lüders, in 1951, rejected this theory and presented one which is the same as Dirac's. Treatments of observables with continuous spectra by both Dirac and von Neumann are criticised, and the possibility of a generalised version of the projection postulate for this case is considered. The paper concludes with a discussion of the status of the projection postulate (in its various forms) as a separate postulate (independent of the other postulates of quantum mechanics) and as a separate form of time development (in addition to the time-dependent Schrödinger equation).
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.
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.