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Philosophy of Science

Quantum Logic and the Projection Postulate

Geoffrey Paul Hellman

Published Sep 1, 1981• 23 citations
Publisher

Abstract

This paper explores the status of the von Neumann-Lüders state transition rule (the “projection postulate”) within “real-logic” quantum logic. The entire discussion proceeds from a reading of the Lüders rule according to which, although idealized in applying only to “minimally disturbing” measurements, it nevertheless makes empirical claims and is not a purely mathematical theorem. An argument (due to Friedman and Putnam) is examined to the effect that QL has an explanatory advantage over Copenhagen and other interpretations which relativize truth-value assignments to experimental arrangements. Two versions of QL, the lattice-theoretic (LT) and partial-Boolean-algebra (PBA), are considered. It turns out that the projection postulate is intimately connected with the choice of conditional connective for QL. The effect of the projection postulate is obtained with the Sasaki conditional. However, this choice is found to require extra assumptions, on both the LT and PBA versions, which are either just as ad hoc as the projection postulate itself or indefensible from within the real-logic QL framework.

DOI
10.1086/289011

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This paper explores the status of the von Neumann-Lüders state transition rule (the “projection postulate”) within “real-logic” quantum logic. The entire discussion proceeds from a reading of the Lüders rule according to which, although idealized in applying only to “minimally disturbing” measurements, it nevertheless makes empirical claims and is not a purely mathematical theorem. An argument (due to Friedman and Putnam) is examined to the effect that QL has an explanatory advantage over Copenhagen and other interpretations which relativize truth-value assignments to experimental arrangements. Two versions of QL, the lattice-theoretic (LT) and partial-Boolean-algebra (PBA), are considered. It turns out that the projection postulate is intimately connected with the choice of conditional connective for QL. The effect of the projection postulate is obtained with the Sasaki conditional. However, this choice is found to require extra assumptions, on both the LT and PBA versions, which are either just as ad hoc as the projection postulate itself or indefensible from within the real-logic QL framework.
In a recent paper, Michael Friedman and Hilary Putnam argued that the Lüders rule is ad hoc from the point of view of the Copenhagen interpretation but that it receives a natural explanation within realist quantum logic as a probability conditionalization rule. Geoffrey Hellman maintains that quantum logic cannot give a non-circular explanation of the rule, while Jeffrey Bub argues that the rule is not ad hoc within the Copenhagen interpretation. As I see it, all four are wrong. Given that there is to be a projection postulate, there are at least two natural arguments which the Copenhagen advocate can offer on behalf of the Lüders rule, contrary to Friedman and Putnam. However, the argument which Bub offers is not a good one. At the same time, contrary to Hellman, quantum logic really does provide an explanation of the Lüders rule, and one which is superior to that of the Copenhagen account, since it provides an understanding of why there should be a projection postulate at all.
Friedman and Putnam have argued (Friedman and Putnam 1978) that the quantum logical interpretation of quantum mechanics gives us an explanation of interference that the Copenhagen interpretation cannot supply without invoking an additional ad hoc principle, the projection postulate. I show that it is possible to define a notion of equivalence of experimental arrangements relative to a pure state φ, or (correspondingly) equivalence of Boolean subalgebras in the partial Boolean algebra of projection operators of a system, which plays a role in the Copenhagen explanation of interference analogous to the role played by the material equivalence, given φ, of certain propositions in the Friedman-Putnam quantum logical analysis. I also show that the quantum logical interpretation and the Copenhagen interpretation are equally capable of avoiding the paradoxical conclusion of the Einstein-Podolsky-Rosen argument (Einstein, Podolsky, and Rosen 1935). Thus, neither interference phenomena nor the correlations between separated systems provide a test case for distinguishing between the relative acceptability of the Copenhagen interpretation and the quantum logical interpretation as explanations of quantum effects.
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).
In this paper I formalize the notion of minimal disturbance, as this seems to be required by usual interpretations of the theory of quantum mechanics, and construct a quantum logical (lattice) model of the type of situation that seems to be at the root of the problem of the interpretation of Luders’ projection rule as a criterion of minimal disturbance for individual state transformations. What is particularly interesting in the situation to be depicted here is that, on the basis of a simple model, which depends only on some very general features of the lattice structure of the theory (and its semantical interpretation), usual interpretive assumptions on minimal disturbance appear to be wanting. If we restrict our attention to the statistics of measurement results, Luders’ rule can, easily be interpreted as a formula describing ‘minimal change’ for statistical states.
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 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.