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Crítica (México D F En línea)

Mediciones ideales en la mecánica cuántica

Sergio F. Martínez

Published Dec 10, 1988• 0 citations
Publisher

Abstract

As a series of investigations have shown, the interpretation of the change upon measurement in quantum mechanics described by the "projection pastulate", as a purely statistical formula, is clear. This formula, here denoted by VNL, is a version of the conditional expectation in Hilbert spaces, and this mathematical result can be given a salid physical interpretation in terlDl of quantum statistics. The problem of interpretation arises when the formula comes in for interpretation, as a description of what happens to (states of) individual systems in measurement. Usual interpretations see the formula VNL as a description of a class of measurement transformations which are "minimally disturbing". However, a series of arguments show that such an interpretation is seriouslf flawed. (See Teller (1983), Martínez (1988) and references therein.] In Martínez (1987) I have derived the formula VNL from simple physical assumptions in a lattice theoretical framework. The formula so derived describes individual state transformations of a certain type. The states involved are states that describe the properties a system has relative to magnitudes (measuring situations). In this paper I propase that the change of state on measurement deseribed by the formula VNL, as derived in (1987), can be understood as a change on individual states along the lines of Von Neumann's initial proposal for interpreting non-maximal measurements. I discuss the objections raised by Lüders and others to Von Neumann's idea and show that they do not apply to the interpretation proposed here. This proposal has the advantage, among others, that it does not need to reify a controversial class of "minimally disturbing measurement transformations".

DOI
10.22201/iifs.18704905e.1988.680

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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.
Usual derivations of Lüders's projection rule show that Lüders's rule is the rule required by quantum statistics to calculate the final state after an ideal (minimally disturbing) measurement. These derivations are at best inconclusive, however, when it comes to interpreting Lüders's rule as a description of individual state transformations. In this paper, I show a natural way of deriving Lüders's rule from well-motivated and explicit physical assumptions referring to individual systems. This requires, however, the introduction of a concept of individual state which is not standard.
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.
Much of the recent discussion of problematic aspects of quantum-mechanical measurement centers around that feature of quantum theory which is called “the projection postulate.” This is roughly the claim that a change of a certain sort occurs in the state of a physical system when a measurement is made on the system. In this paper an argument for the projection postulate due to von Neumann is considered. Attention is focused on trying to provide an understanding of the notion of “the state of a physical system” which is compatible with the argument von Neumann offers. An attempt is made to formulate the argument in terms of an objectivistic interpretation of probability concepts. It is seen that such an interpretation does not provide a suitable way of understanding the argument. An attempt is made to illustrate the source of this failure in terms of a non-quantum-mechanical example.
An alternative to the Lüders generalization of the von Neumann projection is formulated allowing a generalization to measurements represented by positive operator-valued measures. This generalization is based on the notions of non-orthogonal projections and bi-orthonormal systems in Hilbert-Schmidt space, and is closely related to phase-space representations of quantum mechanics. A number of applications will be discussed. It is demonstrated that the alternative projection has a natural interpretation as a representation of the information on the initial state, to be obtained by the measurement.