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Philosophy of Science

Lüders's Rule as a Description of Individual State Transformations

Sergio F. Martínez

Published Sep 1, 1991• 15 citations
Publisher

Abstract

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.

DOI
10.1086/289622

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An interpretation of quantum mechanics that rejects hidden variables has to say something about the way measurement can be understood as a transformation on states of individual systems, and that leads to the core of the interpretive problems posed by Luders' projection rule: What, if any, is its physical content? In this paper I explore one suggestion which is implicit in usual interpretations of the rule and show that this view does not stand on solid ground. In the process, important aspects of the role played by the projection postulate in the conceptual structure of quantum mechanics will be clarified. It will be shown in particular that serious objections can be raised against the (often implicit) view that identifies the physical relation of compatibility preserved by Luders' rule with the relation of simultaneous measurability. Preparatory measurements are measurements for which we can con- clude on the basis of the result of measurement that the system is (immediately after measurement) in an eigenstate of the eigenvalue corresponding to the given result. Preparatory measurements are first of all experimental processes that select an ensemble to which a quantum (statistical) state can be assigned. Statistical interpretations of the state vector take as fundamental this assignment of a quantum state to (sub)ensembles selected by experimental procedures. But if the exist- ence of hidden variables is denied, then it seems that it should be possible to interpret preparatory measurements in terms of individual state transformations. This can be done by thinking of these measure- ments as idealized measurement transformations satisfying the so-called first kind condition: an immediate repetition of the same measurement gives the same result as the initial measurement. This idealization (taken here for the sake of simplicity) will be assumed here to be non-controversial. Furthermore, it will be assumed here that every measurement can be represented as a first kind measurement: thus, a 'measurement' in this investigation is always to be a 'first kind measure- ment'. For measurements of maximal magnitudes (i.e., magnitudes with no degenerate eigenvalue) Von Neumann argued (1955) that the final state is given by the (unique) eigenstate of the eigenvalue representing the
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Are there consistent and physically reasonable alternatives to the projection postulate? Does it have unique properties compared to acceptable alternatives? We answer these questions by systematically investigating hypothetical state-update rules for quantum systems that Nature could have chosen over the Lüders rule. Among other basic properties, any prospective rule must define unique post-measurement states and not allow for superluminal signalling. Particular attention will be paid to consistently defining post-measurement states when performing local measurements in composite systems. Explicit examples of valid unconventional update rules are presented, each resulting in a distinct, well-defined foil of quantum theory. This framework of state-update rules allows us to identify operational properties that distinguish the projective update rule from all others and to put earlier derivations of the projection postulate into perspective.
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