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Physical Review Letters

A No-Go Theorem for Joint Property Ascriptions in Modal Interpretations of Quantum Mechanics

Pieter E. Vermaas

Published Mar 17, 1997• 43 citations
Publisher

Abstract

With an interpretation of quantum mechanics one can not only predict probabilities for outcomes of measurements performed on quantum systems but also ascribe properties to these systems themselves. I consider interpretations (notably modal interpretations) that ascribe at least those properties to a system which are represented by the eigenprojections of the reduced state. Such interpretations solve the measurement problem for ideal measurements. Here, however, I prove that these interpretations cannot define joint property ascriptions to three or more systems.

DOI
10.1103/physrevlett.78.2033

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With an interpretation of quantum mechanics one can not only predict probabilities for outcomes of measurements performed on quantum systems but also ascribe properties to these systems themselves. I consider interpretations (notably modal interpretations) that ascribe at least those properties to a system which are represented by the eigenprojections of the reduced state. Such interpretations solve the measurement problem for ideal measurements. Here, however, I prove that these interpretations cannot define joint property ascriptions to three or more systems.
The aim of this paper is to introduce a new member of the family of the modal interpretations of quantum mechanics. In this modal-Hamiltonian interpretation, the Hamiltonian of the quantum system plays a decisive role in the property-ascription rule that selects the definite-valued observables whose possible values become actual. We show that this interpretation is effective for solving the measurement problem, both in its ideal and its non-ideal versions, and we argue for the physical relevance of the property-ascription rule by applying it to well-known physical situations. Moreover, we explain how this interpretation supplies a description of the elemental categories of the ontology referred to by the theory, where quantum systems turn out to be bundles of possible properties.
Modal interpretations are hidden-variable, no-collapse interpretations of quantum mechanics that were designed to solve the measurement problem and reconcile this theory with relativity. Yet, as no-go theorems by Dickson and Clifton (1998), Arntzenius (1998) and Myrvold (2002) demonstrate, current modal interpretations are incompatible with relativity. In the mainstream modal interpretations, properties of composite systems are generally unrelated to the properties of their subsystems. We propose holistic and relational interpretations of properties to explain this failure of property composition. Based on these interpretations, we consider strategies for circumventing Myrvold's theorem, which are also effective against the other two theorems.
In this chapter I introduce the different versions of the modal interpretation, including the three on which I focus in this book. The property ascription of these versions is characterised by the map W α ↦ {〈p j , C α j 〉} j which I call the core property ascription. In the next chapter I discuss how this core property ascription determines the full property ascription W α ↦ The best modal interpretation The best imaginable modal interpretation is, I guess, an interpretation which (A) ascribes at all times all the properties to a system which pertain to that system, and (B) ascribes these properties such that the classical logical relations between the negation, conjunction and disjunction of properties are satisfied. The content of the first requirement (A) is clear: assuming that every projection onto a subspace of a Hilbert space ℋ α represents one and only one property of α, it follows that all sets of definite-valued projections should contain all the projections in ℋ α and that all the maps {. j } j should be maps from all the projections in ℋ α to the values {0,1}. The content of requirement (B) is, however, less clear because there is consensus neither about how to define the negation, conjunction and disjunction of properties in quantum mechanics nor about how to impose the logical relations. In Section 5.1 I present my choice for the definitions of the negation, conjunction and disjunction.
We describe an interpretation of quantum mechanics based on reduced density matrices of sub-systems from which the standard Copenhagen interpretation emerges as an effective description for macro-systems. The interpretation is a modal one, but does not suffer from the range of problems that plague other modal interpretations. The key feature is that quantum states carry an additional property assignment in the form of one the eigenvectors of the reduced density matrix which evolves evolves according to a stochastic process driven by the unmodified Schrodinger equation, but it is usually hidden from the emergent classical description due to the ergodic nature of its dynamics. However, during a quantum measurement, ergodicity is broken by decoherence and definite outcomes occur with probabilities that agree with the Born rule.
Orthodox quantum mechanics includes the principle that an observable of a system possesses a well-defined value if and only if the presence of that value in the system is certain to be confirmed on measurement. Modal interpretations reject the controversial ‘only if’ half of this principle to secure definite outcomes for quantum measurements that leave the apparatus entangled with the object it has measured. However, using a result that turns on the construction of a Kochen–Specker contradiction, I argue that modal interpretations cannot deliver a metaphysically tenable conception of properties in quantum mechanics unless they also abandon the less controversial ‘if’ half of the orthodox principle.