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A Dilemma for the Traditional Interpretation of Quantum Mixtures

Published online by Cambridge University Press:  28 February 2022

Nancy Cartwright
Nancy Cartwright*
Affiliation:
University of Maryland
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Extract

This paper argues that the ignorance interpretation of mixtures is physically unrealistic. The ignorance interpretation is the orthodox interpretation for mixtures, and should not be confused with the ignorance interpretation for superpositions, which has been largely abandoned. Mixtures, unlike superpositions, do not interfere. They are represented by mixed (or non-idempotent, i.e. W2 ≠ W) operators; superpositions, by pure (or idempotent) operators or by vectors. In the minimal interpretation both pure and mixed operators may be taken to describe collections. Any pure state, ψ, may be expressed as a sum of other pure states, ϕ 1, ϕ2, …, ϕn. Yet we cannot postulate that the members of the collection described by ψ are each in one of the pure states ϕ1, ϕ2, …, ϕn. This is because of the interference between the pure ϕ1, …, ϕn. On the other hand, if we have a mixture of ϕ1, …, ϕn, we can consistently postulate that the members of the collections are each in one of the pure states ϕ1, …, ϕn.

Type
Part VII Philosophical Problems of the Physical Sciences
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1972 , 1972 , pp. 251 - 258
Copyright
Copyright © 1974 by D. Reidel Publishing Company

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References

Notes

1 Neumanm, John von, The Mathematical Foundations of Quantum Mechanics, (trans, by Beyer, R.), Princeton University Press, Princeton, 1955.Google Scholar

2 It has been suggested that the difficulties with interaction operators arise not from the ignorance interpretation, but rather from the principle - often called the axiom of reduction - which tells us that the components are each in a mixed state at the conclusion of an interaction. It is, however, empirically well-confirmed that the statistical results in a collection of components formed in the manner suggested will be described by the mixed operator predicted by the axiom of reduction. So if the intent of the suggestion is to deny that such collections are described by mixed operators, it must be rejected on empirical grounds. On the other hand, the suggestion may be intended to point out that, even though the collection is represented by a mixed operator, we cannot use this fact to make inferences about the state of the individual members in the usual way. But this is simply to urge that the ordinary model for mixtures does not apply to interaction operators. So, without more details, denying the axiom of reduction is tantamount to denying the applicability of the ignorance interpretation to interaction mixtures. The real question to be answered at this point is not which of the two principles is at fault, but rather what are the inferences about individuals which can be drawn in interaction cases.

3 Feyerabend, P. K., ‘On the Quantum Theory of Measurement’, Observation and Interpretation, (ed. by Körner, S.), Academic Press, New York, 1957, pp. 121-130.Google Scholar

4 Fano, U., ‘Description of States in Quantum Mechanics by Density Matrix and Operator Techniques’, Reviews of Modern Physics 29 (1957) 74ff.CrossRefGoogle Scholar

5 The de-excited atoms cannot be in an eigenstate of L′z, for z′≠z. For no member of a collection represented by W can be in state which assigns a positive probability to an outcome to which W assigns a zero probability. But any eigenstate of L′z, z′≠z, will assign non-zero probability to the prohibited outcomes mz = ±2.