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Philosophy of Science

The Applicability of Shannon Information in Quantum Mechanics and Zeilinger's Foundational Principle

Christopher G. Timpson

Published Dec 1, 2003• 11 citations
Publisher

Abstract

Recently, Brukner and Zeilinger have presented a number of arguments suggesting that the Shannon information is not well defined as a measure of information in quantum mechanics. If established, this result would be highly significant, as the Shannon information is fundamental to the way we think about information not only in classical but also in quantum information theory. On consideration, however, these arguments are found unsuccessful; I go on to suggest how they might be arising as a consequence of Zeilinger's proposed foundational principle for quantum mechanics.

DOI
10.1086/377403

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In Phys. Rev. A 63 022113 (2001), Brukner and Zeilinger claim that the Shannon information is not well defined as a measure of information in quantum mechanics, adducing arguments that seek to show that it is inextricably tied to classical notions of measurement. It is shown here that these arguments do not succeed: the Shannon information does not have problematic ties to classical concepts. In a further argument, Brukner and Zeilinger compare the Shannon information unfavourably to their preferred information measure, I(p), with regard to the definition of a notion of `total information content'. This argument is found unconvincing and the relationship betwen measures of information and notions of `total information content' investigated.

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Recently, Brukner and Zeilinger have presented a number of arguments suggesting that the Shannon information is not well defined as a measure of information in quantum mechanics. If established, this result would be highly significant, as the Shannon information is fundamental to the way we think about information not only in classical but also in quantum information theory. On consideration, however, these arguments are found unsuccessful; I go on to suggest how they might be arising as a consequence of Zeilinger's proposed foundational principle for quantum mechanics.
In Phys. Rev. A 63 022113 (2001), Brukner and Zeilinger claim that the Shannon information is not well defined as a measure of information in quantum mechanics, adducing arguments that seek to show that it is inextricably tied to classical notions of measurement. It is shown here that these arguments do not succeed: the Shannon information does not have problematic ties to classical concepts. In a further argument, Brukner and Zeilinger compare the Shannon information unfavourably to their preferred information measure, I(p), with regard to the definition of a notion of `total information content'. This argument is found unconvincing and the relationship betwen measures of information and notions of `total information content' investigated.
In Brukner and Zeilinger's interpretation of quantum mechanics, information is introduced as the most fundamental notion and the finiteness of information is considered as an essential feature of quantum systems. They also define a new measure of information which is inherently different from the Shannon information and try to show that the latter is not useful in defining the information content in a quantum object. Here, we show that there are serious problems in their approach which make their efforts unsatisfactory. The finiteness of information does not explain how objective results appear in experiments and what an instantaneous change in the so-called information vector (or catalog of knowledge) really means during the measurement. On the other hand, Brukner and Zeilinger's measure of information is derived basically from a definition of variance for a binomial distribution. This quantifies the dispersion of methods for putting N constituents of a system into two different groups and illustrates, principally, the distinction between the frequency of occurrence and the probability of occurrence of an event. So, it does not qualify as a proper measure of information.
It is pointed out that the case for Shannon entropy and von Neumann entropy, as measures of uncertainty in quantum mechanics, is not as bleak as suggested in quant-ph/0006087. The main argument of the latter is based on one particular interpretation of Shannon's H-function (related to consecutive measurements), and is shown explicitly to fail for other physical interpretations. Further, it is shown that Shannon and von Neumann entropies have in fact a common fundamental significance, connected to the existence of a unique geometric measure of uncertainty for classical and quantum ensembles. Some new properties of the ``total information'' measure proposed in quant-ph/0006087 are also given.
This thesis is a contribution to the debate on the implications of quantum information theory for the foundations of quantum mechanics. In Part 1, the logical and conceptual status of various notions of information is assessed. It is emphasized that the everyday notion of information is to be firmly distinguished from the technical notions arising in information theory; however it is maintained that in both settings `information' functions as an abstract noun, hence does not refer to a particular or substance (the worth of this point is illustrated in application to quantum teleportation). The claim that `Information is Physical' is assessed and argued to face a destructive dilemma. Accordingly, the slogan may not be understood as an ontological claim, but at best, as a methodological one. The reflections of Bruckner and Zeilinger (2001) and Deutsch and Hayden (2000) on the nature of information in quantum mechanics are critically assessed and some results presented on the characterization of entanglement in the Deutsch-Hayden formalism. Some philosophical aspects of quantum computation are discussed and general morals drawn concerning the nature of quantum information theory. In Part II, following some preliminary remarks, two particular information-theoretic approaches to the foundations of quantum mechanics are assessed in detail. It is argued that Zeilinger's (1999) Foundational Principle is unsuccessful as a foundational principle for quantum mechanics. The information-theoretic characterization theorem of Clifton, Bub and Halvorson (2003) is assessed more favourably, but the generality of the approach is questioned and it is argued that the implications of the theorem for the traditional foundational problems in quantum mechanics remains obscure.
Some approaches which seek to make progress with the conceptual troubles of quantum theory using information-theoretic ideas are investigated. These include Zeilinger’s Foundational Principle and the information-theoretic characterization theorem of Clifton, Bub and Halvorson. It is shown that Zeilinger’s Foundational Principle does not provide a foundational principle for quantum mechanics and fails to underwrite explanations of the randomness of quantum measurement and the existence of entanglement. The Clifton- Bub-Halvorson theorem is more positively assessed: here an axiomatisation of quantum mechanics has been achieved. However some questions are raised concerning the C ∗-algebraic starting point of the theorem, which seems not a sufficiently neutral theoretical framework for the axiomatic project. Moreover, it is argued that rather than the Clifton-Bub-Halvorson result motivating an information-theoretic approach (principle theory approach) to understanding the quantum world which obviates our traditional concerns, such an approach simply fails to engage with the crucial interpretational issues.