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arXiv:quant-ph/0112178 · Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

On a supposed conceptual inadequacy of the Shannon information in quantum mechanics

Christopher G. Timpson

Published Jun 30, 2003• 74 citations
arXivHTMLPDF
DOI
10.1016/s1355-2198(03)00037-6

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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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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.
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.