induced by a measurement, and encompasses the fact that this von Neumann entropy reduction is concave in the initial state.

From the concavity of the mutual information and the von Neumann entropy reduction, it follows that these quantities are Schur-concave (the former naturally for pure-state ensembles) for completely symmetric classical measurements, and for unitarily covariant quantum measurements. Thus the possibility that either of these kinds of information gain [*decreases*]{} with increasing initial uncertainty is associated with the asymmetry of the measurement in question.

Acknowledgments {#acknowledgments .unnumbered}
===============

The author would like to thank Gerard Jungman, Howard Barnum, Howard Wiseman, Terry Rudolph and Michael Hall for helpful discussions. The author is also grateful both to Vlatko Vedral for hospitality during a visit to Imperial College, and Lucien Hardy for hospitality during a visit to the Perimeter Institute where some of the initial stages of this work was carried out. This work was supported by the Australian Research Council and the State of Queensland. Note added: After submitting this manuscript, which was first posted as eprint quant-ph/0412006, it was brought to my attention that the work presented here overlaps with concurrent work by Barchielli and Lupeiri (quant-ph/0409019 and quant-ph/0412116).

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