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The Holevo bound is a bound on the mutual information for a given quantumencoding. In 1996 Schumacher, Westmoreland and Wootters [Schumacher,Westmoreland and Wootters, Phys. Rev. Lett. 76, 3452 (1996)] derived a boundwhich reduces to the Holevo bound for complete measurements, but which istighter for incomplete measurements. The most general quantum operations may beboth incomplete and inefficient. Here we show that the bound derived by SWW canbe further extended to obtain one which is yet again tighter for inefficientmeasurements. This allows us in addition to obtain a generalization of a boundderived by Hall, and to show that the average reduction in the von Neumannentropy during a quantum operation is concave in the initial state, for allquantum operations. This is a quantum version of the concavity of the mutualinformation. We also show that both this average entropy reduction, and themutual information for pure state ensembles, are Schur-concave for unitarilycovariant measurements; that is, for these measurements, information gainincreases with initial uncertainty.Comment: 7 pages, revtex4. v2: published version; note added regarding similar  work by Barchielli and Lupier

A bound on the mutual information, and properties of entropy reduction, for quantum channels with inefficient measurements