When a measurement is made on a quantum system in which classical information is encoded, the measurement reduces the observers' average Shannon entropy for the encoding ensemble. This reduction, being the mutual information, is always non-negative. For efficient measurements the state is also purified; that is, on average, the observers' von Neumann entropy for the state of the system is also reduced by a non-negative amount. Here we point out that by rewriting a bound derived by Hall [Phys. Rev. A 55, 100 (1997)], which is dual to the Holevo bound, one finds that for efficient measurements, the mutual information is bounded by the reduction in the von Neumann entropy. We also show that this result, which provides a physical interpretation for Hall's bound, may be derived directly from the Schumacher-Westmoreland-Wootters theorem [Phys. Rev. Lett. 76, 3452 (1996)]. We discuss these bounds, and their relationship to another bound, valid for efficient measurements on pure state ensembles, which involves the subentropy.