  We generalize recent work of Massar and Popescu dealing with the amount of
classical data that is produced by a quantum measurement on a quantum state
ensemble. In the previous work it was shown how spurious randomness generally
contained in the outcomes can be eliminated without decreasing the amount of
knowledge, to achieve an amount of data equal to the von Neumann entropy of the
ensemble. Here we extend this result by giving a more refined description of
what constitute equivalent measurements (that is measurements which provide the
same knowledge about the quantum state) and also by considering incomplete
measurements. In particular we show that one can always associate to a POVM
with elements a_j, an equivalent POVM acting on many independent copies of the
system which produces an amount of data asymptotically equal to the entropy
defect of an ensemble canonically associated to the ensemble average state and
the initial measurement (a_j). In the case where the measurement is not
maximally refined this amount of data is strictly less than the von Neumann
entropy, as obtained in the previous work. We also show that this is the best
achievable, i.e. it is impossible to devise a measurement equivalent to the
initial measurement (a_j) that produces less data. We discuss the
interpretation of these results. In particular we show how they can be used to
provide a precise and model independent measure of the amount of knowledge that
is obtained about a quantum state by a quantum measurement. We also discuss in
detail the relation between our results and Holevo's bound, at the same time
providing a new proof of this fundamental inequality.
