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To maximize average information gain for a classical measurement, alloutcomes of an observation must be equally likely. The condition of equallylikely outcomes may be enforced in quantum theory by ensuring that one's state$\rho$ is maximally different, or complementary, to the measured observable.This requires the ability to perform unitary operations on the state,conditioned on the results of prior measurements. We consider the case ofmeasurement of a component of angular momentum for a qudit (a $D$-dimensionalsystem, with $D=2J+1$). For weak or continuous-in-time (i.e. repeated weak)measurements, we show that the complementarity condition ensures an averageimprovement, in the rate of purification, of only 2. However, we show that bychoosing the optimal control protocol of this type, one can attain the bestpossible scaling, $O(D^{2})$, for the average improvement. For this protocolthe acquisition of information is nearly deterministic. Finally we contrastthese results with those for complementarity-based protocols in a register ofqbits.

Maximum information gain in weak or continuous measurements of qudits: complementarity is not enough