  To maximize average information gain for a classical measurement, all
outcomes of an observation must be equally likely. The condition of equally
likely 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 of
measurement of a component of angular momentum for a qudit (a $D$-dimensional
system, with $D=2J+1$). For weak or continuous-in-time (i.e. repeated weak)
measurements, we show that the complementarity condition ensures an average
improvement, in the rate of purification, of only 2. However, we show that by
choosing the optimal control protocol of this type, one can attain the best
possible scaling, $O(D^{2})$, for the average improvement. For this protocol
the acquisition of information is nearly deterministic. Finally we contrast
these results with those for complementarity-based protocols in a register of
qbits.

Quantum Physics