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 the system’s state ρ 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, where J is the total angular momentum). 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 locally optimal control protocol of this type, one can attain the best possible scaling, O(D2), 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 qubits.
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