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.
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机译：为了最大程度地提高经典测量的平均信息增益，观测结果的所有结果也必须具有同等可能性。在量子理论中，可以通过确保一个人的状态与测得的可观测值最大不同或互补来实施同样可能的结果条件。这要求以先验结果为条件，对该状态执行统一运算的能力。测量。我们考虑测量一个量角动量分量的情况（一个$ D $维系统，其中$ D = 2J + 1 $）。对于弱的或连续的（即反复的弱）测量，我们表明互补条件确保纯化率的平均提高仅为2。但是，我们表明通过选择这种类型的最佳控制方案，可以达到最佳的缩放比例$ O（D ^ {2}）$，以实现平均改善。对于此协议，信息的获取几乎是确定性的。最后，我们将这些结果与qbit寄存器中基于互补性协议的结果进行对比。
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