Global versus local optimality in feedback-controlled qubit purification: new insights from minimizing R\'enyi entropies

It was first shown by Jacobs, in 2003, that the process of qubit state purification by continuous measurement of one observable can be enhanced, on average, by unitary feedback control. Here, we quantify this by the reduction in any one of the family of R\'enyi entropies $S_\alpha$, with $0< \alpha < \infty$, at some terminal time, revealing the rich structure of stochastic quantum control even for this simple problem. We generalize Jacobs' original argument, which was for (the unique) impurity measure with a linear evolution map under his protocol, by replacing linearity with convexity, thereby making it applicable to R\'enyi entropies $S_\alpha$ for $\alpha$ in a finite interval about $1$. Even with this generalization, Jacobs' argument fails to identify the surprising fact, which we prove by Bellman's principle of dynamic programming, that his protocol is globally optimal for all R\'enyi entropies whose decrease is locally maximized by that protocol. Also surprisingly, even though there is a range of R\'enyi entropies whose decrease is always locally maximized by the null-control protocol, that null-control protocol cannot be shown to be globally optimal in any instance. These results highlight the non-intuitive relation between local and global optimality in stochastic quantum control.