  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.
