is pure then the system can be controlled precisely - that is, any pure state can be prepared. Thus the entropy measures the extent to which a pure state, or pure evolution can be obtained, and thus the level of predictability which can be achieved over the future behavior of the system [@note3]. The primary role of measurement in feedback control is therefore to reduce the entropy of the system. As such the average reduction in von Neumann entropy provides a ranking of the effectiveness of different measurements for feedback control, other things being equal. Further details regarding quantum feedback control and von Neumann entropy can be found in reference. [@DJJ]

The entropy reduction is also relevant to the transformation of pure-state entanglement, since the von Neumann entropy measures the entanglement of pure states. As a result this quantity gives the amount by which pure-state entanglement is broken by a local measurement.

We give two corollaries of the general information bound derived in the first part which involve $\langle
\Delta S(\rho) \rangle$. The first is a generalization of a bound derived by Hall [@Hall; @ib] to inefficient measurements. Hall’s bound states that for efficient measurements the mutual information is bounded 