that the initial state is $\rho$. Since the von Neumann entropy is a measure of how much we know about the state of the system, this is the difference between what we knew about the system state before we made the measurement, and what we know (on average) about the system state at the end of the measurement; it thus measures how much we learn about the final state of the system. Equivalently, it can be said to measure the degree of “state-reduction” which the measurement induces. While it is the mutual information which is important for communication, the reduction in the von Neumann entropy is important for feedback control. Feedback control is the process of performing a sequence of measurements on a system, and applying unitary operations after each measurement in order control the evolution of the system. Such a procedure is useful for controlling systems which are driven by noise. If the ability to perform unitary operations is unlimited, then the von Neumann entropy provides a measure of the level of control which can be achieved: if the system has maximal entropy then the unitary operations have no effect on the system state whatsoever; conversely, if the state 