  A $D$-dimensional Markovian open quantum system will undergo quantum jumps
between pure states, if we can monitor the bath to which it is coupled with
sufficient precision. In general these jumps, plus the between-jump evolution,
create a trajectory which passes through infinitely many different pure states.
Here we show that, for any ergodic master equation, one can expect to find an
{\em adaptive} monitoring scheme on the bath that can confine the system state
to jumping between only $K$ states, for some $K \geq (D-1)^2+1$. For $D=2$ we
explicitly construct a 2-state ensemble for any ergodic master equation,
showing that one bit is always sufficient to track a qubit.
