How Many Bits Does It Take to Track an Open Quantum System?

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 adaptive monitoring scheme on the bath that can confine the system state to jumping between only K states, for some K >= (D - 1)(2) + 1. For D = 2 we explicitly construct a two-state ensemble for any ergodic master equation, showing that one bit is always sufficient to track a qubit.