  In this paper, we study the recursion of measurement outcomes for open
quantum networks under sequential measurements. Open quantum networks are
networked quantum subsystems (e.g., qubits) with the state evolutions described
by a continuous Lindblad master equation. When measurements are performed
sequentially along such continuous dynamics, the quantum network states undergo
random jumps and the corresponding measurement outcomes can be described by a
vector of probabilistic Boolean variables. The induced recursion of the Boolean
vectors forms a probabilistic Boolean network. First of all, we show that the
state transition of the induced Boolean networks can be explicitly represented
through realification of the master equation. Next, when the open quantum
dynamics is relaxing in the sense that it possesses a unique equilibrium as a
global attractor, structural properties including absorbing states,
reducibility, and periodicity for the induced Boolean network are direct
consequences of the relaxing property. Particularly, we show that generically,
relaxing quantum dynamics leads to irreducible and aperiodic chains for the
measurement outcomes. Finally, we show that for quantum consensus networks as a
type of non-relaxing open quantum network dynamics, the communication classes
of the measurement-induced Boolean networks are encoded in the quantum
Laplacian of the underlying interaction graph.
