  We characterize the set of generalized quantum measurements that can be
decomposed into a continuous measurement process using a stream of probe qubits
and a tunable interaction Hamilto- nian. Each probe in the stream interacts
weakly with the target quantum system, then is measured projectively in a
standard basis. This measurement result is used in a closed feedback loop to
tune the interaction Hamiltonian for the next probe. The resulting evolution is
a stochastic process with the structure of a one-dimensional random walk. To
maintain this structure, and require that at long times the measurement
outcomes be independent of the path, the allowed interaction Hamil- tonians
must lie in a restricted set, such that the Hamiltonian terms on the target
system form a finite dimensional Jordan algebra. This algebraic structure of
the interaction Hamiltonians yields a large class of generalized measurements
that can be continuously performed by our scheme, and we fully describe this
set.
