  We examine most-likely paths between initial and final states for diffusive
quantum trajectories in continuously monitored pure-state qubits, obtained as
extrema of a stochastic path integral. We demonstrate the possibility of
"multipaths" in the dynamics of continuously-monitored qubit systems, wherein
multiple most-likely paths travel between the same pre- and post-selected
states over the same time interval. Most-likely paths are expressed as
solutions to a Hamiltonian dynamical system. The onset of multipaths may be
determined by analyzing the evolution of a Lagrange manifold in this phase
space, and is mathematically analogous to the formation of caustics in ray
optics or semiclassical physics. Additionally, we develop methods for finding
optimal traversal times between states, or optimal final states given an
initial state and evolution time; both give insight into the measurement
dynamics of continuously-monitored quantum states. We apply our methods in two
systems: a qubit with two non-commuting observables measured simultaneously,
and a qubit measured in one observable while subject to Rabi drive. In the
two-observable case we find multipaths due to caustics, bounded by a diverging
Van-Vleck determinant, and their onset time. We also find multipaths generated
by paths with different "winding numbers" around the Bloch sphere in both
systems.
