  A fundamental requirement for the emergence of classical behavior from an
underlying quantum description is that certain observed quantum systems make a
transition to chaotic dynamics as their action is increased relative to
$\hbar$. While experiments have demonstrated some aspects of this transition,
the emergence of quantum trajectories with a positive Lyapunov exponent has
never been observed directly. Here, we remove a major obstacle to achieving
this goal by showing that, for the Duffing oscillator, the transition to a
positive Lyapunov exponent can be resolved clearly from observed trajectories
even with measurement efficiencies as low as 20%. We also find that the
positive Lyapunov exponent is robust to highly mixed, low purity states and to
variations in the parameters of the system.
