  The standard quantum formalism introduced at the undergraduate level treats
measurement as an instantaneous collapse. In reality however, no physical
process can occur over a truly infinitesimal time interval. A more subtle
investigation of open quantum systems lead to the theory of continuous
measurement and quantum trajectories, in which wave function collapse occurs
over a finite time scale associated with an interaction. Within this formalism,
it becomes possible to ask many new questions that would be trivial or even
ill-defined in the context of the more basic measurement model. In this thesis,
we investigate both theoretically and experimentally what fundamentally new
capabilities arise when an experimental apparatus can resolve the continuous
dynamics of a measurement. Theoretically, we show that when one can perform
feedback operations on the timescale of the measurement process, the resulting
tools provide significantly more control over entanglement generation, and in
some settings can generate it optimally. We derive these results using a novel
formalism which encompasses most known quantum feedback protocols.
Experimentally, we show that continuous measurement allows one to observe the
dynamics of a system undergoing simultaneous non-commuting measurements, which
provides a reinterpretation of the Heisenberg uncertainty principle. Finally,
we combine the theoretical focus on quantum feedback with the experimental
capabilities of superconducting circuits to implement a feedback controlled
quantum amplifier. The resulting system is capable of adaptive measurement,
which we use to perform the first canonical phase measurement.
