= Solution
The target specifies the desired output; the controller selects a <control input>; an actuator realizes it on the system; and the sensor returns a measured output. The environment supplies additional interactions and disturbances. A simple loop is:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-52-control-loop.png]
{title=Quantum control loop, showing the measurement record and measurement backaction}
For a quantum sensor, a <quantum measurement> has statistical outcomes and generally changes the state through <measurement backaction>. For example, outcome $m$ of an instrument with operator $K_m$ has <probability> and conditional state
$$
p_m=\operatorname{Tr}(K_m\rho K_m^\dagger),\qquad
\rho_m=\frac{K_m\rho K_m^\dagger}{p_m}.
$$
A single unknown <quantum state> cannot generally supply its complete instantaneous state vector to a controller. Noncommuting <observables> cannot be read simultaneously with arbitrary precision, and reliable <quantum state tomography> normally needs many reproducible preparations. A conditional state estimate can instead be updated from a continuous record, with its disturbance included in the model.
Quantum actuators change a <Quantum Hamiltonian> through applied electromagnetic fields, tunable interactions or couplings to auxiliary systems. A closed-system actuator implements <unitary operators>, so it preserves the <spectrum> and <purity of a density operator>; it cannot arbitrarily overwrite a state as a classical assignment operation could. Dissipative actuators or <quantum measurements> can change these invariants, but their dynamics must be accounted for.
The environment can entangle with the system, causing <decoherence> and dissipation when unobserved degrees of freedom are traced out. It is consequently part of the dynamical model, rather than just an additive classical disturbance. Conversely, <Markovian reservoir engineering> uses such coupling constructively to prepare and stabilize states.
In <open-loop control> the applied waveform is prescribed without using measurements from the current evolution. In <closed-loop control> it depends on the available measurement record or inferred state. <Measurement-based quantum feedback> is one realization of the latter; experimental adaptation across repeated preparations is another. The measurement record is classical information, but its production has quantum backaction.
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