Quantum control theory and applications: A survey <s> Continuous measurement <s> We consider a two-level quantum system (qubit) that is continuously measured by a detector. The information provided by the detector is taken into account to describe the evolution during a particular realization of the measurement process. We discuss the Bayesian formalism for such ``selective'' evolution of an individual qubit and apply it to several solid-state setups. In particular, we show how to suppress qubit decoherence using continuous measurement and a feedback loop. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Continuous measurement <s> We consider the use of feedback control during a measurement to increase the rate at which a quantum system is purified, and more generally the rate at which near-pure states may be prepared. We derive the optimal bang-bang algorithm for rapid state preparation from an initial completely mixed state when the measurement basis is unrestricted, and evaluate its performance numerically. We also consider briefly the case in which the measurement basis is fixed with respect to the state to be prepared, and describe the qualitative structure of the optimal bang-bang algorithm. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Continuous measurement <s> We present a pedagogical treatment of the formalism of continuous quantum measurement. Our aim is to show the reader how the equations describing such measurements are derived and manipulated in a direct manner. We also give elementary background material for those new to measurement theory, and describe further various aspects of continuous measurements that should be helpful to those wanting to use such measurements in applications. Specifically, we use the simple and direct approach of generalized measurements to derive the stochastic master equation describing the continuous measurements of observables, give a tutorial on stochastic calculus, treat multiple observers and inefficient detection, examine a general form of the measurement master equation, and show how the master equation leads to information gain and disturbance. To conclude, we give a detailed treatment of imaging the resonance fluorescence from a single atom as a concrete example of how a continuous position measurement arises in a physical system. <s> BIB003
Projective measurements are treated as instantaneous and this model is reasonable when the strength of the measurement (i.e., the coupling strength between the measurement apparatus and the measured system) is large enough and the measurement time-scale is much shorter than all other related time-scales for a given task BIB002 . However, this framework may not be sufficient to describe some important situations such as continuously monitoring some aspects of a quantum system  , BIB003 . In quantum feedback control, it is important to continuously extract feedback information to adjust the system evolution. Hence, there is an essential requirement to develop a continuous measurement theory for quantum feedback control. Fortunately, recent results have shown that continuous measurements are experimentally realizable for practical quantum systems such as solid-state qubits BIB001 . In continuous measurements, one can continuously monitor an observable of a quantum system and the evolution of the system described in terms of the measurement record can be obtained from a stochastic master equation BIB003 . Continuous measurements can be derived from projective measurements under appropriate assumptions. For example, to implement a continuous measurement on an atom, we let the atom interact with an auxiliary quantum system (e.g., an electromagnetic field), and then make projective measurements on the auxiliary system (e.g., using a photodetector). In a short time interval, we extract only partial information about the atom. Hence, we also refer to continuous measurement as weak measurement or continuous weak measurement. For a detailed introduction to continuous measurement the reader can refer to BIB003 . Also, the stochastic master equation which describes a quantum system under continuous measurement will be presented in the following subsection.