Compared with many other methods which only give time sub-optimal designs, the quantum brachistochrone equation has a great potential to provide accurate time-optimal protocols for essentially any quantum control problem. So far it has been of limited use, however, due to the inadequacy of conventional numerical methods to solve it. Here, using differential geometry, we reformulate the quantum brachistochrone curves as geodesics on the unitary group. This identification allows us to design a numerical method that can efficiently solve the brachistochrone problem by first solving a family of geodesic equations.
Quantum systems may be used to transmit classical information. To do this the sender encodes information by preparing the system in one of an alphabet of possible states, and sends it to the receiver. The receiver then performs a measurement on the system in order to obtain information about which state was sent. Here we describe a general bound on the information which is accessible to the receiver in such a channel when the receivers measurement is noisy. In addition to extracting classical information, measurements also reduce the entropy of a quantum system. This is important, for example, in quantum feedback control. We discuss two corollaries of the information bound that involve this entropy reduction.
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.
Nano-electro-mechanical devices are now rapidly approaching the point where it will be possible to observe quantum mechanical behavior. However, for such behavior to be visible it is necessary to reduce the thermal motion of these devices down to temperatures in the millikelvin range. Here we consider the use of feedback control for this purpose. We analyze an experimentally realizable situation in which the position of the resonator is continuously monitored by a Single-Electron Transistor. Because the resonator is harmonic, it is possible to use a classical description of the measurement process, and we discuss both the quantum and classical descriptions. Because of this the optimal feedback algorithm can be calculated using classical control theory. We examine the quantum state of the controlled oscillator, and the achievable effective temperature. Our estimates indicate that with current experimental technology, feedback cooling is likely to bring the required milliKelvin temperatures within reach.