A qubit which is prepared in one of two non-orthogonal states and subjected to bit-flipping noise is considered. The objective is to use measurement and feedback control to correct the state of the qubit. Three classical schemes using projective measurements, i.e. discrimination and re-preparation, do nothing and random preparation, have been discussed, and are not optimal with respect to a performance which is quantified by the average fidelity of the corrected state compared to the initial state. In addition, one quantum scheme using a non-projective measurement with an optimum measurement strength achieves the best trade-off between gaining information about the system and disturbing it through measurement back-action. The performance of a quantum control scheme outperforms the classical schemes. Furthermore, no universal corrected scheme is discussed.
A qubit which is prepared in one of two non-orthogonal states and subjected to bit-flipping noise is considered. The objective is to use measurement and feedback control to correct the state of the qubit. Three classical schemes using projective measurements, i.e. discrimination and re-preparation, do nothing and random preparation, have been discussed, and are not optimal with respect to a performance which is quantified by the average fidelity of the corrected state compared to the initial state. In addition, one quantum scheme using a non-projective measurement with an optimum measurement strength achieves the best trade-off between gaining information about the system and disturbing it through measurement back-action. The performance of a quantum control scheme outperforms the classical schemes. Furthermore, no universal corrected scheme is discussed.
Measurements in quantum mechanics cannot perfectly distinguish all states and necessarily disturb the measured system. We present and analyze a proposal to demonstrate fundamental limits on quantum control of a single qubit arising from these properties of quantum measurements. We consider a qubit prepared in one of two nonorthogonal states and subsequently subjected to dephasing noise. The task is to use measurement and feedback control to attempt to correct the state of the qubit. We demonstrate that projective measurements are not optimal for this task, and that there exists a nonprojective measurement with an optimum measurement strength which achieves the best trade-off between gaining information about the system and disturbing it through measurement backaction. We study the performance of a quantum control scheme that makes use of this weak measurement followed by feedback control, and demonstrate that it realizes the optimal recovery from noise for this system. We contrast this approach with various classically inspired control schemes.
We consider an open quantum system subjected to a noise channel under measurement-based feedback control and two prototypical classes of decoherence channels are considered: phase damping and generalized amplitude damping. Based on quantum trajectory theory, we obtain an extended master equation for the dynamics of the reduced system in the presence of feedback control. For a qubit system we analytically solve this master equation and obtain the solution of the state vector dynamics. Then we propose an effective feedback control scheme for preparing an arbitrary quantum pure state. We also study how to protect two nonorthogonal states effectively, and find that projective measurement with unbiased basis is not optimal for this task, while weak measurement with biased basis could realize the best protection of two nonorthogonal states. Furthermore, the inefficiencies in the feedback process are also discussed.
This thesis addresses the of developing a counter-part of the well established classical theory of control. We dwell on the fundamental fact that states are generally not perfectly distinguishable, and measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the setting. We center our investigations around the of transforming the state of a system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the quantum tracking problem and show how it can be formulated as an optimization that can be approached both numerically and analytically. This provides a simple route to the characterization of the trade-off between information gain and disturbance, and is seen to have several applications in information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of measurements. We also demonstrate that our optimal controllers are immediately applicable in several information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.
The advance of experimental techniques in circuit quantum electrodynamics over last decadehas made it possible to monitor the evolution of qubits under continuous measurements in\nreal time . This kind of measurements, as compared to the discrete ones, provides a stream\nof information for feedback to alter the system's quantum dynamics continuously in a desired\nmanner, thus allowing broader possibilities. We obtain the information at a cost, the backaction on the states. However, the back-action turns out not to be just something disturbing \nand harmful to the control but instead can be harnessed, together with the control, to steer the system to our targets. \nThis thesis discusses one such application, namely to use measurements to create entanglement from remote qubits under locally optimal control. It is illustrated that by carefully designing symmetric measurement and feedback operators, we can construct efficient Markovian or non-Markovian protocols which generate N-qubit W,\ngeneral Dicke and Greenberger-Horne-Zeilinger states with high fidelity above 94% up to\nN = 100. In some cases, the fidelity even approaches unity. Following that, we describe how\nto take advantages of the continuous stream of information to protect the unitary dynamics of\na quantum system subject to noises. We develop a practical protocol for continuous operation\nof a quantum error correcting code for protection of coherent evolution due to an encoded\nHamiltonian against environmental errors, using the three qubit bit-flip code and bit-flip\nerrors as a canonical example. We demonstrate, for both quantum memory and adiabatic\nHamiltonians, that the protocol, once optimized, can significantly reduce logical errors with\na performance which is close to the theoretical limit of the bit-flip code's capability. In the\nadiabatic Hamiltonian case, we also propose some novel correction operators instead of the\nconventional ones to further reduce the logical errors.
Communicating classical information with a quantum system involves the receiver making a measurement on the system so as to distinguish as well as possible the alphabet of states used by the sender. We consider the situation in which this measurement takes an appreciable time. In this case the measurement must be described by a continuous measurement process. We consider a continuous implementation of the optimal measurement for distinguishing between two non-orthogonal states, and show that feedback control can be used during this measurement to increase the rate at which the information regarding the initial preparation is obtained. We show that while the maximum obtainable increase is modest, the effect is purely quantum mechanical in the sense that the enhancement is only possible when the initial states are non-orthogonal. We find further that the enhancement in the rate of information gain is achieved at the expense of reducing the total information which the measurement can extract in the long-time limit.
In this paper, we propose feedback designs for manipulating a quantum state to a target state by performing sequential measurements. In light of Belavkin's quantum feedback control theory, for a given set of (projective or nonprojective) measurements and a given time horizon, we show that finding the measurement selection policy that maximizes the probability of successful state manipulation is an optimal control problem for a controlled Markovian process. The optimal policy is Markovian and can be solved by dynamical programming. Numerical examples indicate that making use of feedback information significantly improves the success probability compared to classical scheme without taking feedback. We also consider other objective functionals including maximizing the expected fidelity with the target state as well as minimizing the expected arrival time. The connections and differences among these objectives are also discussed.