We discuss the possibility of protecting the state of a quantum system that goes through noise by measurements and operations before and after the noise process. The aim is to seek the optimal protocol that makes the input and output states as close as possible and to clarify the role of the measurements therein. We consider two cases: one can perform quantum measurements and operations (i) only after the noise process and (ii) both before and after. We prove in a two-dimensional Hilbert space that, in case (i), the noise suppression is essentially impossible for all types of noise and, in case (ii), the optimal protocol for the depolarizing noise is either the ``do nothing'' protocol or the ``discriminate and reprepare'' protocol. These protocols are not ``truly quantum'' and can be considered as classical. They involve no measurement or only use the measurement outcomes. These results describe the fundamental limitations in quantum mechanics from the viewpoint of control theory. Finally, we conjecture that a statement similar to case (ii) holds for higher-dimensional Hilbert spaces and present some numerical evidence.
We discuss the possibility of protecting the state of a quantum system that goes through noise by measurements and operations before and after the noise process. The aim is to seek the optimal protocol that makes the input and output states as close as possible and to clarify the role of the measurements therein. We consider two cases: one can perform quantum measurements and operations (i) only after the noise process and (ii) both before and after. We prove in a two-dimensional Hilbert space that, in case (i), the noise suppression is essentially impossible for all types of noise and, in case (ii), the optimal protocol for the depolarizing noise is either the do nothing'' protocol or the discriminate and reprepare'' protocol. These protocols are not ``truly quantum'' and can be considered as classical. They involve no measurement or only use the measurement outcomes. These results describe the fundamental limitations in quantum mechanics from the viewpoint of control theory. Finally, we conjecture that a statement similar to case (ii) holds for higher-dimensional Hilbert spaces and present some numerical evidence.
We discuss protection of a quantum state that goes through a noise process by measurements and operations before and after the noise process. In our previous work, we showed the nonexistence of ``truly quantum'' protocols that protect an unknown qubit state against depolarizing noise better than ``classical'' ones. Toward identifying the class of noise processes that is optimally suppressed by such a ``classical'' protocol, we extend our previous result in two directions. First, we show that the statement is also true in any finite-dimensional Hilbert spaces, which was previously conjectured; the optimal protocol is either the do nothing protocol or the discriminate and reprepare protocol, depending on the strength of the noise. Second, in the case of a single qubit, we show that essentially the same conclusion holds for any unital noise. Thus, the noise must be nonunital for a control protocol beyond ``classical'' ones to exist.
Due to the interaction with the environment, a quantum state is often affected by the different types of noises which becomes to one of the biggest problems for practical quantum computation. We study the possibility of protecting the mixed state of a quantum system that goes through noise by weak measurements and control operations. The aim is to find the optimal measurement strength and control operations and make the input and output states as close as possible. We show that our scheme can effectively protect arbitrary mixed states against typical types of noise sources: amplitude damping, phase damping and amplitude-phase damping. The optimal measurement and control operators are deduced in different bases of the Bloch sphere to find the best control scheme for each type of noise. The effectiveness of our control scheme is demonstrated by simulation results.
Measurement noise is a major source of noise in quantum metrology.Here, we explore preprocessing protocols that apply quantum controls to the quantum sensor state prior to the final noisy measurement (but after the unknown parameter has been imparted), aiming to maximize the estimation precision.We define the quantum preprocessing-optimized Fisher information, which determines the ultimate precision limit for quantum sensors under measurement noise, and conduct a thorough investigation into optimal preprocessing protocols.First, we formulate the preprocessing optimization problem as a biconvex optimization using the error observable formalism, based on which we prove that unitary controls are optimal for pure states and derive analytical solutions of the optimal controls in several practically relevant cases.Then we prove that for classically mixed states (whose eigenvalues encode the unknown parameter) under commuting-operator measurements, coarse-graining controls are optimal, while unitary controls are suboptimal in certain cases.Finally, we demonstrate that in multiprobe systems where noisy measurements act independently on each probe, the noiseless precision limit can be asymptotically recovered using global controls for a wide range of quantum states and measurements.Applications to noisy Ramsey interferometry and thermometry are presented, as well as explicit circuit constructions of optimal controls.
Known as decoherence, the unavoidable interaction of a quantum system with its surrounding environment is usually considered to be detrimental for quantum information processing. In this thesis the coherent, open loop control of such open systems is studied. Concepts from quantum control theory and the theory of open quantum system are adopted in order to fight decoherence and implement quantum gates in a noiseless manner. In particular, Lie algebraic methods and numerical optimization tools are used to investigate the control properties of a single spin interacting with a spin environment. We show that, independent of the size of the environment, every unitary transformation can be implemented on the system spin through a single control field. We proceed by investigating dynamical decoupling, a method to suppress the interactions with the environment, for finite- and for in- finite dimensional systems. We prove that every finite dimensional system can be protected from decoherence, even if the environment is infinite dimensional, whereas for noise described by a Lindblad master equation dynamical decoupling will never succeed. This will lead to a new method to distinguish decoherence from intrinsic noise terms. We further prove that not every infinite dimensional system can be protected from decoherence through dynamical decoupling. Afterwards we investigate dynamical decoupling of systems that are described by quadratic Hamiltonians, showing that such interactions can always be suppressed with two simple operations. In the last part we investigate the coherent control of a Lindblad master equation. We show that a strong noise process exhibiting a decoherence free subspace can substantially increase the number of unitary operations that can be implemented, allowing us to fully control parts of the system. Afterwards we develop a scheme to make Hamiltonians and Lindbladians commutative by adding an auxiliary system. The old, possibly non-commutative dynamics, is recovered through a non-selective measurement.
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.
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.