This letter investigates parameter estimation in quantum systems that undergo dynamical evolution. Optimal control problems are formulated to maximize the information, about an unknown parameter, extracted by a given quantum measurement apparatus. This letter introduces the concept of “admissible controls”—control laws that do not depend on the unknown parameter they elicit. For scalar parameter estimation in unital quantum systems interrogated by binary measurements, this letter derives a necessary and sufficient condition on quantum measurement operators so that an information maximizing control law is admissible. When the admissibility condition is satisfied, it is shown that the resulting optimal control problem may be solved using well-established techniques.
This letter investigates parameter estimation in quantum systems that undergo dynamical evolution. Optimal control problems are formulated to maximize the information, about an unknown parameter, extracted by a given quantum measurement apparatus. This letter introduces the concept of “admissible controls”—control laws that do not depend on the unknown parameter they elicit. For scalar parameter estimation in unital quantum systems interrogated by binary measurements, this letter derives a necessary and sufficient condition on quantum measurement operators so that an information maximizing control law is admissible. When the admissibility condition is satisfied, it is shown that the resulting optimal control problem may be solved using well-established techniques.
We present system-theoretic quantum state reconstruction methods that minimize estimation error by combining optimal quantum control with asymptotically efficient estimation. Introducing the notion of optimal observability of a quantum dynamical system---a concept that does not exist in classical control theory---we formulate and solve the Pareto optimal control problem of maximizing state estimation accuracy while minimizing the expenditure of available control and measurement resources. Necessary and sufficient conditions for optimal observability, based on the quantum optimal observability Gramian, are presented. We examine the finite sample efficiency of the estimation methodology for two- and three-level systems using ideal and noisy control fields, and demonstrate the advantages of state reconstruction schemes based on optimal observability theory for experimentally realistic sample sizes. These results indicate that the optimal control of quantum measurement bases can be used to minimize state reconstruction errors by fully exploiting the information geometry of quantum states.
Information control and online estimation are studied in this paper, focusing on unital quantum systems. To infer the value of an unknown parameter characterizing a quantum state, control is applied to the quantum system and the resulting state is measured by a fixed apparatus. The control is selected with the aim of maximizing the Fisher information, about the unknown parameter, extracted from the fixed apparatus. We introduce the notion of "controlled Fisher information (CFI)" to describe the maximum Fisher information attainable by control. A method for computing the CFI is designed and an analytic upper bound on the CFI is derived. Necessary and sufficient conditions for the admissibility (i.e., parameter independence) of controls achieving the CFI are given. For scenarios in which the admissibility conditions are not met, we devise an online joint estimation and control algorithm that, with high probability, achieves the CFI after a sufficient number of measurements.
Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.
Quantum control plays a crucial role in quantum inference tasks such as parameter estimation and state discrimination. Despite this, the fields of quantum control and quantum inference have remained largely disjoint. We bridge these fields, revealing that (i) quantum control determines the fundamental performance limits of quantum estimation and state discrimination, and (ii) control of statistical information measures enables performance approaching these limits. This thesis begins by establishing an axiomatic formalization of quantum control theory and developing a mathematical framework for analyzing, approximating, and controlling the evolution of quantum systems. A self-contained treatment of optimal quantum control is presented. This treatment exploits the unique structure of quantum dynamics rather than merely borrowing notions from classical control. Quantum control systems are shown to be universally contractive, which plays an important role in the treatment of optimal quantum control and in the derivation of various approximation results. Next, the thesis proceeds by addressing statistical inference problems, with emphasis on parameter estimation and state discrimination. Controlled information measures are introduced, providing more sensible alternatives to existing information measures for quantum inference, e.g., quantum Fisher information. Controlled information measures are shown to exhibit useful properties, e.g., data-processing inequalities. Methods for deriving achievability and converse bounds on controlled information measures are developed, and adaptive control techniques for information maximization are proposed. This thesis leverages a cross-pollination of control theory, statistical inference, and quantum information towards the development of an application-focused theory of quantum inference.
Quantum parameter estimation has been long pursued in various physics areas, such as quantum optics, quantum information processing, gravitational wave detection, and so on. Generally, the quantum system dynamics is usually assumed to be fixed and the system is closed, the highest precision can be achieved by preparing the optimal probe state and performing the quantum measurement based on ideal basis that saturates the quantum Cramer-Rao inequality. However, how to extend this problem to a more general open quantum system is still an open question. In this paper, we apply the optimal control theory, Pontrygin optimal principle, to enhance the parameter estimation precision in open quantum systems. The control laws and the control algorithm are exactly presented. Simulation results demonstrated the efficiency of the optimal control laws.
We discuss the problem of finding the best measurement strategy for estimating the value of a quantum system parameter. In general the optimum quantum measurement, in the sense that it maximizes the quantum Fisher information and hence allows one to minimize the estimation error, can only be determined if the value of the parameter is already known. A modification of the quantum Van Trees inequality, which gives a lower bound on the error in the estimation of a random parameter, is proposed. The suggested inequality allows us to assert if a particular quantum measurement, together with an appropriate estimator, is optimal. An adaptive strategy to estimate the value of a parameter, based on our modified inequality, is proposed.