Optimal state estimation of controllable quantum dynamical systems

Abstract

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.