Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion

For successful experimental implementation of any quantum protocol, the quantum states and operations involved must be confirmed to be sufficiently closed to their theoretical targets. One way to obtain such a confirmation is to perform another experiment and from the obtained data make an estimate of the quantum operator involved. Statistically, this is a constrained multiparameter estimation problem - the quantum estimation problem - where we assume we are given a finite number of identical copies of a quantum state or operation, we perform measurements whose mathematical description is assumed to be known, and from the outcome statistics we make our estimate. Due to the probabilistic behavior of the measurement outcomes and the finiteness of the number of measurement trials, there always exist statistical errors in any quantum estimate. The size of the error depends on the choice of measurements and the estimation procedure. In statistics, the former is called an experimental design, while the latter is called an estimator. It is, therefore, a key aim of both classical and quantum estimation theory to find a combination of experimental design and estimator which gives us more precise estimation results using fewer measurement trials. A standard combination in quantum information experiments is that of quantum tomography and maximum likelihood estimator. Although the term "quantum tomography" can be used in several different contexts, we use it to mean an experimental design in which an independently and identically prepared set of measurements are used throughout the entire experiment [1].