Convex Optimization-Based Parameter Estimation and Experiment Design for Pauli Channels

In this technical note, we study the problem of parameter estimation for quantum channels, which are (completely positive) maps acting on quantum states. We focus on an important subclass of channels called Pauli channels which are characterized by a certain set of vectors (directions) and a number of scalar parameters. For the case where the directions are known, a special parametrization turns the parameter estimation problem into a convex optimization problem. For the case of unknown directions we give a simple algorithm to estimate the directions for qubit Pauli channels. These results assume that the identification experiment configuration is given, namely, a set of quantum states are given on which the channel acts, and a set of positive measurement (operator)s are fixed from which information is gathered. In the second part of the technical note we consider the problem of determining the experiment design, namely, determining states and measurements for optimal parameter identification of the channels. We formalize this problem as a maximization problem for Fisher information and, assuming known channel directions, prove that this problem is convex. We also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal. For qubit Pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions. We illustrate the results of the technical note with numerical examples and simulations.