Maximum-confidence measurement for qubit states

Abstract

In quantum state discrimination, one aims to identify unknown states from a given ensemble by performing measurements. Different strategies such as minimum-error discrimination or unambiguous state identification find different optimal measurements. Maximum-confidence measurements (MCMs) maximize the confidence with which inputs can be identified given the measurement outcomes. This unifies a range of discrimination strategies including minimum-error and unambiguous state identification, which can be understood as limiting cases of MCM. In this work we investigate MCMs for general ensembles of qubit states. We present a method for finding MCMs for qubit-state ensembles by exploiting their geometry and apply it to several interesting cases, including ensembles of two and four mixed states and ensembles of an arbitrary number of pure states. We also compare MCMs to minimum-error and unambiguous discrimination for qubits. Our results provide interpretations of various qubit measurements in terms of MCM and can be used to devise qubit protocols.