Maximum confidence measurement for quantum information processing

Abstract

The problem of quantum state discrimination has its origin in one of the key results in quantum information theory: a single-shot measurement cannot perfectly distinguish non-orthogonal states. It aims to construct strategies to correctly guess which state from an ensemble of states is prepared. For this purpose, one may consider distinct figures of merits, such as an average error in the guessing task. The present thesis considers a maximum-confidence measurement (MCM) that optimizes guessing a state {\it per} a detection event. We present a construction of an MCM for ensembles of quantum states and apply it to the certification of quantum measurements in a semi-device-independent (sDI) scenario, in which details of measurements are not assumed but the working principle must be consistent with an underlying physical theory. We first develop the theoretical framework of finding an MCM for an ensemble. We apply it to qubit states such as geometrically uniform states and noisy symmetric, informationally complete states, which are of particular interest in quantum communication protocols. We then consider an sDI scenario of an MCM in quantum and noncontextual ontological theories and show that an MCM of quantum states can be certified by finding a discrepancy between the abovementioned theories. The outcome statistics that quantum theory can only predict, whereas classical theories fail, may be used to achieve quantum advantages in related information tasks.