Informationally complete quantum measurements - Andrew Scott

An informationally-complete quantum measurement (IC-POVM) is a measurement described by a positive operator-valued measure (POVM) which has the property that every quantum state is uniquely determined by its measurement statistics. Consequently, given multiple copies of an unknown state, a sequence of measurements will give an estimate of the statistics, and hence, identify the state itself. This process is called quantum state tomography. In the context of quantum information theory, two important examples of IC-POVMs have been studied: complete sets of mutually unbiased bases (MUBs) and symmetric informationally-complete POVMs (SIC-POVMs). The former is prescribed by a set of orthogonal von Neumann measurements while the latter is a POVM consisting of subnormalized pure-state projectors. Both are statistically unbiased and both have the minimum number of elements possible for an IC-POVM of their type: d+1 bases and d2 POVM elements, respectively, where d is the dimension of the quantum system under study. In this talk I will present an introduction to informationally-complete quantum measurements and then a detailed study of the two above-mentioned types. We will make use of frame theory to give IC-POVMs a mathematical structure and relate the two above-mentioned types to complex projective designs. Open problems and other types of IC-POVMs will also be considered.