In my research I am especially interested in getting a better understanding of the foundations of quantum mechanics from quantum information investigations. One way is to study different representations of quantum states. Clever representations will guide us when trying to understand what a quantum state is. Density matrices are most commonly used to represent quantum states. From the density matrix predications about outcomes for all possible measurements upon a system can be calculated. So far so good, but it is hard to see how these mathematical objects, the density matrices, should be interpreted, and what the equations of quantum mechanics mean. An alternative representation has been suggested by Christopher A. Fuchs in the Quantum Foundations group at Perimeter. We study how to represent quantum states as probabilities related to special standard measurements: symmetric informationally complete measurements (with acronym SIC-POVM or just SIC). To define and understand probabilities is not straightforward, but probabilities are at least more intuitive than density matrices. Expressing quantum states as probability distributions gives a new way of comparing quantum theory with classical theory. In the 'language of SICs' the Born rule states a relation between the probabilities of two different possible experiments â?? this is a part of quantum interference and one of the most central features of quantum mechanics.
There are two main areas of my research. One is to find SICs and prove they exists in all dimensions. The problem is easy to define but it has turned out to be a very hard mathematical problem which is connected to several different areas of mathematics, e.g. group theory, Lie algebra, combinatorics, frames and spherical designs, and extension fields and Galois theory. The second is to characterize the set of quantum states in the SIC-representation. It will be a set of probabilities which is constrained by implications of the Born rule.