The Group Inverse for Quantum Information Theory

In the tradition of subjectivist probability theory, QBism views quantum states as beliefs on the part of users of quantum theory. What quantum theory adds is a new normative rule, equivalent to the Born rule, for transferring beliefs from one experiment to another. In the case of a so-called minimal informationally complete reference device, implementing this rule amounts to inverting a full-rank conditional probability matrix. In the more general case, frame theory offers a framework for constructing the appropriate linear transformation. We show that this construction turns out to be equivalent to taking the group inverse of a conditional probability matrix, and thus can be calculated without invoking a prior Hilbert space representation. We discuss the group inverse more generally, its history, its properties, and methods of calculation from the point of view of quantum information theory.