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.