Transition probabilities are an important and useful tool in quantum mechanics. However, in their present form, they are limited in scope and only apply to pure quantum states. In this article we extend their applicability to mixed states and to transitions between quantum effects. We also present their dependence on a measured operation or instrument. We begin by defining our concepts on a general quantum effect algebra. These concepts are illustrated using Holevo operations and instruments. We then present transition probabilities in the special case of the Hilbert space formulation of quantum mechanics. We show that for pure states and particular types of operations the transition probabilities reduce to their usual form. We give examples in terms of Lüders operations and instruments.
The concepts microscopic reversibility, detailed balancing, equal probabilities, equilibrium,
H theorem are often used in (coarse grained) quantum statistics. It is pointed out that each has a (fine grained) quantum-mechanical counterpart. These concepts are formulated, and their interrelations discussed, at an abstract mathematical level. The results are then interpreted by taking first the fine grained and then the coarse grained point of view. A type of transition matrix arises which does not seem to have been investigated so far, and some of its properties are discussed.
Motivated by physically plausible axioms, the concept of a transition amplitude space (tas) is defined. Various connections between this framework and other axiomatic approaches to the foundations of quantum mechanics are derived. In particular, transition probability spaces, the algebraic approach, the operational statistics and quantum logic approaches, and traditional Hilbert space quantum mechanics are considered. It is shown that a tas always admits a Hilbert space representation. Results are obtained concerning isomorphisms and automorphisms for tas’s. The concept of an A-form is introduced and the relationship between certain A-forms and bounded linear operators on a Hilbert space are studied. Finally, it is shown that sums and tensor products of tas’s can be formulated in a natural way.