Sequential Products of Quantum Subtests

The sequential product A ◦ B of two subtests A and B on a sequential σ-effect algebra E with an order determining set of σ-states is defined. The concepts of sharp, compatible and coexistent subtests are defined. It is shown that a subtest is repeatable if and only if it is sharp. It is also shown that if A is sharp and A, B coexist, then A and B are compatible. A partial order on subtests is defined using the notion of refinement. It is shown that if A is sharp, then the infimum A ∧ B exists if and only if A and B are compatible and in this case A ∧ B = A ◦ B. A generalization of a stochastic matrix called a transition effect matrix (TEM) on E is defined. A TEM is then used to describe a quantum Markov chain. These chains possess two types of natural dynamics called state and matrix dynamics. It is demonstrated that these two types of dynamics are statistically equivalent.