Sequential products of quantum measurements

Sequential products of quantum measurements are defined and studied. Two types of measurement equivalence are considered and their relationships with compatibility and the sequential product are discussed. It is shown that a measurement A is sharp if and only if A is equivalent to the sequential product of A with itself. Refinements of measurements are defined and it is shown that they produce a partial order on the set of measurements. Lattice properties of this partially ordered set are briefly discussed. Finally we consider convex combinations and conditioning for quantum measurements.