Sequential quantum measurements

A quantum effect is an operator A on a complex Hilbert space H that satisfies 0 less than or equal toA less than or equal toI. We denote the set of quantum effects by epsilon (H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B is an element of epsilon (H) by A circleB=A(1/2)BA(1/2). The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A circleB imply that A and B commute for the usual operator product. For example, if A circleB satisfies certain distributive or associative laws, then AB=BA. Moreover, if A circleB is an element ofP(H), then AB=BA and A circleB=B or B circleA=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied. (C) 2001 American Institute of Physics.