Sequential product of quantum effects

Abstract

Abstract. Unsharp quantum measurements can be modelled by means of the class E(H) of positive contractions on a Hilbert space H, in brief, quantum effects. For A, B ∈ E(H) the operation of sequential product A•B = A 1/2 BA 1/2 was proposed as a model for sequential quantum measurements. We continue these investigations on sequential product and answer positively the following question: the assumption A • B ≥ B implies AB = BA = B.Then we propose a geometric approach of quantum effects and their sequential product by means of contractively contained Hilbert spaces and operator ranges. This framework leads us naturally to consider lattice properties of quantum effects, sums and intersections, and to prove that the sequential product is left distributive with respect to the intersection.

A uniqueness problem of the sequence product on operator effect algebra {\cal E}(H)

“…These yielded by B and C has important physics meaning. Professor Gudder called it the sequential product of B and C, and denoted it by B • C. It represents the quantum effect produced by fist measuring A then measuring B ( [6][7][8]). This sequential product has also been generalized to an algebraic structure called a sequential effect algebra ( [7]).…”

“…The composition Φ 2 • Φ 1 is a new operation, called a sequential operation as it is obtained by performing first Φ 1 and then Φ 2 . In general, [6][7][8]). This sequential product has also been generalized to an algebraic structure called a sequential effect algebra ( [7]).…”

Characterization of the sequential product on quantum effects

“…Definition 1: A sequential product ‫ؠ‬ on E͑H͒ is a binary operation on E͑H͒ satisfying conditions (1)- (5), namely, for all A , B E͑H͒.…”

Topologies on quantum effects

“…Quantum effects play an important role in quantum measurement theory(see [1,2,3,4,5,6,7,8]). In the Hilbert space model of quantum mechanics, effects for a physical system S are represented by positive operators on a complex Hilbert space H that are bounded above by the identity operator I.…”