A Note on Sequential Product of Quantum Effects

Abstract EN

The quantum effects for a physical system can be described by the set ℰ(ℋ) of positive operators on a complex Hilbert space ℋ that are bounded above by the identity operator I.

For A,B∈ℰ(ℋ), let A∘B=A1/2BA1/2 be the sequential product and let A*B=(AB+BA)/2 be the Jordan product of A,B∈ℰ(ℋ).

The main purpose of this note is to study some of the algebraic properties of effects.

Many of our results show that algebraic conditions on A∘B and A*B imply that A and B have 3×3 diagonal operator matrix forms with Iℛ(A)¯∩ℛ(B)¯ as an orthogonal projection on closed subspace ℛ(A)¯∩ℛ(B)¯ being the common part of A and B.

Moreover, some generalizations of results known in the literature and a number of new results for bounded operators are derived.