An explicit expression of supremum of bounded quantum observables
Xiaoming Xu, Hong-Ke Du, Xiaochun Fang · Journal of Mathematical Physics · 2009
Bounded observables corresponding to a quantum system are usually represented by S(H), the set of all bounded linear self-adjoint operators on a Hilbert space H. In 2006, Gudder introduced a logic order, ≼, on S(H). For A,B∊S(H), A≼B if and only if there exists C∊S(H) such that AC=0 and A+C=B. Given A,B∊S(H), let A∨B be the least upper bound (supremum) for A and B with respect to the Gudder order. In 2007, Pulmannová and Vincenková proved that A∨B exists if and only if A and B have an upper bound for the Gudder order. In this paper, we present some new necessary and sufficient conditions for which A∨B exists and give an explicit representation of A∨B (if A∨B exists).