The orthogonality postulate in axiomatic quantum mechanics

Abstract

Letp(A,α,E) be the probability that a measurement of an observableA for the system in a stateα will lead to a value in a Borel setE. An experimental function is a function f from the set of all statesI into [0,1] for which there are an observableA and a Borel setE such thatf(α)=p(A, α, E) for allα ∈I. A sequencef₁,f₂,... of experimental functions is said to be orthogonal if there is an experimental functiong such thatg+f₁+f₂+...=1, and it is said to be pairwise orthogonal iff_i+f_j⩽ 1 fori≠j. It is shown that if we assume both notions to be equivalent then the setL of all experimental functions is an orthocomplemented partially ordered set with respect to the natural order of real functions with the complementationf′=1−f, each observableA can be identified with anL-valued measureμ_A, each stateα can be identified with a probability measurem_α onL and we havep(A,α,E)=m_α oμA(E). Thus we obtain the abstract setting of axiomatic quantum mechanics as a consequence of a single postulate.