Effect

The term effect was introduced by G. Ludwig [1] as a technical term in his axiomatic reconstruction of quantum mechanics. Intuitively, this term refers to the “effect” of a physical object on a measuring device. Every experiment is understood to be carried out on a particular ensemble (“Gesamtheit”) of objects (► ensembles in quantum mechanics), all of which are subjected to the same preparation procedure; each object interacting with the measuring device triggers one of the different possible measurement outcomes. Technically, preparation procedures and effects are used as primitive concepts to postulate the existence of probability assignments: each measurement outcome, identified by its effect, and each preparation procedure are assumed to determine a unique probability which represents the probability of the occurrence of that particular outcome. Thus, an effect can be taken to be the probability assignment, associated with a given outcome, to an ensemble of objects, or the preparation procedure applied to this ensemble [3].

In Hilbert space quantum mechanics, an effect is defined as an affine map from the set of states to the interval [0,1], or equivalently, as a linear operator E whose expectation value tr[ρE] for any state (► density operator) ρ lies within [0,1]. From this it follows that E is a positive bounded, hence selfadjoint, ► operator.