Jauch–Piron states on quantum logics

Abstract

We show in this note that if [Formula: see text] is a Boolean subalgebra of the lattice quantum logic [Formula: see text], then each state on [Formula: see text] can be extended over [Formula: see text] as a Jauch–Piron state provided [Formula: see text] is Jauch–Piron unital with respect to [Formula: see text] (i.e. for each nonzero [Formula: see text], there is a Jauch–Piron state [Formula: see text] on [Formula: see text] such that [Formula: see text]). We then discuss this result for the case of [Formula: see text] being the Hilbert space logic [Formula: see text] and [Formula: see text] being a set-representable logic.